- 3110/31/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 0111/01/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Bubble instability of mIIA on AdS_4 x S^6
Abstract: Recently, a set of non-supersymmetric AdS_4 vacua of massive type IIA string theory has been constructed. These vacua are perturbatively stable with respect to the full KK spectrum of type mIIA supergravity and furthermore, they are stable against a variety of non-perturbative decay channels. Hence, at this point, they represent a serious challenge to the AdS swampland conjecture. In my talk, I will review in detail the construction of these vacua as well as introduce a new decay channel, ultimately sealing their fate as being unstable.
- 0211/02/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Counting invariant curves on a Calabi-Yau threefold with an involution
Abstract: Gopakumar-Vafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a Calabi-Yau threefold. In this talk, I will give a general overview of two of the sheaf-theoretic approaches to defining these invariants: via stable pairs a la Pandharipande-Thomas (PT) and via perverse sheaves a la Maulik-Toda (MT). I will then outline a parallel theory of Gopakumar-Vafa invariants for a Calabi-Yau threefold X with an involution. They are integers n_beta(g,h) which give a virtual count of curves of genus g in the class beta which are invariant under the involution and whose quotient by the involution has genus h. I will give two definitions of n_beta(g,h) which are conjectured to be equivalent, one in terms of a version of PT theory, and one in terms of a version of MT theory. These invariants can be computed and the conjecture proved in the case where X=SxC where S is an Abelian or K3 surface with a symplectic involution. In these cases, the invariants are given by formulas expressed with Jacobi modular forms. In the case where S is an Abelian surface, the specialization of n_beta(g,h) to h=0 recovers the count of hyperelliptic curves on Abelian surfaces first computed by B-Oberdieck-Pandharipande-Yin. This is joint work with Stephen Pietromonaco.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero
Abstract: We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields. This is a joint work with J. Cao and R. Villaflor. (arXiv:2109.00587)
- 0311/03/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Non-Invertible Duality Defects in 3+1 Dimensions
Speaker: Clay Cordova (U Chicago)
Title: Non-Invertible Duality Defects in 3+1 Dimensions
Abstract: For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions and determine the fusion rule. From modular invariance and a direct analysis of one-form symmetry-protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. By further assuming time-reversal symmetry, we find that the presence of certain duality defects implies that the low-energy phase has to be gapless unless the one-form symmetry is spontaneously broken. We give an explicit realization of this duality defect in the free Maxwell theory where the duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides.
When Computer Algebra Meets Satisfiability: A New Approach to Combinatorial Mathematics
Abstract: Solvers for the Boolean satisfiability (SAT) problem have been increasingly used to resolve problems in mathematics due to their excellent search algorithms. This talk will describe a new method for mathematical search that couples SAT solvers with computer algebra systems (CAS), thereby combining the expressiveness of CASs with the search power of SAT solvers. This paradigm has led to a number of results on long-standing mathematical questions such as the first computer-verifiable resolution of Lam’s problem and the discovery of a new infinite class of Williamson matrices.
- 0411/04/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Fusion Category Symmetries in Quantum Field Theory
Speaker: Yifan Wang (NYU)
Title: Fusion Category Symmetries in Quantum Field Theory
Abstract: Topological defects provide a modern perspective on symmetries in quantum field theory. They generalize the familiar inverti
ble symmetries described by groups to non-invertible symmetries described by fusion categories. Such generalized symmetries are ubiquitous in quantum field theory and provide new constraints on renormalization group flows and the IR phase diagram. In this talk I’ll review some recent progress in identifying and understanding fusion category symmetries in 1+1d conformal field theories. Time permitting, I’ll also comment on higher dimensional generalizations.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The stability of charged black holes
Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
11/4/21 CMSA Interdisciplinary Science Seminar
Title: Exploring Invertibility in Image Processing and Restoration
Abstract: Today’s smartphones have enabled numerous stunning visual effects from denoising to beautification, and we can share high-quality JPEG images easily on the internet, but it is still valuable for photographers and researchers to keep the original raw camera data for further post-processing (e.g., retouching) and analysis. However, the huge size of raw data hinders its popularity in practice, so can we almost perfectly restore the raw data from a compressed RGB image and thus avoid storing any raw data? This question leads us to design an invertible image signal processing pipeline. Then we further explore invertibility in other image processing and restoration tasks, including image compression, reversible image conversion (e.g., image-to-video conversion), and embedding novel views in a single JPEG image. We demonstrate that customized invertible neural networks are highly effective in these inherently non-invertible tasks.
- 0511/05/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
The Greene-Plesser Construction Revisited
Member Seminar
Speaker: Chuck Doran
Title: The Greene-Plesser Construction Revisited
Abstract: The first known construction of mirror pairs of Calabi-Yau manifolds was the Greene-Plesser “quotient and resolve” procedure which applies to pencils of hypersurfaces in projective space. We’ll review this approach, uncover the hints it gives for some more general mirror constructions, and describe a brand-new variant that applies to pencils of hypersurfaces in Grassmannians. This last is joint work with Tom Coates and Elana Kalashnikov (arXiv:2110.0727).
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 0611/06/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 3110/31/2021
Quantum Matter Workshop
Please note: this workshop has been postponed to a later date. Details will be posted to this page when they are available.
Throughout the summer, scheduled speakers for the Quantum Matter Workshop will give talks on Zoom for the Quantum Matter/Condensed Matter seminar.
The CMSA will be hosting our second workshop on Quantum Matter. Both of these workshops are part of our program on Quantum Matter in Mathematics and Physics. The first workshop took place in December 2019, and was extremely successful, attracting participants worldwide. Learn more about the first workshop here.
Organizers: Du Pei, Ryan Thorngren, Juven Wang, Yifan Wang, and Shing-Tung Yau.
Speakers:
- Xiao Chen, Rutgers University
- Bert Halperin, Harvard Physics
- Daniel Harlow, MIT
- Michael Hopkins, Harvard Math
- Chang-Tse Hsieh, Kavli IMPU
- Philip Kim, Harvard Physics
- Ethan Lake, University of Utah
- Hotat Lam, Princeton University
- Mikhail Lukin, Harvard Physics
- Subir Sachdev, Harvard Physics
- Anders Sandvik, Boston University
- Nati Seiberg, IAS
- Husan Shapourian, Harvard
- Xue-Yang Song, Harvard
- Nat Tantivasadakarn, Harvard
- Juven Wang, CMSA
- Yifan Wang, CMSA
- Frank Wilczek, MIT
Big Data Conference 2021
On August 24, 2021, the CMSA hosted our seventh annual Conference on Big Data. The Conference features many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The 2021 Big Data Conference took place virtually on Zoom.
Organizers:
- Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University
- Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business
- Horng-Tzer Yau, Professor of Mathematics, Harvard University
- Sergiy Verstyuk, CMSA, Harvard University
Speakers:
- Andrew Blumberg, University of Texas at Austin
- Moran Koren, Harvard CMSA
- Hima Lakkaraju, Harvard University
- Katrina Ligett, The Hebrew University of Jerusalem
Time (ET; Boston time) Speaker Title/Abstract 9:00AM Conference Organizers Introduction and Welcome 9:10AM – 9:55AM Andrew Blumberg, University of Texas at Austin Title: Robustness and stability for multidimensional persistent homology Abstract: A basic principle in topological data analysis is to study the shape of data by looking at multiscale homological invariants. The idea is to filter the data using a scale parameter that reflects feature size. However, for many data sets, it is very natural to consider multiple filtrations, for example coming from feature scale and density. A key question that arises is how such invariants behave with respect to noise and outliers. This talk will describe a framework for understanding those questions and explore open problems in the area.
10:00AM – 10:45AM Katrina Ligett, The Hebrew University of Jerusalem Title: Privacy as Stability, for Generalization Abstract: Many data analysis pipelines are adaptive: the choice of which analysis to run next depends on the outcome of previous analyses. Common examples include variable selection for regression problems and hyper-parameter optimization in large-scale machine learning problems: in both cases, common practice involves repeatedly evaluating a series of models on the same dataset. Unfortunately, this kind of adaptive re-use of data invalidates many traditional methods of avoiding overfitting and false discovery, and has been blamed in part for the recent flood of non-reproducible findings in the empirical sciences. An exciting line of work beginning with Dwork et al. in 2015 establishes the first formal model and first algorithmic results providing a general approach to mitigating the harms of adaptivity, via a connection to the notion of differential privacy. In this talk, we’ll explore the notion of differential privacy and gain some understanding of how and why it provides protection against adaptivity-driven overfitting. Many interesting questions in this space remain open.
Joint work with: Christopher Jung (UPenn), Seth Neel (Harvard), Aaron Roth (UPenn), Saeed Sharifi-Malvajerdi (UPenn), and Moshe Shenfeld (HUJI). This talk will draw on work that appeared at NeurIPS 2019 and ITCS 2020
10:50AM – 11:35AM Hima Lakkaraju, Harvard University Title: Towards Reliable and Robust Model Explanations Abstract: As machine learning black boxes are increasingly being deployed in domains such as healthcare and criminal justice, there is growing emphasis on building tools and techniques for explaining these black boxes in an interpretable manner. Such explanations are being leveraged by domain experts to diagnose systematic errors and underlying biases of black boxes. In this talk, I will present some of our recent research that sheds light on the vulnerabilities of popular post hoc explanation techniques such as LIME and SHAP, and also introduce novel methods to address some of these vulnerabilities. More specifically, I will first demonstrate that these methods are brittle, unstable, and are vulnerable to a variety of adversarial attacks. Then, I will discuss two solutions to address some of the vulnerabilities of these methods – (i) a framework based on adversarial training that is designed to make post hoc explanations more stable and robust to shifts in the underlying data; (ii) a Bayesian framework that captures the uncertainty associated with post hoc explanations and in turn allows us to generate explanations with user specified levels of confidences. I will conclude the talk by discussing results from real world datasets to both demonstrate the vulnerabilities in post hoc explanation techniques as well as the efficacy of our aforementioned solutions.
11:40AM – 12:25PM Moran Koren, Harvard CMSA Title: A Gatekeeper’s Conundrum Abstract: Many selection processes contain a “gatekeeper”. The gatekeeper’s goal is to examine an applicant’s suitability to a proposed position before both parties endure substantial costs. Intuitively, the introduction of a gatekeeper should reduce selection costs as unlikely applicants are sifted out. However, we show that this is not always the case as the gatekeeper’s introduction inadvertently reduces the applicant’s expected costs and thus interferes with her self-selection. We study the conditions under which the gatekeeper’s presence improves the system’s efficiency and those conditions under which the gatekeeper’s presence induces inefficiency. Additionally, we show that the gatekeeper can sometimes improve selection correctness by behaving strategically (i.e., ignore her private information with some probability).
12:25PM Conference Organizers Closing Remarks Some remarks on contact Calabi-Yau 7-manifolds
Abstract: In geometry and physics it has proved useful to relate G2 and Calabi-Yau geometry via circle bundles. Contact Calabi-Yau 7-manifolds are, in the simplest cases, such circle bundles over Calabi-Yau 3-orbifolds. These 7-manifolds provide testing grounds for the study of geometric flows which seek to find torsion-free G2-structures (and thus Ricci flat metrics with exceptional holonomy). They also give useful backgrounds to examine the heterotic G2 system (also known as the G2-Hull-Strominger system), which is a coupled set of PDEs arising from physics that involves the G2-structure and gauge theory on the 7-manifold. I will report on recent progress on both of these directions in the study of contact Calabi-Yau 7-manifolds, which is joint work with H. Sá Earp and J. Saavedra.
CMSA Math-Science Literature Lecture: Kunihiko Kodaira and complex manifolds
Yujiro Kawamata (University of Tokyo)
Title: Kunihiko Kodaira and complex manifolds
Abstract: Kodaira’s motivation was to generalize the theory of Riemann surfaces in Weyl’s book to higher dimensions. After quickly recalling the chronology of Kodaira, I will review some of Kodaira’s works in three sections on topics of harmonic analysis, deformation theory and compact complex surfaces. Each topic corresponds to a volume of Kodaira’s collected works in three volumes, of which I will cover only tiny parts.
Talk chair: Baohua Fu
Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants for algebraic surfaces
During the Spring 2021 Semester Artan Sheshmani (CMSA/ I.M. A.U.) will be teaching a CMSA special lecture series on Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants for algebraic surfaces.
In order to attend this series, please fill out this form.
The lectures will be held Mondays from 8:00 – 9:30 AM ET and Wednesdays from 8:00 – 9:00 AM ET beginning January 25 on Zoom.
You can watch Prof. Sheshmani describe the series here.
CMSA Math-Science Literature Lecture: Michael Atiyah: Geometry and Physics
Nigel Hitchin (University of Oxford)
Title: Michael Atiyah: Geometry and Physics
Abstract: In mid-career, as an internationally renowned mathematician, Michael Atiyah discovered that some problems in physics responded to current work in algebraic geometry and this set him on a path to develop an active interface between mathematics and physics which was formative in the links which are so active today. The talk will focus, in a fairly basic fashion, on some examples of this interaction, which involved both applying physical ideas to solve mathematical problems and introducing mathematical ideas to physicists.
Talk chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: Is relativity compatible with quantum theory?
Arthur Jaffe (Harvard University)
Title: Is relativity compatible with quantum theory?
Abstract: We review the background, mathematical progress, and open questions in the effort to determine whether one can combine quantum mechanics, special relativity, and interaction together into one mathematical theory. This field of mathematics is known as “constructive quantum field theory.” Physicists believe that such a theory describes experimental measurements made over a 70 year period and now refined to 13-decimal-point precision—the most accurate experiments ever performed.
Talk chair: Zhengwei Liu
CMSA Math-Science Literature Lecture: Noncommutative Geometry, the Spectral Aspect
Alain Connes (Collège de France)
Title: Noncommutative Geometry, the Spectral Aspect
Abstract: This talk will be a survey of the spectral side of noncommutative geometry, presenting the new paradigm of spectral triples and showing its relevance for the fine structure of space-time, its large scale structure and also in number theory in connection with the zeros of the Riemann zeta function.
Talk chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: Homotopy spectra and Diophantine equations
Yuri Manin (Max Planck Institute for Mathematics)
Title: Homotopy spectra and Diophantine equations
Abstract: For a long stretch of time in the history of mathematics, Number Theory and Topology formed vast, but disjoint domains of mathematical knowledge. Origins of number theory can be traced back to the Babylonian clay tablet Plimpton 322 (about 1800 BC) that contained a list of integer solutions of the “Diophantine” equation $a^2+b^2=c^2$: archetypal theme of number theory, named after Diophantus of Alexandria (about 250 BC). Topology was born much later, but arguably, its cousin — modern measure theory, — goes back to Archimedes, author of Psammites (“Sand Reckoner”), who was approximately a contemporary of Diophantus. In modern language, Archimedes measures the volume of observable universe by counting the number of small grains of sand necessary to fill this volume. Of course, many qualitative geometric models and quantitative estimates of the relevant distances precede his calculations. Moreover, since the estimated numbers of grains of sand are quite large (about $10^{64}$), Archimedes had to invent and describe a system of notation for large numbers going far outside the possibilities of any of the standard ancient systems. The construction of the first bridge between number theory and topology was accomplished only about fifty years ago: it is the theory of spectra in stable homotopy theory. In particular, it connects $Z$, the initial object in the theory of commutative rings, with the sphere spectrum $S$. This connection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory. In this talk based upon the authors’ (Yu. Manin and M. Marcolli) joint research project, I suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds.
Talk chair: Michael Hopkins
CMSA Math-Science Literature Lecture: Log Calabi-Yau fibrations
Caucher Birkar (University of Cambridge)
Title: Log Calabi-Yau fibrations
Abstract: Fano and Calabi-Yau varieties play a fundamental role in algebraic geometry, differential geometry, arithmetic geometry, mathematical physics, etc. The notion of log Calabi-Yau fibration unifies Fano and Calabi-Yau varieties, their fibrations, as well as their local birational counterparts such as flips and singularities. Such fibrations can be examined from many different perspectives. The purpose of this talk is to introduce the theory of log Calabi-Yau fibrations, to remind some known results, and to state some open problems.
CMSA Math-Science Literature Lecture: Classical and quantum integrable systems in enumerative geometry
Andrei Okounkov (Columbia University)
Title: Classical and quantum integrable systems in enumerative geometry
Abstract: For more than a quarter of a century, thanks to the ideas and questions originating in modern high-energy physics, there has been a very fruitful interplay between enumerative geometry and integrable system, both classical and quantum. While it is impossible to summarize even the most important aspects of this interplay in one talk, I will try to highlight a few logical points with the goal to explain the place and the role of certain more recent developments.
Talk chair: Cumrun Vafa
Workshop on Quantum Information
The Center of Mathematical Sciences and Applications will be hosting a workshop on Quantum Information on April 23-24, 2018. In the days leading up to the conference, the American Mathematical Society will also be hosting a sectional meeting on quantum information on April 21-22. You can find more information here.
The following speakers are confirmed:
- Fernando G.S.L Brandão (CalTech)
- Jacob Biamonte (Skoltech)
- Isaac Chuang (MIT)
- Iris Cong (Harvard)
- Aram Harrow (MIT)
- Ke Li (HIT)
- Mikhail D. Lukin (Harvard)
- Shunlong Luo (AMSS)
- Renato Renner (ETH Zürich)
- Peter Shor (MIT)
CMSA Math-Science Literature Lecture: Knot Invariants From Gauge Theory in Three, Four, and Five Dimensions
Edward Witten (IAS)
Title: Knot Invariants From Gauge Theory in Three, Four, and Five Dimensions
Abstract: I will explain connections between a sequence of theories in two, three, four, and five dimensions and describe how these theories are related to the Jones polynomial of a knot and its categorification.
Talk chair: Cliff Taubes
F-Theory Conference
The CMSA will be hosting an F-Theory workshop September 29-30, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Click here for videos of the talks.
Organizers:
- Paolo Aluffi (Florida State)
- Lara B. Anderson (Virginia Tech)
- Mboyo Esole (Northeastern)
- Shing-Tung Yau (Harvard)
Speakers:
- Mirjam Cvetic, University of Pennsylvania
- Tommaso de Fernex, University of Utah
- James Gray, Virginia Tech
- Jonathan Heckman, University of Pennsylvania
- Monica Kang, Harvard University
- Sándor Kovács, University of Washington
- Anatoly Libgober, UIC
- Matilde Marcolli, Caltech, University of Toronto, and Perimeter Institute
- Washington Taylor, MIT
- Cumrun Vafa, Harvard University
Morphogenesis: Geometry and Physics
Just over a century ago, the biologist, mathematician and philologist D’Arcy Thompson wrote “On growth and form”. The book – a literary masterpiece – is a visionary synthesis of the geometric biology of form. It also served as a call for mathematical and physical approaches to understanding the evolution and development of shape. In the century since its publication, we have seen a revolution in biology following the discovery of the genetic code, which has uncovered the molecular and cellular basis for life, combined with the ability to probe the chemical, structural, and dynamical nature of molecules, cells, tissues and organs across scales. In parallel, we have seen a blossoming of our understanding of spatiotemporal patterning in physical systems, and a gradual unveiling of the complexity of physical form. So, how far are we from realizing the century-old vision that “Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed” ?
To address this requires an appreciation of the enormous ‘morphospace’ in terms of the potential shapes and sizes that living forms take, using the language of mathematics. In parallel, we need to consider the biological processes that determine form in mathematical terms is based on understanding how instabilities and patterns in physical systems might be harnessed by evolution.
In Fall 2018, CMSA will focus on a program that aims at recent mathematical advances in describing shape using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems.
The first workshop will focus on the interface between Morphometrics and Mathematics, while the second will focus on the interface between Morphogenesis and Physics.The workshop is organized by L. Mahadevan (Harvard), O. Pourquie (Harvard), A. Srivastava (Florida).As part of the program on Mathematical Biology a workshop on Morphogenesis: Geometry and Physics will take place on December 3-5, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Videos
Please Register Here
PDF of the Schedule
Speakers:
- Arkhat Abzhanov, Imperial College
- Yohanns Bellaiche, Paris
- Cheng Ming Chuong, USC
- Zev Gartner, UCSF
- Thomas Gregor, Princeton
- Dagmar Iber, Zurich
- Ian Jermyn, Durham University
- Raymond Keller, UVA
- Allon Klein, HMS
- Lisa Manning, Syracuse
- Cristina Marchetti, UCSB
- Sean Megason, HMS
- Elliot Meyerowitz, Caltech
- Michel Milinkovitch, Geneva
- Leonardo Morsut, USC
- Olivier Pourquié, HMS
- Eric Siggia, Rockefeller University
- Ben Simons, Cambridge
- Sebastian Streichan, UCSB
- Aryeh Warmflash, Rice
2019 Big Data Conference
1 Oxford Street, Cambridge MA 02138On August 19-20, 2019 the CMSA will be hosting our fifth annual Conference on Big Data. The Conference will feature many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The talks will take place in Science Center Hall D, 1 Oxford Street.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants.
Videos can be found in this Youtube playlist or in the schedule below.
Workshop on Foundations of Computational Science
On August 29-31, 2019 the Center of Mathematical Sciences and Applications will be hosting a workshop on Foundations of Computational Science. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by David Xianfeng Gu.
Please register here.
Speakers:
- Sarah Adel Bargal, Boston University
- Jianfeng Chen, Harvard
- Tat Seng Chua, National University of Singapore
- Ke Deng, Tsinghua
- David Xianfeng Gu, Stony Brook
- Yike Guo, Imperial College London
- Minlie Huang, Tsinghua
- Scott Kominers, Harvard
- Brian Kulis, Boston University
- Wee Sun Lee, National University of Singapore
- Qianxiao Li, National University of Singapore
- Hanzhong Liu, Tsinghua
- Jun Liu, Harvard
- Xiao-Li Meng, Harvard
- Cengiz Pehlevan, Harvard
- Donald Rubin, Harvard
- Suproteem Sarkar, Harvard
- Zuowei Shen, National University of Singapore
- Yuanchun Shi, Tsinghua
- Justin Solomon, MIT
- Hang Su, Tsinghua
- Maosong Sun, Tsinghua
- Mirac Suzgun, Harvard
- Sergiy Verstyuk, CMSA
- Xiaoqin Wang, Tsinghua
- Bin Xu, Tsinghua
- Jun Zhu, Tsinghua
- Wenwu Zhu, Tsinghua
Videos of the talks are contained in the Youtube playlist below. They can also be found through links in the schedule.
Angular momentum in general relativity
Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960′, due to the discovery of “supertranslation ambiguity”: the angular momentums recorded by two distant observers of the same system may not be the same. In this talk, I shall show how the mathematical theory of optimal isometric embedding and quasilocal angular momentum identifies a correction term, and leads to a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau.
From Algebraic Geometry to Vision and AI: A Symposium Celebrating the Mathematical Work of David Mumford
On August 18 and 20, 2018, the Center of Mathematic Sciences and Applications and the Harvard University Mathematics Department hosted a conference on From Algebraic Geometry to Vision and AI: A Symposium Celebrating the Mathematical Work of David Mumford. The talks took place in Science Center, Hall B.
Saturday, August 18th: A day of talks on Vision, AI and brain sciencesMonday, August 20th: a day of talks on MathSpeakers:
- Stuart Geman, Brown
- Janos Kollar, Princeton
- Tai Sing Lee, CMU
- Emanuele Macri, Northeastern
- Jitendra Malik, Berkeley / FAIR
- Peter Michor, University of Vienna
- Michael Miller, Johns Hopkins
- Aaron Pixton, MIT
- Jayant Shah, Northeastern
- Josh Tenenbaum, MIT
- Burt Totaro, UCLA
- Avi Wigderson, IAS
- Ying Nian Wu, UCLA
- Laurent Younes, Johns Hopkins
- Song-Chun Zhu, UCLA
Organizers:
- Ching-Li Chai, University of Pennsylvania
- David Gu, Stony Brook University
- Amnon Neeman, Australian National University
- Mark Nitzberg, University of California at Berkeley
- Yang Wang, Hong Kong University of Science and Technology
- Shing-Tung Yau, Harvard University
- Song-Chun Zhu, University of California, Los Angeles
Publication:
Pure and Applied Mathematics Quarterly
Special Issue: In Honor of David Mumford
Guest Editors: Ching-Li Chai, Amnon Neeman
Geometric Analysis Approach to AI Workshop
Due to inclement weather on Sunday, the second half of the workshop has been moved forward one day. Sunday and Monday’s talks will now take place on Monday and Tuesday.
On January 18-21, 2019 the Center of Mathematical Sciences and Applications will be hosting a workshop on the Geometric Analysis Approach to AI.
This workshop will focus on the theoretic foundations of AI, especially various methods in Deep Learning. The topics will cover the relationship between deep learning and optimal transportation theory, DL and information geometry, DL Learning and information bottle neck and renormalization theory, DL and manifold embedding and so on. Furthermore, the recent advancements, novel methods, and real world applications of Deep Learning will also be reported and discussed.
The workshop will take place from January 18th to January 23rd, 2019. In the first four days, from January 18th to January 21, the speakers will give short courses; On the 22nd and 23rd, the speakers will give conference representations. This workshop is organized by Xianfeng Gu and Shing-Tung Yau.
The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Please register here
Speakers:
- Sarah Adel Bargal, Boston University
- Guy Bresler, MIT
- Tina Eliassi-Rad, Northeastern
- Yun Raymond Fu, Northeastern
- Brian Kulis, Boston University
- Na Lei, Dalian University of Technology
- Yi Ma, UC Berkeley
- Minh Hoai Nguyen, Stony Brook
- Francesco Orabona, Boston University
- Cengiz Pehlevan, Harvard SEAS
- Tomaso Poggio, MIT
- Zhiwei Qin, DiDi Research America
- Kate Saenko, Boston University
- Dimitris Samaras, Stony Brook
- Johannes Schmidt-Hieber, University of Twente
- Steven Skiena, Stony Brook
- Vivienne Sze, MIT
- Naftali Tishby, ICNC
- Jiajun Wu, MIT
- Ying Nian Wu, UCLA
- Gangqiang Xia, Morgan Stanley
- Eric Xing, Carnegie Mellon
- Donghui Yan, UMass Dartmouth
- Alan Yuille, Johns Hopkins
- Juhua Zhu, Argus
Workshop on Aspects of General Relativity
The Center of Mathematical Sciences and Applications will be hosting a workshop on General Relativity from May 23 – 24, 2016. The workshop will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138. The workshop will start on Monday, May 23 at 9am and end on Tuesday, May 24 at 4pm.
Speakers:
- Po-Ning Chen, Columbia University
- Piotr T. Chruściel, University of Vienna
- Justin Corvino, Lafayette College
- Greg Galloway, University of Miami
- James Guillochon, Harvard University
- Lan-Hsuan Huang, University of Connecticut
- Dan Kapec, Harvard University
- Dan Lee, CUNY
- Alex Lupsasca, Harvard University
- Pengzi Miao, University of Miami
- Prahar Mitra, Harvard University
- Lorenzo Sironi, Harvard University
- Jared Speck, MIT
- Mu-Tao Wang, Columbia University
Please click Workshop Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Please click here for registration – Registration is capped at 70 participants.
Schedule:
May 23 – Day 1 8:30am Breakfast 8:55am Opening remarks 9:00am – 9:45am Greg Galloway, “Some remarks on photon spheres and their uniqueness“ 9:45am – 10:30am Prahar Mitra, “BMS supertranslations and Weinberg’s soft graviton theorem“ 10:30am – 11:00am Break 11:00am – 11:45am Dan Kapec, “Area, Entanglement Entropy and Supertranslations at Null Infinity“ 11:45am – 12:30pm Piotr T. Chruściel, “The cosmological constant and the energy of gravitational radiation” 12:30pm – 2:00pm Lunch 2:00pm – 2:45pm James Guillochon, “Tidal disruptions of stars by supermassive black holes: dynamics, light, and relics” 2:45pm – 3:30pm Mu-Tao Wang, “Quasi local conserved quantities in general relativity“ 3:30pm – 4:00pm Break 4:00pm – 4:45pm Po-Ning Chen, “Quasi local energy in presence of gravitational radiations” 4:45pm – 5:30pm Pengzi Miao, “Total mean curvature, scalar curvature, and a variational analog of Brown York mass“ May 24 – Day 2 8:45am Breakfast 9:00am – 9:45am Justin Corvino, “Scalar curvature deformation and the Bartnik mass“ 9:45am – 10:30am Lan-Hsuan Huang, “Constraint Manifolds with the Dominant Energy Condition“ 10:30am – 11:00am Break 11:00am – 11:45am Dan Lee, “Lower semicontinuity of Huisken’s isoperimetric mass“ 11:45am – 12:30pm Jared Speck, “Shock Formation in Solutions to the Compressible Euler Equations“ 12:30pm – 2:00pm Lunch 2:00pm – 2:45pm Lorenzo Sironi, “Electron Heating and Acceleration in the Vicinity of Supermassive Black Holes“ 2:45pm – 3:30pm Alex Lupsasca, “Near Horizon Extreme Kerr Magnetospheres“ * Click titles for talk videos. All videos are also available on “Harvard CMSA” channel on Youtube, grouped into playlist “Workshop on Aspects on General Relativity“.
* This event is sponsored by National Science Foundation (NSF) and CMSA Harvard University.
Organizers: Piotr T. Chruściel and Shing-Tung Yau
Big Data Conference 2018
1 Oxford Street, Cambridge MA 02138On August 23-24, 2018 the CMSA will be hosting our fourth annual Conference on Big Data. The Conference will feature many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The talks will take place in Science Center Hall B, 1 Oxford Street.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants.
Confirmed Speakers:
- Mohammad Akbarpour, Stanford
- Emily Breza, Harvard
- Francesca Dominici, Harvard
- Chiara Farronato, Harvard
- Kobi Gal, Ben Gurion
- Jonah Kallenbach, Reverie Labs
- Samuel Kou, Harvard
- Laura Kreidberg, Harvard
- Danielle Li, MIT
- Libby Mishkin, Uber
- Josh Speagle, Harvard
- William Stein, University of Washington
- Alex Teyltelboym, University of Oxford
- Sergiy Verstyuk, CMSA/Harvard
Organizers:
- Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University
- Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business
- Richard Freeman, Herbert Ascherman Professor of Economics, Harvard University
- Jun Liu, Professor of Statistics, Harvard University
- Horng-Tzer Yau, Professor of Mathematics, Harvard University
Workshop on Morphometrics, Morphogenesis and Mathematics
In Fall 2018, the CMSA will host a Program on Mathematical Biology, which aims to describe recent mathematical advances in using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems.
The plethora of natural shapes that surround us at every scale is both bewildering and astounding – from the electron micrograph of a polyhedral virus, to the branching pattern of a gnarled tree to the convolutions in the brain. Even at the human scale, the shapes seen in a garden at the scale of a pollen grain, a seed, a sapling, a root, a flower or leaf are so numerous that “it is enough to drive the sanest man mad,” wrote Darwin. Can we classify these shapes and understand their origins quantitatively?
In biology, there is growing interest in and ability to quantify growth and form in the context of the size and shape of bacteria and other protists, to understand how polymeric assemblies grow and shrink (in the cytoskeleton), and how cells divide, change size and shape, and move to organize tissues, change their topology and geometry, and link multiple scales and connect biochemical to mechanical aspects of these problems, all in a self-regulated setting.
To understand these questions, we need to describe shape (biomathematics), predict shape (biophysics), and design shape (bioengineering).
For example, in mathematics there are some beautiful links to Nash’s embedding theorem, connections to quasi-conformal geometry, Ricci flows and geometric PDE, to Gromov’s h principle, to geometrical singularities and singular geometries, discrete and computational differential geometry, to stochastic geometry and shape characterization (a la Grenander, Mumford etc.). A nice question here is to use the large datasets (in 4D) and analyze them using ideas from statistical geometry (a la Taylor, Adler) to look for similarities and differences across species during development, and across evolution.
In physics, there are questions of generalizing classical theories to include activity, break the usual Galilean invariance, as well as isotropy, frame indifference, homogeneity, and create both agent (cell)-based and continuum theories for ordered, active machines, linking statistical to continuum mechanics, and understanding the instabilities and patterns that arise. Active generalizations of liquid crystals, polar materials, polymers etc. are only just beginning to be explored and there are some nice physical analogs of biological growth/form that are yet to be studied.
The CMSA will be hosting a Workshop on Morphometrics, Morphogenesis and Mathematics from October 22-24 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
The workshop is organized by L. Mahadevan (Harvard), O. Pourquie (Harvard), A. Srivastava (Florida).
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Videos of the talks
Confirmed Speakers:
- Arkhat Abzhanov, Imperial College
- Siobhan Braybrook, UCLA
- Cassandra Extavour, Harvard
- Anjali Goswami, University College London
- David Gu, Stony Brook
- Jukka Jernvall, Helsinki
- Eric Klassen, Florida State
- Sayan Mukherjee, Duke
- Peter Olver, U Minnesota
- Nipam Patel, Berkeley
- Stephanie Pierce, Harvard
- Karen Sears, UCLA
- Alain Trouve, ENS-Cachan, France
- Laurent Younes, Johns Hopkins
2015 Conference on Big Data
1 Oxford Street, Cambridge MA 02138The Center of Mathematical Sciences and Applications will be having a conference on Big Data August 24-26, 2015, in Science Center Hall B at Harvard University. This conference will feature many speakers from the Harvard Community as well as many scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
For more info, please contact Sarah LaBauve at slabauve@math.harvard.edu.
Registration for the conference is now closed.
Please click here for a downloadable version of this schedule.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found here.
Monday, August 24
Time Speaker Title 8:45am Meet and Greet 9:00am Sendhil Mullainathan Prediction Problems in Social Science: Applications of Machine Learning to Policy and Behavioral Economics 9:45am Mike Luca Designing Disclosure for the Digital Age 10:30 Break 10:45 Jianqing Fan Big Data Big Assumption: Spurious discoveries and endogeneity 11:30am Daniel Goroff Privacy and Reproducibility in Data Science 12:15pm Break for Lunch 2:00pm Ryan Adams Exact Markov Chain Monte Carlo with Large Data 2:45pm David Dunson Scalable Bayes: Simple algorithms with guarantees 3:30pm Break 3:45pm Michael Jordan Computational thinking, inferential thinking and Big Data 4:30pm Joel Tropp Applied Random Matrix Theory 5:15pm David Woodruff Input Sparsity and Hardness for Robust Subspace Approximation Tuesday, August 25
Time Speaker Title 8:45am Meet and Greet 9:00am Gunnar Carlsson Persistent homology for qualitative analysis and feature generation 9:45am Andrea Montanari Semidefinite Programming Relaxations for Graph and Matrix Estimation: Algorithms and Phase Transitions 10:30am Break 10:45am Susan Athey Machine Learning and Causal Inference for Policy Evaluation 11:30am Denis Nekipelov Robust Empirical Evaluation of Large Competitive Markets 12:15pm Break for Lunch 2:00pm Lucy Colwell Using evolutionary sequence variation to make inferences about protein structure and function: Modeling with Random Matrix Theory 2:45pm Simona Cocco Inverse Statistical Physics approaches for the modeling of protein families 3:30pm Break 3:45pm Remi Monasson Inference of top components of correlation matrices with prior informations 4:30pm Sayan Mukherjee Random walks on simplicial complexes and higher order notions of spectral clustering A Banquet from 7:00 – 8:30pm will follow Tuesday’s talks. This event is by invitation only.
Wednesday, August 26
Time Speaker Title 8:45am Meet and Greet 9:00am Ankur Moitra Beyond Matrix Completion 9:45am Florent Krzakala Optimal compressed sensing with spatial coupling and message passing 10:30am Break 10:45am Piotr Indyk Fast Algorithms for Structured Sparsity 11:30am Guido Imbens Exact p-values for network inference 12:15pm Break for lunch 2:00pm Edo Airoldi Some fundamental ideas for causal inference on large networks 2:45pm Ronitt Rubinfeld Something for almost nothing: sublinear time approximation algorithms 3:30pm Break 3:45pm Lenka Zdeborova Clustering of sparse networks: Phase transitions and optimal algorithms 4:30pm Jelani Nelson Dimensionality reductions via sparse matrices The number of n-queens configurations
Speaker: Michael Simkin, Harvard CMSA
Title: The number of n-queens configurations
Abstract: The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. The problem has a storied history and was studied by such eminent mathematicians as Gauss and Polya. The problem has also found applications in fields such as algorithm design and circuit development.
Despite much study, until recently very little was known regarding the asymptotics of Q(n). We apply modern methods from probabilistic combinatorics to reduce understanding Q(n) to the study of a particular infinite-dimensional convex optimization problem. The chief implication is that (in an appropriate sense) for a~1.94, Q(n) is approximately (ne^(-a))^n. Furthermore, our methods allow us to study the typical “shape” of n-queens configurations.
CMSA Math-Science Literature Lecture: Discrepancy Theory and Randomized Controlled Trials
Dan Spielman (Yale University)
Title: Discrepancy Theory and Randomized Controlled Trials
Abstract: Discrepancy theory tells us that it is possible to partition vectors into sets so that each set looks surprisingly similar to every other. By “surprisingly similar” we mean much more similar than a random partition. I will begin by surveying fundamental results in discrepancy theory, including Spencer’s famous existence proofs and Bansal’s recent algorithmic realizations of them. Randomized Controlled Trials are used to test the effectiveness of interventions, like medical treatments. Randomization is used to ensure that the test and control groups are probably similar. When we know nothing about the experimental subjects, uniform random assignment is the best we can do. When we know information about the experimental subjects, called covariates, we can combine the strengths of randomization with the promises of discrepancy theory. This should allow us to obtain more accurate estimates of the effectiveness of treatments, or to conduct trials with fewer experimental subjects. I will introduce the Gram-Schmidt Walk algorithm of Bansal, Dadush, Garg, and Lovett, which produces random solutions to discrepancy problems. I will then explain how Chris Harshaw, Fredrik Sävje, Peng Zhang, and I use this algorithm to improve the design of randomized controlled trials. Our Gram-Schmidt Walk Designs have increased accuracy when the experimental outcomes are correlated with linear functions of the covariates, and are comparable to uniform random assignments in the worst case.
Talk chair: Salil Vadhan
The n-queens problem
Abstract: The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n).
In this talk we’ll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost’ configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.
This is joint work with Peter Keevash.
CMSA Math-Science Literature Lecture: Quantum topology and new types of modularity
Don Zagier (Max Planck Institute for Mathematics and International Centre for Theoretical Physics)
Title: Quantum topology and new types of modularity
Abstract: The talk concerns two fundamental themes of modern 3-dimensional topology and their unexpected connection with a theme coming from number theory. A deep insight of William Thurston in the mid-1970s is that the vast majority of complements of knots in the 3-sphere, or more generally of 3-manifolds, have a unique metric structure as hyperbolic manifolds of constant curvature -1, so that 3-dimensional topology is in some sense not really a branch of topology at all, but of differential geometry. In a different direction, the work of Vaughan Jones and Ed Witten in the late 1980s gave rise to the field of Quantum Topology, in which new types of invariants of knot complements and 3-manifolds are introduced that have their origins in ideas coming from quantum field theory. These two themes then became linked by Kashaev’s famous Volume Conjecture, now some 25 years old, which says that the Kashaev invariant _N of a hyperbolic knot K (this is a quantum invariant defined for each positive integer N and whose values are algebraic numbers) grows exponentially as N tends to infinity with an exponent proportional to the hyperbolic volume of the knot complement. About 10 years ago, I was led by numerical experiments to the discovery that Kashaev’s invariant could be upgraded to an invariant having rational numbers as its argument (with the original invariant being the value at 1/N) and that the Volume Conjecture then became part of a bigger story saying that the new invariant has some sort of strange transformation property under the action x -> (ax+b)/(cx+d) of the modular group SL(2,Z) on the argument. This turned out to be only the beginning of a fascinating and multi-faceted story relating quantum invariants, q-series, modularity, and many other topics. In the talk, which is intended for a general mathematical audience, I would like to recount some parts of this story, which is joint work with Stavros Garoufalidis (and of course involving contributions from many other authors). The “new types of modularity” in the title refer to a specific byproduct of these investigations, namely that there is a generalization of the classical notion of holomorphic modular form – which plays an absolutely central role in modern number theory – to a new class of holomorphic functions in the upper half-plane that no longer satisfy a transformation law under the action of the modular group, but a weaker extendability property instead. This new class, called “holomorphic quantum modular forms”, turns out to contain many other functions of a more number-theoretical nature as well as the original examples coming from quantum invariants.
Talk chair: Mark Kisin
Machine Learning for Multiscale Model Reduction Workshop
The Machine Learning for Multiscale Model Reduction Workshop will take place on March 27-29, 2019. This is the second of two workshops organized by Michael Brenner, Shmuel Rubinstein, and Tom Hou. The first, Fluid turbulence and Singularities of the Euler/ Navier Stokes equations, will take place on March 13-15, 2019. Both workshops will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Speakers:
- Joan Bruna, Courant Institute
- Predrag Cvitanovic, Georgia Tech
- Stephan Hoyer, Google Research
- De Huang, Caltech
- George Karniadakis, Brown University
- Richard Kerswell, Cambridge University
- Stephane Mallat, ENS
- Stanley Osher, UCLA
- Jacob Page, Cambridge University
- Houman Owhadi, Caltech
- Zuowei Shen, National University of Singapore
- Jack Xin, UC Irvine
- Jinchao Xu, Penn State University
- Lexing Ying, Stanford University and Facebook AI Research
- Pengchuan Zhang, Microsoft Research
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Workshop on Coding and Information Theory
The workshop on coding and information theory will take place April 9-13, 2018 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
This workshop will focus on new developments in coding and information theory that sit at the intersection of combinatorics and complexity, and will bring together researchers from several communities — coding theory, information theory, combinatorics, and complexity theory — to exchange ideas and form collaborations to attack these problems.
Squarely in this intersection of combinatorics and complexity, locally testable/correctable codes and list-decodable codes both have deep connections to (and in some cases, direct motivation from) complexity theory and pseudorandomness, and recent progress in these areas has directly exploited and explored connections to combinatorics and graph theory. One goal of this workshop is to push ahead on these and other topics that are in the purview of the year-long program. Another goal is to highlight (a subset of) topics in coding and information theory which are especially ripe for collaboration between these communities. Examples of such topics include polar codes; new results on Reed-Muller codes and their thresholds; coding for distributed storage and for DNA memories; coding for deletions and synchronization errors; storage capacity of graphs; zero-error information theory; bounds on codes using semidefinite programming; tensorization in distributed source and channel coding; and applications of information-theoretic methods in probability and combinatorics. All these topics have attracted a great deal of recent interest in the coding and information theory communities, and have rich connections to combinatorics and complexity which could benefit from further exploration and collaboration.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
Click here for a list of registrants.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Confirmed participants include:
- Emmanuel Abbe, Princeton University
- Simeon Ball, Universitat Politècnica de Catalunya
- Boris Bukh, Carnegie Mellon University
- Mahdi Cheraghchi, Imperial College London
- Sivakanth Gopi, Princeton University
- Elena Grigorescu, University of Purdue
- Hamed Hassani, University of Pennsylvania
- Navin Kashyap, Indian Institute of Science
- Young-Han Kim, University of California, San Diego
- Swastik Kopparty, Rutgers University
- Nati Linial, Hebrew University of Jerusalem
- Shachar Lovett, University of California, San Diego
- William Martin, Worcester Polytechnic Institute
- Arya Mazumdar, University of Massachusetts at Amherst
- Or Meir, University of Haifa
- Olgica Milenkovic, ECE Illinois
- Chandra Nair, Chinese University of Hong Kong
- Yuval Peres, Microsoft Research
- Yury Polyanskiy, Massachusetts Institute of Technology
- Maxim Raginsky, University of Illinois at Urbana-Champaign
- Sankeerth Rao Karingula, UC San Diego
- Ankit Singh Rawat, MIT
- Noga Ron-Zewi, University of Haifa
- Ron Roth, Israel Institute of Technology
- Atri Rudra, State University of New York, Buffalo
- Alex Samorodnitsky, Hebrew University of Jerusalem
- Itzhak Tamo, Tel Aviv University
- Amnon Ta-Shma, Tel Aviv University
- Himanshu Tyagi, Indian Institute of Science
- David Zuckerman, University of Texas at Austin
CMSA Math-Science Literature Lecture: Area-minimizing integral currents and their regularity
Camillo De Lellis (IAS)
Title: Area-minimizing integral currents and their regularity
Abstract: Caccioppoli sets and integral currents (their generalization in higher codimension) were introduced in the late fifties and early sixties to give a general geometric approach to the existence of area-minimizing oriented surfaces spanning a given contour. These concepts started a whole new subject which has had tremendous impacts in several areas of mathematics: superficially through direct applications of the main theorems, but more deeply because of the techniques which have been invented to deal with related analytical and geometrical challenges. In this lecture I will review the basic concepts, the related existence theory of solutions of the Plateau problem, and what is known about their regularity. I will also touch upon several fundamental open problems which still defy our understanding.
Talk Chair: William Minicozzi
CMSA Math-Science Literature Lecture: From Deep Learning to Deep Understanding
Harry Shum (Tsinghua University)
Title: From Deep Learning to Deep Understanding
Abstract: In this talk I will discuss a couple of research directions for robust AI beyond deep neural networks. The first is the need to understand what we are learning, by shifting the focus from targeting effects to understanding causes. The second is the need for a hybrid neural/symbolic approach that leverages both commonsense knowledge and massive amount of data. Specifically, as an example, I will present some latest work at Microsoft Research on building a pre-trained grounded text generator for task-oriented dialog. It is a hybrid architecture that employs a large-scale Transformer-based deep learning model, and symbol manipulation modules such as business databases, knowledge graphs and commonsense rules. Unlike GPT or similar language models learnt from data, it is a multi-turn decision making system which takes user input, updates the belief state, retrieved from the database via symbolic reasoning, and decides how to complete the task with grounded response.
Talk chair: Shing-Tung Yau
CMSA Math-Science Literature Lecture: Hodge structures and the topology of algebraic varieties
Claire Voisin (Collège de France)
Title: Hodge structures and the topology of algebraic varieties
Abstract: We review the major progress made since the 50’s in our understanding of the topology of complex algebraic varieties. Most of the results we will discuss rely on Hodge theory, which has some analytic aspects giving the Hodge and Lefschetz decompositions, and the Hodge-Riemann relations. We will see that a crucial ingredient, the existence of a polarization, is missing in the general Kaehler context. We will also discuss some results and problems related to algebraic cycles and motives.
Talk chair: Joe Harris
Workshop on Optimization in Image Processing
The Center of Mathematical Sciences and Applications will be hosting a workshop on Optimization in Image Processing on June 27 – 30, 2016. This 4-day workshop aims to bring together researchers to exchange and stimulate ideas in imaging sciences, with a special focus on new approaches based on optimization methods. This is a cutting-edge topic with crucial impact in various areas of imaging science including inverse problems, image processing and computer vision. 16 speakers will participate in this event, which we think will be a very stimulating and exciting workshop. The workshop will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
Titles, abstracts and schedule will be provided nearer to the event.
Speakers:
- Antonin Chambolle, CMAP, Ecole Polytechnique
- Raymond Chan, The Chinese University of Hong Kong
- Ke Chen, University of Liverpool
- Patrick Louis Combettes, Université Pierre et Marie Curie
- Mario Figueiredo, Instituto Superior Técnico
- Alfred Hero, University of Michigan
- Ronald Lok Ming Lui, The Chinese University of Hong Kong
- Mila Nikolova, Ecole Normale Superieure Cachan
- Shoham Sabach, Israel Institute of Technology
- Martin Benning, University of Cambridge
- Jin Keun Seo, Yonsei University
- Fiorella Sgallari, University of Bologna
- Gabriele Steidl, Kaiserslautern University of Technology
- Joachim Weickert, Saarland University
- Isao Yamada, Tokyo Institute of Technology
- Wotao Yin, UCLA
Please click Workshop Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Please click here for registration – Registration Deadline: June 7, 2016; Registration is capped at 70 participants.
Schedule:
June 27 – Day 1 9:00am Breakfast 9:20am Opening remarks 9:30am – 10:20am Joachim Weickert, “FSI Schemes: Fast Semi-Iterative Methods for Diffusive or Variational Image Analysis Problems” 10:20am – 10:50am Break 10:50am – 11:40pm Patrick Louis Combettes, “Block-Iterative Asynchronous Variational Image Recovery” 11:40am – 12:30pm Isao Yamada, “Spicing up Convex Optimization for Certain Inverse Problems” 12:30pm – 2:00pm Lunch 2:30pm – 3:20pm Fiorella Sgallari, “Majorization-Minimization for Nonconvex Optimization” 3:20pm – 3:50pm Break 3:50pm – 4:40pm Shoham Sabach, “A Framework for Globally Convergent Methods in Nonsmooth and Nonconvex Problems” June 28 – Day 2 9:00am Breakfast 9:30am – 10:20am Antonin Chambolle, “Acceleration of alternating minimisations” 10:20am – 10:50am Break 10:50am – 11:40am Mario Figueiredo, “ADMM in Image Restoration and Related Problems: Some History and Recent Advances” 11:40am – 12:30pm Ke Chen, “Image Restoration and Registration Based on Total Fractional-Order Variation Regularization” 12:30pm – 2:30pm Lunch 2:30pm – 4:40pm Discussions June 29 – Day 3 9:00am Breakfast 9:30am – 10:20am Alfred Hero, “Continuum relaxations for discrete optimization” 10:20am – 10:50am Break 10:50am – 11:40am Wotao Yin, “Coordinate Update Algorithms for Computational Imaging and Machine Learning” 11:40am – 12:30pm Mila Nikolova, “Limits on noise removal using log-likelihood and regularization” 12:30pm – 2:30pm Lunch 2:30pm – 3:20pm Martin Benning, “Nonlinear spectral decompositions and the inverse scale space method” 3:20pm – 3:50pm Break 3:50pm – 4:40pm Ronald Ming Lui, “TEMPO: Feature-endowed Teichmuller extremal mappings of point cloud for shape classification” June 30 – Day 4 9:00am Breakfast 9:30am – 10:20am Jin Keun Seo, “Mathematical methods for biomedical impedance imaging” 10:20am – 10:50am Break 10:50am – 11:40am Gabriele Steidl, “Iterative Multiplicative Filters for Data Labeling” 11:40am – 12:30pm Raymond Chan, “Point-spread function reconstruction in ground-based astronomy” * This event is sponsored by CMSA Harvard University.
Organizers: Raymond Chan and Shing-Tung Yau
CMSA Math-Science Literature Lecture: Moment maps and the Yang-Mills functional
Frances Kirwan (University of Oxford)
Title: Moment maps and the Yang-Mills functional
Abstract: In the early 1980s Michael Atiyah and Raoul Bott wrote two influential papers, ‘The Yang-Mills equations over Riemann surfaces’ and ‘The moment map and equivariant cohomology’, bringing together ideas ranging from algebraic and symplectic geometry through algebraic topology to mathematical physics and number theory. The aim of this talk is to explain their key insights and some of the new directions towards which these papers led.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
Talk chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: Quantum error correcting codes and fault tolerance
Peter Shor (MIT)
Title: Quantum error correcting codes and fault tolerance
Abstract: We will go over the fundamentals of quantum error correction and fault tolerance and survey some of the recent developments in the field.
Talk chair: Zhengwei LiuWorkshop on Probabilistic and Extremal Combinatorics
The workshop on Probabilistic and Extremal Combinatorics will take place February 5-9, 2018 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
Extremal and Probabilistic Combinatorics are two of the most central branches of modern combinatorial theory. Extremal Combinatorics deals with problems of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects satisfying certain requirements. Such problems are often related to other areas including Computer Science, Information Theory, Number Theory and Geometry. This branch of Combinatorics has developed spectacularly over the last few decades. Probabilistic Combinatorics can be described informally as a (very successful) hybrid between Combinatorics and Probability, whose main object of study is probability distributions on discrete structures.
There are many points of interaction between these fields. There are deep similarities in methodology. Both subjects are mostly asymptotic in nature. Quite a few important results from Extremal Combinatorics have been proven applying probabilistic methods, and vice versa. Such emerging subjects as Extremal Problems in Random Graphs or the theory of graph limits stand explicitly at the intersection of the two fields and indicate their natural symbiosis.
The symposia will focus on the interactions between the above areas. These topics include Extremal Problems for Graphs and Set Systems, Ramsey Theory, Combinatorial Number Theory, Combinatorial Geometry, Random Graphs, Probabilistic Methods and Graph Limits.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Confirmed participants include:
- Jozsef Balogh, University of Illinois, Urbana
- Fan Chung (Graham), University of California, San Diego
- Asaf Ferber, Massachusetts Institute of Technology
- Jacob Fox, Stanford Unviersity
- David Gamarnik, Massachusetts Institute of Technology
- Penny Haxell, University of Waterloo
- Hao Huang, Emory University
- Jeff Kahn, Rutgers University
- Peter Keevash, Oxford University
- Michael Krivelevich, Tel Aviv University
- Daniela Kühn, University of Birmingham
- Shoham Letzer, ITS Zürich
- Shachar Lovett, University of California, San Diego
- Eyal Lubetzky, Courant Institute
- Rob Morris, IMPA
- Bhargav Narayanan, Rutgers University
- Deryk Osthus, University of Birmingham
- Janos Pach, NYU
- Yuval Peres, Microsoft Redmond
- Alexey Pokryovskyi, ETH Zürich
- Wojciech Samotij, Tel Aviv University
- Lisa Sauermann, Stanford University
- Mathias Schacht, University of Hamburg
- Alexander Scott, University of Oxford
- Asaf Shapira, Tel Aviv University
- Jozef Skokan, London School of Economics
- Joel Spencer, New York University
- Angelika Steger, ETH Zurich
- Jacques Verstraete, University of California, San Diego
- Yufei Zhao, Massachusetts Institute of Technology
- David Zuckerman, University of Texas at Austin
Co-organizers of this workshop include Benny Sudakov and David Conlon. More details about this event, including participants, will be updated soon.
CMSA Math-Science Literature Lecture: Isadore Singer’s Work on Analytic Torsion
Edward Witten (IAS)
Title: Isadore Singer’s Work on Analytic Torsion
Abstract: I will review two famous papers of Ray and Singer on analytic torsion written approximately half a century ago. Then I will sketch the influence of analytic torsion in a variety of areas of physics including anomalies, topological field theory, and string theory.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer.
Talk chair: Cumrun Vafa
CMSA Math-Science Literature Lecture: Homological (homotopical) algebra and moduli spaces in Topological Field theories
Kenji Fukaya (Simons Center for Geometry and Physics)
Title: Homological (homotopical) algebra and moduli spaces in Topological Field theories
Abstract: Moduli spaces of various gauge theory equations and of various versions of (pseudo) holomorphic curve equations have played important role in geometry in these 40 years. Started with Floer’s work people start to obtain more sophisticated object such as groups, rings, or categories from (system of) moduli spaces. I would like to survey some of those works and the methods to study family of moduli spaces systematically.
Talk chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: The Atiyah-Singer Index Theorem
Dan Freed (The University of Texas at Austin)
Title: The Atiyah-Singer Index Theorem
Abstract: The story of the index theorem ties together the Gang of Four—Atiyah, Bott, Hirzebruch, and Singer—and lies at the intersection of analysis, geometry, and topology. In the first part of the talk I will recount high points in the early developments. Then I turn to subsequent variations and applications. Throughout I emphasize the role of the Dirac operator.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
Talk chair: Cumrun Vafa
Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer
In 2021, the CMSA hosted a lecture series on the literature of the mathematical sciences. This series highlights significant accomplishments in the intersection between mathematics and the sciences. Speakers include Edward Witten, Lydia Bieri, Simon Donaldson, Michael Freedman, Dan Freed, and many more.
Videos of these talks can be found in this Youtube playlist.
https://youtu.be/vb_JEhUW9t4
In the Spring 2021 semester, the CMSA hosted a sub-program on this series titled A Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer. Below is the schedule for talks in that subprogram
April 6, 2021 | 9:00 – 10:30am ET
Edward Witten (IAS) April 13, 2021 | 9:00 – 10:30am ET
Claire Voisin (College de France) April 20, 2021 | 9:00 – 10:30am ET
Dan Freed (the University of Texas at Austin) April 27, 2021 | 9:00 – 10:30am ET
Frances Kirwan (University of Oxford) 10/12/2021 Combinatorics, Physics and Probability Seminar
Title: On counting algebraically defined graphs
Abstract: For many classes of graphs that arise naturally in discrete geometry (for example intersection graphs of segments or disks in the plane), the edges of these graphs can be defined algebraically using the signs of a finite list of fixed polynomials. We investigate the number of n-vertex graphs in such an algebraically defined class of graphs. Warren’s theorem (a variant of a theorem of Milnor and Thom) implies upper bounds for the number of n-vertex graphs in such graph classes, but all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving a lower bound for this number (under some reasonable assumptions on the fixed list of polynomials), and this lower bound essentially matches the upper bound from Warren’s theorem.
Simons Collaboration Workshop, Jan. 10-13, 2018
The CMSA will be hosting a four-day Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory on January 10-13, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Please click here to register for this event. We have space for up to 30 registrants on a first come, first serve basis.
We may be able to provide some financial support for grad students and postdocs interested in this event. If you are interested in funding, please send a letter of support from your mentor to Hansol Hong at hansol84@gmail.com.Confirmed Participants:
- Mohammed Abouzaid (Columbia University)
- Sergueï Barannikov (Paris Diderot University)
- Cheol-Hyun Cho (Seoul National University)
- Young-Hoon Kiem (Seoul National University)
- Thomas Lam (University of Michigan)
- Siu-Cheong Lau (Boston University)
- Radu Laza (Stony Brook University)
- Si Li (Tsinghua University)
- Kaoru Ono (Kyoto University)
- Tony Pantev (University of Pennsylvania)
- Colleen Robles (Duke University)
- Yan Soibelman (Kansas State University)
- Kazushi Ueda (University of Tokyo)
- Chenglong Yu (Harvard University)
- Eric Zaslow (Northwestern University)
CMSA Math-Science Literature Lecture: On the History of quantum cohomology and homological mirror symmetry
Maxim Kontsevich (IHÉS)
Title: On the History of quantum cohomology and homological mirror symmetry
Abstract: About 30 years ago, string theorists made remarkable discoveries of hidden structures in algebraic geometry. First, the usual cup-product on the cohomology of a complex projective variety admits a canonical multi-parameter deformation to so-called quantum product, satisfying a nice system of differential equations (WDVV equations). The second discovery, even more striking, is Mirror Symmetry, a duality between families of Calabi-Yau varieties acting as a mirror reflection on the Hodge diamond.
Later it was realized that the quantum product belongs to the realm of symplectic geometry, and a half of mirror symmetry (called Homological Mirror Symmetry) is a duality between complex algebraic and symplectic varieties. The search of correct definitions and possible generalizations lead to great advances in many domains, giving mathematicians new glasses, through which they can see familiar objects in a completely new way.
I will review the history of major mathematical advances in the subject of HMS, and the swirl of ideas around it.
Talk chair: Paul Seidel
Quantum Matter Workshop
CMSA, 20 Garden Street, Cambridge, MA 02138 USAOn December 2-4, 2019 the CMSA will be hosting a workshop on Quantum Matter as part of our program on Quantum Matter in Mathematics and Physics. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Pictures can be found here.
Organizers: Juven Wang (CMSA), Xiao-Gang Wen (MIT), and Shing-Tung Yau (Harvard)
Confirmed Speakers:
- Zhen Bi, MIT | Video
- Claudio Chamon, BU | Video
- Trithep Devakul, Princeton | Video
- Anushya Chandran, BU
- Liang Fu, MIT
- Andrey Gromov, Brown | Video
- Daniel Louis Jafferis, Harvard | Video
- Eslam Khalaf, Harvard | Video
- Hong Liu, MIT
- Shang Liu, Harvard | Video
- Emil Prodan, Yeshiva | Video
- Subir Sachdev, Harvard | Video
- Dries Sels, Harvard | Video
- Yuya Tanizaki, NCSU | Video
- Senthil Todadri, MIT | Video
- Juven Wang, CMSA | Video
- Yifan Wang, CMSA | Video
- Xiao-Gang Wen, MIT
- Xueda Wen, MIT | Video
- Xi Yin, Harvard | Video
- Yizhi You, Princeton | Video
- Yunqin Zheng, Princeton | Video
Mini-school on Nonlinear Equations, December 3-4, 2016
The Center of Mathematical Sciences and Applications will be hosting a Mini-school on Nonlinear Equations on December 3-4, 2016. The conference will have speakers and will be hosted at Harvard CMSA Building: Room G10 20 Garden Street, Cambridge, MA 02138.
The mini-school will consist of lectures by experts in geometry and analysis detailing important developments in the theory of nonlinear equations and their applications from the last 20-30 years. The mini-school is aimed at graduate students and young researchers working in geometry, analysis, physics and related fields.
Please click here to register for this event.
Speakers:
- Cliff Taubes (Harvard University)
- Valentino Tosatti (Northwestern University)
- Pengfei Guan (McGill University)
- Jared Speck (MIT)
Schedule:
December 3rd – Day 1 9:00am – 10:30am Cliff Taubes, “Compactness theorems in gauge theories” 10:45am – 12:15pm Valentino Tosatti, “Complex Monge-Ampère Equations” 12:15pm – 1:45pm LUNCH 1:45pm – 3:15pm Pengfei Guan, “Monge-Ampère type equations and related geometric problems” 3:30pm – 5:00pm Jared Speck, “Finite-time degeneration of hyperbolicity without blowup for solutions to quasilinear wave equations” December 4th – Day 2 9:00am – 10:30am Cliff Taubes, “Compactness theorems in gauge theories” 10:45am – 12:15pm Valentino Tosatti, “Complex Monge-Ampère Equations” 12:15pm – 1:45pm LUNCH 1:45pm – 3:15pm Pengfei Guan, “Monge-Ampère type equations and related geometric problems” 3:30pm – 5:00pm Jared Speck, “Finite-time degeneration of hyperbolicity without blowup for solutions to quasilinear wave equations” Please click Mini-School Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
* This event is sponsored by National Science Foundation (NSF) and CMSA Harvard University.
Simons Collaboration on Homological Mirror Symmetry
The Center of Mathematical Sciences and Applications will be hosting a 3-day workshop on Homological Mirror Symmetry and related areas on May 6 – May 8, 2016 at Harvard CMSA Building: Room G10 20 Garden Street, Cambridge, MA 02138
Organizers:
D. Auroux, S.C. Lau, N.C. Leung, Bong Lian, C.C. Liu, S.T. Yau
Speakers:
- Netanel Blaier (MIT)
- Kwokwai Chan (CUHK)
- Bohan Fang (Peking University)
- Amanda Francis (BYU)
- Hansol Hong (CUHK)
- Heather Lee (Purdue University)
- Si Li (Tsinghua University)
- Yu-Shen Lin (Stanford University)
- Alex Perry (Harvard University)
- Hiro Tanaka (Harvard University)
- Sara Tukachinsky (HUJ)
- Michael Viscardi (MIT)
- Eric Zaslow (Northwestern University)
- Jingyu Zhao (Columbia University)
Please click here for the conference Main Website.
Please click Simons Workshop Schedule with Abstract for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Schedule:
May 6 – Day 1 9:00am Breakfast 9:35am Opening remarks 9:45am – 10:45am Si Li, “Quantum master equation, chiral algebra, and integrability” 10:45am – 11:15am Break 11:15am – 12:15pm Sara Tukachinsky, “Point like bounding chains and open WDVV“ 12:15pm – 1:45pm Lunch 1:45pm – 2:45pm Bohan Fang, “Mirror B model for toric Calabi Yau 3 folds“ 2:45pm – 3:00pm Break 3:00pm – 4:00pm Hiro Tanaka, “Toward Fukaya categories over arbitrary coefficients“ 4:00pm – 4:15pm Break 4:15pm – 5:15pm Hansol Hong, “Noncommutative mirror functors“ May 7 – Day 2 9:00am Breakfast 9:45am – 10:45am Eric Zaslow, “Lagrangian fillings what does the sheaf say?“ 10:45am – 11:15am Break 11:15am – 12:15pm Alex Perry, “Derived categories of Gushel Mukai varieties“ 12:15pm – 1:45pm Lunch 1:45pm – 2:45pm Amanda Francis, “A Landau Ginzburg mirror theorem inspired by Borcea Voisin symmetry“ 2:45pm – 3:00pm Break 3:00pm – 4:00pm Heather Lee, “Homological mirror symmetry for open Riemann surfaces from pair of pants decompositions“ 4:00pm – 4:15pm Break 4:15pm – 5:15pm Yu-Shen Lin, “Counting Holomorphic Discs via Tropical Discs on K3 Surfaces“ May 8 – Day 3 9:00am Breakfast 9:45am – 10:45am Kwokwai Chan, “HMS for local CY manifolds via SYZ“ 10:45am – 11:15am Break 11:15am – 12:15pm Netanel Blaier, “The quantum Johnson homomorphism, formality and symplectic isotopy“ 12:15pm – 1:45pm Lunch 1:45pm – 2:45pm Jingyu Zhao, “Periodic symplectic cohomology and the Hodge filtration“ 2:45pm – 3:00pm Break 3:00pm – 4:00pm Michael Viscardi, “Equivariant quantum cohomology and the geometric Satake equivalence“ * Click titles for talk videos. All videos are also available on “Harvard CMSA” channel on Youtube, grouped into playlist “Simons Collaboration on Homological Mirror symmetry“.
This event is sponsored by the Simons Foundation and CMSA Harvard University.
10/5/2021 Combinatorics, Physics and Probability Seminar
Title: Geodesic Geometry on Graphs
Abstract: In a graph G = (V, E) we consider a system of paths S so that for every two vertices u,v in V there is a unique uv path in S connecting them. The path system is said to be consistent if it is closed under taking subpaths, i.e. if P is a path in S then any subpath of P is also in S. Every positive weight function w: E–>R^+ gives rise to a consistent path system in G by taking the paths in S to be geodesics w.r.t. w. In this case, we say w induces S. We say a graph G is metrizable if every consistent path system in G is induced by some such w.
We’ll discuss the concept of graph metrizability, and, in particular, we’ll see that while metrizability is a rare property, there exists infinitely many 2-connected metrizable graphs.
Joint work with Nati Linial.
10/19/2021 Combinatorics, Physics and Probability Seminar
Title: Ising model, total positivity, and criticality
Abstract: The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality.
The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig’s theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes.
I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with P. Pylyavskyy. Building on this connection, I will give a new boundary correlation formula for the critical Ising modelCMSA Math-Science Literature Lecture: Theorems of Torelli type
Eduard Jacob Neven Looijenga (Tsinghua University & Utrecht University)
Title: Theorems of Torelli type
Abstract: Given a closed manifold of even dimension 2n, then Hodge showed around 1950 that a kählerian complex structure on that manifold determines a decomposition of its complex cohomology. This decomposition, which can potentially vary continuously with the complex structure, extracts from a non-linear given, linear data. It can contain a lot of information. When there is essentially no loss of data in this process, we say that the Torelli theorem holds. We review the underlying theory and then survey some cases where this is the case. This will include the classical case n=1, but the emphasis will be on K3 manifolds (n=2) and more generally, on hyperkählerian manifolds. These cases stand out, since one can then also tell which decompositions occur.
Talk chair: Gerard van der Geer
Workshop on Geometry, Imaging, and Computing
On March 24-26, The Center of Mathematical Sciences and Applications will be hosting a workshop on Geometry, Imaging, and Computing, based off the journal of the same name. The workshop will take place in CMSA building, G10.
The organizing committee consists of Yang Wang (HKUST), Ronald Lui (CUHK), David Gu (Stony Brook), and Shing-Tung Yau (Harvard).
Please click here to register for the event.
Confirmed Speakers:
- Jianfeng Cai (HKUST)
- Shikui Chen (Stony Brook)
- Jerome Darbon (Brown University)
- Laurent Demanet (MIT)
- David Gu (Stony Brook)
- Monica Hurdal (Florida State University)
- Rongjie Lai (RPI)
- Yue Lu (Harvard)
- Ronald Lok Ming Lui (CUHK)
- Lakshminarayanan Mahadevan (Harvard)
- Eric Miller (Tufts)
- Ashley Prater (AFOSR)
- Lixin Shen (Syracuse University)
- Allen Tannenbaum (Stony Brook)
- Guowei Wei (Michigan State)
- Stephen Wong (Houston Methodist)
- Jun Zhang (University of Michigan, Ann Arbor)
- Song Zhang (Purdue University)
- Hongkai Zhao (University of California, Irvine)
Mirror symmetry, gauged linear sigma models, matrix factorizations, and related topics
CMSA, 20 Garden Street, Cambridge, MA 02138 USAOn March 4-6, 2020 the CMSA will be hosting a three-day workshop on Mirror symmetry, Gauged linear sigma models, Matrix factorizations, and related topics as part of the Simons Collaboration on Homological Mirror Symmetry. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Speakers:
- Andrei Căldăraru, University of Wisconsin
- David Favero, University of Alberta
- Elana Kalashnikov, Harvard University
- Tsung-Ju Lee, CMSA
- Conan Leung, CUHK
- David Morrison, University of California, Santa Barbara
- Mauricio Romo, YMSC
- Yun Shi, CMSA
- Mark Shoemaker, Colorado State University
- Rachel Webb, University of Michigan
- Chris Woodward, Rutgers University
- Guangbo Xu, Texas A&M University
- Chenglong Yu, University of Pennsylvania
Videos from the workshop are available in the Youtube playlist.
The 2017 Charles River Lectures
The 2017 Charles River Lectures
Charles River with Bench at SunsetJointly organized by Harvard University, Massachusetts Institute of Technology, and Microsoft Research New England, the Charles River Lectures on Probability and Related Topics is a one-day event for the benefit of the greater Boston area mathematics community.
The 2017 lectures will take place 9:15am – 5:30pm on Monday, October 2 at Harvard University in the Harvard Science Center.
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UPDATED LOCATION
Harvard University
Harvard Science Center (Halls C & E)
1 Oxford Street, Cambridge, MA 02138 (Map)
Monday, October 2, 2017
9:15 AM – 5:30 PM
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Please note that registration has closed.
Speakers:
- Paul Bourgade (Courant Institute, NYU)
- Massimiliano Gubinelli (University of Bonn)
- Andrea Montanari (Stanford University)
- Roman Vershynin (University of California, Irvine)
- Ofer Zeitouni (Weizmann Institute)
Agenda:
In Harvard Science Center Hall C:
8:45 am – 9:15 am: Coffee/light breakfast
9:15 am – 10:15 am: Ofer Zeitouni
Title:
Abstract:
10:20 am – 11:20 am: Andrea Montanari
Title:
Abstract:
11:20 am – 11:45 am: Break
11:45 am – 12:45 pm: Paul Bourgade
Title:
Abstract:
1:00 pm – 2:30 pm: Lunch
In Harvard Science Center Hall E:
2:45 pm – 3:45 pm: Roman Vershynin
Title: Deviations of random matrices and applications
Abstract: Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This lecture will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in high dimensional geometry and data science.
3:45 pm – 4:15 pm: Break
4:15 pm – 5:15 pm: Massimiliano Gubinelli
Title:
Abstract:
Poster:
2017 Charles River Lectures Poster
Organizers:
Alexei Borodin, Henry Cohn, Vadim Gorin, Elchanan Mossel, Philippe Rigollet, Scott Sheffield, and H.T. Yau
Workshop on Invariance and Geometry in Sensation, Action and Cognition
As part of the program on Mathematical Biology a workshop on Invariance and Geometry in Sensation, Action and Cognition will take place on April 15-17, 2019.
Legend has it that above the door to Plato’s Academy was inscribed “Μηδείς άγεωµέτρητος είσίτω µον τήν στέγην”, translated as “Let no one ignorant of geometry enter my doors”. While geometry and invariance has always been a cornerstone of mathematics, it has traditionally not been an important part of biology, except in the context of aspects of structural biology. The premise of this meeting is a tantalizing sense that geometry and invariance are also likely to be important in (neuro)biology and cognition. Since all organisms interact with the physical world, this implies that as neural systems extract information using the senses to guide action in the world, they need appropriately invariant representations that are stable, reproducible and capable of being learned. These invariances are a function of the nature and type of signal, its corruption via noise, and the method of storage and use.
This hypothesis suggests many puzzles and questions: What representational geometries are reflected in the brain? Are they learned or innate? What happens to the invariances under realistic assumptions about noise, nonlinearity and finite computational resources? Can cases of mental disorders and consequences of brain damage be characterized as break downs in representational invariances? Can we harness these invariances and sensory contingencies to build more intelligent machines? The aim is to revisit these old neuro-cognitive problems using a series of modern lenses experimentally, theoretically and computationally, with some tutorials on how the mathematics and engineering of invariant representations in machines and algorithms might serve as useful null models.
In addition to talks, there will be a set of tutorial talks on the mathematical description of invariance (P.J. Olver), the computer vision aspects of invariant algorithms (S. Soatto), and the neuroscientific and cognitive aspects of invariance (TBA). The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by L. Mahadevan (Harvard), Talia Konkle (Harvard), Samuel Gershman (Harvard), and Vivek Jayaraman (HHMI).
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Videos
Tentative Speaker List:
- Alessandro Achille, UCLA
- Vijay Balasubramanian, University of Pennsylvania
- Jeannette Bohg, Stanford
- Ed Connor, Johns Hopkins
- Moira Dillon, NYU
- Jacob Feldman, Rutgers
- Ila Fiete, MIT
- Sam Gershman, Harvard
- Gily Ginosar, Weizmann Institute of Science
- Lucia Jacobs, UC Berkeley
- Vivek Jayaraman, HHMI
- Talia Konkle, Harvard
- L. Mahadevan, Harvard
- Michael McCloskey, Johns Hopkins
- Sam Ocko, Stanford
- Peter Olver, University of Minnesota
- Anitha Pasupathy, University of Washington
- Sandro Romani, Janelia
- Stefano Soatto, UCLA
- Tatyana Sharpee, Salk Institute
- Dagmar Sternad, Northeastern
- Elizabeth Torres, Rutgers
Schedule:
Monday, April 15
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 9:15am Welcome and Introduction 9:15 – 10:00am Vivek Jayaraman Title: Insect cognition: Small tales of geometry & invariance Abstract: Decades of field and laboratory experiments have allowed ethologists to discover the remarkable sophistication of insect behavior. Over the past couple of decades, physiologists have been able to peek under the hood to uncover sophistication in insect brain dynamics as well. In my talk, I will describe phenomena that relate to the workshop’s theme of geometry and invariance. I will outline how studying insects —and flies in particular— may enable an understanding of the neural mechanisms underlying these intriguing phenomena.
10:00 – 10:45am Elizabeth Torres Title: Connecting Cognition and Biophysical Motions Through Geometric Invariants and Motion Variability Abstract: In the 1930s Nikolai Bernstein defined the degrees of freedom (DoF) problem. He asked how the brain could control abundant DoF and produce consistent solutions, when the internal space of bodily configurations had much higher dimensions than the space defining the purpose(s) of our actions. His question opened two fundamental problems in the field of motor control. One relates to the uniqueness or consistency of a solution to the DoF problem, while the other refers to the characterization of the diverse patterns of variability that such solution produces.
In this talk I present a general geometric solution to Bernstein’s DoF problem and provide empirical evidence for symmetries and invariances that this solution provides during the coordination of complex naturalistic actions. I further introduce fundamentally different patterns of variability that emerge in deliberate vs. spontaneous movements discovered in my lab while studying athletes and dancers performing interactive actions. I here reformulate the DoF problem from the standpoint of the social brain and recast it considering graph theory and network connectivity analyses amenable to study one of the most poignant developmental disorders of our times: Autism Spectrum Disorders.
I offer a new unifying framework to recast dynamic and complex cognitive and social behaviors of the full organism and to characterize biophysical motion patterns during migration of induced pluripotent stem cell colonies on their way to become neurons.
10:45 – 11:15am Coffee Break 11:15 – 12:00pm Peter Olver Title: Symmetry and invariance in cognition — a mathematical perspective” Abstract: Symmetry recognition and appreciation is fundamental in human cognition. (It is worth speculating as to why this may be so, but that is not my intent.) The goal of these two talks is to survey old and new mathematical perspectives on symmetry and invariance. Applications will arise from art, computer vision, geometry, and beyond, and will include recent work on 2D and 3D jigsaw puzzle assembly and an ongoing collaboration with anthropologists on the analysis and refitting of broken bones. Mathematical prerequisites will be kept to a bare minimum.
12:00 – 12:45pm Stefano Soatto/Alessandro Achille Title: Information in the Weights and Emergent Properties of Deep Neural Networks Abstract: We introduce the notion of information contained in the weights of a Deep Neural Network and show that it can be used to control and describe the training process of DNNs, and can explain how properties, such as invariance to nuisance variability and disentanglement, emerge naturally in the learned representation. Through its dynamics, stochastic gradient descent (SGD) implicitly regularizes the information in the weights, which can then be used to bound the generalization error through the PAC-Bayes bound. Moreover, the information in the weights can be used to defined both a topology and an asymmetric distance in the space of tasks, which can then be used to predict the training time and the performance on a new task given a solution to a pre-training task.
While this information distance models difficulty of transfer in first approximation, we show the existence of non-trivial irreversible dynamics during the initial transient phase of convergence when the network is acquiring information, which makes the approximation fail. This is closely related to critical learning periods in biology, and suggests that studying the initial convergence transient can yield important insight beyond those that can be gleaned from the well-studied asymptotics.
12:45 – 2:00pm Lunch 2:00 – 2:45pm Anitha Pasupathy Title: Invariant and non-invariant representations in mid-level ventral visual cortex My laboratory investigates how visual form is encoded in area V4, a critical mid-level stage of form processing in the macaque monkey. Our goal is to reveal how V4 representations underlie our ability to segment visual scenes and recognize objects. In my talk I will present results from two experiments that highlight the different strategies used by the visual to achieve these goals. First, most V4 neurons exhibit form tuning that is exquisitely invariant to size and position, properties likely important to support invariant object recognition. On the other hand, form tuning in a majority of neurons is also highly dependent on the interior fill. Interestingly, unlike primate V4 neurons, units in a convolutional neural network trained to recognize objects (AlexNet) overwhelmingly exhibit fill-outline invariance. I will argue that this divergence between real and artificial circuits reflects the importance of local contrast in parsing visual scenes and overall scene understanding.
2:45 – 3:30pm Jacob Feldman Title: Bayesian skeleton estimation for shape representation and perceptual organization Abstract: In this talk I will briefly summarize a framework in which shape representation and perceptual organization are reframed as probabilistic estimation problems. The approach centers around the goal of identifying the skeletal model that best “explains” a given shape. A Bayesian solution to this problem requires identifying a prior over shape skeletons, which penalizes complexity, and a likelihood model, which quantifies how well any particular skeleton model fits the data observed in the image. The maximum-posterior skeletal model thus constitutes the most “rational” interpretation of the image data consistent with the given assumptions. This approach can easily be extended and generalized in a number of ways, allowing a number of traditional problems in perceptual organization to be “probabilized.” I will briefly illustrate several such extensions, including (1) figure/ground and grouping (3) 3D shape and (2) shape similarity.
3:30 – 4:00pm Tea Break 4:00 – 4:45pm Moira Dillon Title: Euclid’s Random Walk: Simulation as a tool for geometric reasoning through development Abstract: Formal geometry lies at the foundation of millennia of human achievement in domains such as mathematics, science, and art. While formal geometry’s propositions rely on abstract entities like dimensionless points and infinitely long lines, the points and lines of our everyday world all have dimension and are finite. How, then, do we get to abstract geometric thought? In this talk, I will provide evidence that evolutionarily ancient and developmentally precocious sensitivities to the geometry of our everyday world form the foundation of, but also limit, our mathematical reasoning. I will also suggest that successful geometric reasoning may emerge through development when children abandon incorrect, axiomatic-based strategies and come to rely on dynamic simulations of physical entities. While problems in geometry may seem answerable by immediate inference or by deductive proof, human geometric reasoning may instead rely on noisy, dynamic simulations.
4:45 – 5:30pm Michael McCloskey Title: Axes and Coordinate Systems in Representing Object Shape and Orientation Abstract: I describe a theoretical perspective in which a) object shape is represented in an object-centered reference frame constructed around orthogonal axes; and b) object orientation is represented by mapping the object-centered frame onto an extrinsic (egocentric or environment-centered) frame. I first show that this perspective is motivated by, and sheds light on, object orientation errors observed in neurotypical children and adults, and in a remarkable case of impaired orientation perception. I then suggest that orientation errors can be used to address questions concerning how object axes are defined on the basis of object geometry—for example, what aspects of object geometry (e.g., elongation, symmetry, structural centrality of parts) play a role in defining an object principal axis?
5:30 – 6:30pm Reception Tuesday, April 16
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 9:45am Peter Olver Title: Symmetry and invariance in cognition — a mathematical perspective” Abstract: Symmetry recognition and appreciation is fundamental in human cognition. (It is worth speculating as to why this may be so, but that is not my intent.) The goal of these two talks is to survey old and new mathematical perspectives on symmetry and invariance. Applications will arise from art, computer vision, geometry, and beyond, and will include recent work on 2D and 3D jigsaw puzzle assembly and an ongoing collaboration with anthropologists on the analysis and refitting of broken bones. Mathematical pre
9:45 – 10:30am Stefano Soatto/Alessandro Achille Title: Information in the Weights and Emergent Properties of Deep Neural Networks Abstract: We introduce the notion of information contained in the weights of a Deep Neural Network and show that it can be used to control and describe the training process of DNNs, and can explain how properties, such as invariance to nuisance variability and disentanglement, emerge naturally in the learned representation. Through its dynamics, stochastic gradient descent (SGD) implicitly regularizes the information in the weights, which can then be used to bound the generalization error through the PAC-Bayes bound. Moreover, the information in the weights can be used to defined both a topology and an asymmetric distance in the space of tasks, which can then be used to predict the training time and the performance on a new task given a solution to a pre-training task.
While this information distance models difficulty of transfer in first approximation, we show the existence of non-trivial irreversible dynamics during the initial transient phase of convergence when the network is acquiring information, which makes the approximation fail. This is closely related to critical learning periods in biology, and suggests that studying the initial convergence transient can yield important insight beyond those that can be gleaned from the well-studied asymptotics.
10:30 – 11:00am Coffee Break 11:00 – 11:45am Jeannette Bohg Title: On perceptual representations and how they interact with actions and physical representations Abstract: I will discuss the hypothesis that perception is active and shaped by our task and our expectations on how the world behaves upon physical interaction. Recent approaches in robotics follow this insight that perception is facilitated by physical interaction with the environment. First, interaction creates a rich sensory signal that would otherwise not be present. And second, knowledge of the regularity in the combined space of sensory data and action parameters facilitate the prediction and interpretation of the signal. In this talk, I will present two examples from our previous work where a predictive task facilitates autonomous robot manipulation by biasing the representation of the raw sensory data. I will present results on visual but also haptic data.
11:45 – 12:30pm Dagmar Sternad Title: Exploiting the Geometry of the Solution Space to Reduce Sensitivity to Neuromotor Noise Abstract: Control and coordination of skilled action is frequently examined in isolation as a neuromuscular problem. However, goal-directed actions are guided by information that creates solutions that are defined as a relation between the actor and the environment. We have developed a task-dynamic approach that starts with a physical model of the task and mathematical analysis of the solution spaces for the task. Based on this analysis we can trace how humans develop strategies that meet complex demands by exploiting the geometry of the solution space. Using three interactive tasks – throwing or bouncing a ball and transporting a “cup of coffee” – we show that humans develop skill by: 1) finding noise-tolerant strategies and channeling noise into task-irrelevant dimensions, 2) exploiting solutions with dynamic stability, and 3) optimizing predictability of the object dynamics. These findings are the basis for developing propositions about the controller: complex actions are generated with dynamic primitives, attractors with few invariant types that overcome substantial delays and noise in the neuro-mechanical system.
12:30 – 2:00pm Lunch 2:00 – 2:45pm Sam Ocko Title: Emergent Elasticity in the Neural Code for Space Abstract: To navigate a novel environment, animals must construct an internal map of space by combining information from two distinct sources: self-motion cues and sensory perception of landmarks. How do known aspects of neural circuit dynamics and synaptic plasticity conspire to construct such internal maps, and how are these maps used to maintain representations of an animal’s position within an environment. We demonstrate analytically how a neural attractor model that combines path integration of self-motion with Hebbian plasticity in synaptic weights from landmark cells can self-organize a consistent internal map of space as the animal explores an environment. Intriguingly, the emergence of this map can be understood as an elastic relaxation process between landmark cells mediated by the attractor network during exploration. Moreover, we verify several experimentally testable predictions of our model, including: (1) systematic deformations of grid cells in irregular environments, (2) path-dependent shifts in grid cells towards the most recently encountered landmark, (3) a dynamical phase transition in which grid cells can break free of landmarks in altered virtual reality environments and (4) the creation of topological defects in grid cells. Taken together, our results conceptually link known biophysical aspects of neurons and synapses to an emergent solution of a fundamental computational problem in navigation, while providing a unified account of disparate experimental observations.
2:45 – 3:30pm Tatyana Sharpee Title: Hyperbolic geometry of the olfactory space Abstract: The sense of smell can be used to avoid poisons or estimate a food’s nutrition content because biochemical reactions create many by-products. Thus, the production of a specific poison by a plant or bacteria will be accompanied by the emission of certain sets of volatile compounds. An animal can therefore judge the presence of poisons in the food by how the food smells. This perspective suggests that the nervous system can classify odors based on statistics of their co-occurrence within natural mixtures rather than from the chemical structures of the ligands themselves. We show that this statistical perspective makes it possible to map odors to points in a hyperbolic space. Hyperbolic coordinates have a long but often underappreciated history of relevance to biology. For example, these coordinates approximate distance between species computed along dendrograms, and more generally between points within hierarchical tree-like networks. We find that both natural odors and human perceptual descriptions of smells can be described using a three-dimensional hyperbolic space. This match in geometries can avoid distortions that would otherwise arise when mapping odors to perception. We identify three axes in the perceptual space that are aligned with odor pleasantness, its molecular boiling point and acidity. Because the perceptual space is curved, one can predict odor pleasantness by knowing the coordinates along the molecular boiling point and acidity axes.
3:30 – 4:00pm Tea Break 4:00 – 4:45pm Ed Connor Title: Representation of solid geometry in object vision cortex Abstract: There is a fundamental tension in object vision between the 2D nature of retinal images and the 3D nature of physical reality. Studies of object processing in the ventral pathway of primate visual cortex have focused mainly on 2D image information. Our latest results, however, show that representations of 3D geometry predominate even in V4, the first object-specific stage in the ventral pathway. The majority of V4 neurons exhibit strong responses and clear selectivity for solid, 3D shape fragments. These responses are remarkably invariant across radically different image cues for 3D shape: shading, specularity, reflection, refraction, and binocular disparity (stereopsis). In V4 and in subsequent stages of the ventral pathway, solid shape geometry is represented in terms of surface fragments and medial axis fragments. Whole objects are represented by ensembles of neurons signaling the shapes and relative positions of their constituent parts. The neural tuning dimensionality of these representations includes principal surface curvatures and their orientations, surface normal orientation, medial axis orientation, axial curvature, axial topology, and position relative to object center of mass. Thus, the ventral pathway implements a rapid transformation of 2D image data into explicit representations 3D geometry, providing cognitive access to the detailed structure of physical reality.
4:45 – 5:30pm L. Mahadevan Title: Simple aspects of geometry and probability in perception Abstract: Inspired by problems associated with noisy perception, I will discuss two questions: (i) how might we test people’s perception of probability in a geometric context ? (ii) can one construct invariant descriptions of 2D images using simple notions of probabilistic geometry? Along the way, I will highlight other questions that the intertwining of geometry and probability raises in a broader perceptual context.
Wednesday, April 17Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 9:45am Gily Ginosar Title: The 3D geometry of grid cells in flying bats Abstract: The medial entorhinal cortex (MEC) contains a variety of spatial cells, including grid cells and border cells. In 2D, grid cells fire when the animal passes near the vertices of a 2D spatial lattice (or grid), which is characterized by circular firing-fields separated by fixed distances, and 60 local angles – resulting in a hexagonal structure. Although many animals navigate in 3D space, no studies have examined the 3D volumetric firing of MEC neurons. Here we addressed this by training Egyptian fruit bats to fly in a large room (5.84.62.7m), while we wirelessly recorded single neurons in MEC. We found 3D border cells and 3D head-direction cells, as well as many neurons with multiple spherical firing-fields. 20% of the multi-field neurons were 3D grid cells, exhibiting a narrow distribution of characteristic distances between neighboring fields – but not a perfect 3D global lattice. The 3D grid cells formed a functional continuum with less structured multi-field neurons. Both 3D grid cells and multi-field cells exhibited an anatomical gradient of spatial scale along the dorso-ventral axis of MEC, with inter-field spacing increasing ventrally – similar to 2D grid cells in rodents. We modeled 3D grid cells and multi-field cells as emerging from pairwise-interactions between fields, using an energy potential that induces repulsion at short distances and attraction at long distances. Our analysis shows that the model explains the data significantly better than a random arrangement of fields. Interestingly, simulating the exact same model in 2D yielded a hexagonal-like structure, akin to grid cells in rodents. Together, the experimental data and preliminary modeling suggest that the global property of grid cells is multiple fields that repel each other with a characteristic distance-scale between adjacent fields – which in 2D yields a global hexagonal lattice while in 3D yields only local structure but no global lattice.
Gily Ginosar 1 , Johnatan Aljadeff 2 , Yoram Burak 3 , Haim Sompolinsky 3 , Liora Las 1 , Nachum Ulanovsky 1
(1) Department of Neurobiology, Weizmann Institute of Science, Rehovot 76100, Israel
(2) Department of Bioengineering, Imperial College London, London, SW7 2AZ, UK
(3) The Edmond and Lily Safra Center for Brain Sciences, and Racah Institute of Physics, The Hebrew
University of Jerusalem, Jerusalem, 91904, Israel
9:45 – 10:30am Sandro Romani Title: Neural networks for 3D rotations Abstract: Studies in rodents, bats, and humans have uncovered the existence of neurons that encode the orientation of the head in 3D. Classical theories of the head-direction (HD) system in 2D rely on continuous attractor neural networks, where neurons with similar heading preference excite each other, while inhibiting other HD neurons. Local excitation and long-range inhibition promote the formation of a stable “bump” of activity that maintains a representation of heading. The extension of HD models to 3D is hindered by complications (i) 3D rotations are non-commutative (ii) the space described by all possible rotations of an object has a non-trivial topology. This topology is not captured by standard parametrizations such as Euler angles (e.g. yaw, pitch, roll). For instance, with these parametrizations, a small change of the orientation of the head could result in a dramatic change of neural representation. We used methods from the representation theory of groups to develop neural network models that exhibit patterns of persistent activity of neurons mapped continuously to the group of 3D rotations. I will further discuss how these networks can (i) integrate vestibular inputs to update the representation of heading, and (ii) be used to interpret “mental rotation” experiments in humans.
This is joint work with Hervé Rouault (CENTURI) and Alon Rubin (Weizmann Institute of Science).
10:30 – 11:00am Coffee Break 11:00 – 11:45am Sam Gershman Title: The hippocampus as a predictive map Abstract: A cognitive map has long been the dominant metaphor for hippocampal function, embracing the idea that place cells encode a geometric representation of space. However, evidence for predictive coding, reward sensitivity and policy dependence in place cells suggests that the representation is not purely spatial. I approach this puzzle from a reinforcement learning perspective: what kind of spatial representation is most useful for maximizing future reward? I show that the answer takes the form of a predictive representation. This representation captures many aspects of place cell responses that fall outside the traditional view of a cognitive map. Furthermore, I argue that entorhinal grid cells encode a low-dimensionality basis set for the predictive representation, useful for suppressing noise in predictions and extracting multiscale structure for hierarchical planning.
11:45 – 12:30pm Lucia Jacobs Title: The adaptive geometry of a chemosensor: the origin and function of the vertebrate nose Abstract: A defining feature of a living organism, from prokaryotes to plants and animals, is the ability to orient to chemicals. The distribution of chemicals, whether in water, air or on land, is used by organisms to locate and exploit spatially distributed resources, such as nutrients and reproductive partners. In animals, the evolution of a nervous system coincided with the evolution of paired chemosensors. In contemporary insects, crustaceans, mollusks and vertebrates, including humans, paired chemosensors confer a stereo olfaction advantage on the animal’s ability to orient in space. Among vertebrates, however, this function faced a new challenge with the invasion of land. Locomotion on land created a new conflict between respiration and spatial olfaction in vertebrates. The need to resolve this conflict could explain the current diversity of vertebrate nose geometries, which could have arisen due to species differences in the demand for stereo olfaction. I will examine this idea in more detail in the order Primates, focusing on Old World primates, in particular, the evolution of an external nose in the genus Homo.
12:30 – 1:30pm Lunch 1:30 – 2:15pm Talia Konkle Title: The shape of things and the organization of object-selective cortex Abstract: When we look at the world, we effortlessly recognize the objects around us and can bring to mind a wealth of knowledge about their properties. In part 1, I’ll present evidence that neural responses to objects are organized by high-level dimensions of animacy and size, but with underlying neural tuning to mid-level shape features. In part 2, I’ll present evidence that representational structure across much of the visual system has the requisite structure to predict visual behavior. Together, these projects suggest that there is a ubiquitous “shape space” mapped across all of occipitotemporal cortex that underlies our visual object processing capacities. Based on these findings, I’ll speculate that the large-scale spatial topography of these neural responses is critical for pulling explicit content out of a representational geometry.
2:15 – 3:00pm Vijay Balasubramanian Title: Becoming what you smell: adaptive sensing in the olfactory system Abstract: I will argue that the circuit architecture of the early olfactory system provides an adaptive, efficient mechanism for compressing the vast space of odor mixtures into the responses of a small number of sensors. In this view, the olfactory sensory repertoire employs a disordered code to compress a high dimensional olfactory space into a low dimensional receptor response space while preserving distance relations between odors. The resulting representation is dynamically adapted to efficiently encode the changing environment of volatile molecules. I will show that this adaptive combinatorial code can be efficiently decoded by systematically eliminating candidate odorants that bind to silent receptors. The resulting algorithm for “estimation by elimination” can be implemented by a neural network that is remarkably similar to the early olfactory pathway in the brain. The theory predicts a relation between the diversity of olfactory receptors and the sparsity of their responses that matches animals from flies to humans. It also predicts specific deficits in olfactory behavior that should result from optogenetic manipulation of the olfactory bulb.
3:00 – 3:45pm Ila Feite Title: Invariance, stability, geometry, and flexibility in spatial navigation circuits Abstract: I will describe how the geometric invariances or symmetries of the external world are reflected in the symmetries of neural circuits that represent it, using the example of the brain’s networks for spatial navigation. I will discuss how these symmetries enable spatial memory, evidence integration, and robust representation. At the same time, I will discuss how these seemingly rigid circuits with their inscribed symmetries can be harnessed to represent a range of spatial and non-spatial cognitive variables with high flexibility.
3:45 – 4:00pm L Mahadevan – summary Topology and Dynamics in Quantum Matter Workshop
On September 10-11, 2019, the CMSA will be hosting a second workshop on Topological Aspects of Condensed Matter.
New ideas rooted in topology have recently had a major impact on condensed matter physics, and have led to new connections with high energy physics, mathematics and quantum information theory. The aim of this program will be to deepen these connections and spark new progress by fostering discussion and new collaborations within and across disciplines.
Topics include i) the classification of topological states ii) topological orders in two and three dimensions including quantum spin liquids, quantum Hall states and fracton phases and iii) interplay of symmetry and topology in quantum many body systems, including symmetry protected topological phases, symmetry fractionalization and anomalies iv) topological phenomena in quantum systems driven far from equlibrium v) quantum field theory approaches to topological matter.
This workshop is part of the CMSA’s program on Program on Topological Aspects of Condensed Matter, and is the second of two workshops, in addition to a visitor program and seminars.
The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Click here for a list of restaurants in the area.
Organizers: Michael Hermele (CU Boulder) and Ashvin Vishwanath (Harvard)
Partial list of speakers:
- Nima Arkani-Hamed, IAS
- Jennifer Cano, Stony Brook
- Meng Cheng, Yale
- Lukasz Fidkowski, UW Seattle
- Daniel Freed, Texas
- Jeongwan Haah, Microsoft Research
- Anton Kapustin, Caltech
- Zohar Komargodski, SCGP/Stony Brook
- John McGreevy, UC San Diego
- Prineha Narang, Harvard
- Ying Ran, Boston College
- Shinsei Ryu, Chicago
- Cumrun Vafa, Harvard
- Chong Wang, Perimeter
- Zhenghan Wang, Microsoft Station Q
Videos of the lectures can be found in the Youtube playlist below. Links to talks are also available on the schedule below.
Kickoff Workshop on Topology and Quantum Phases of Matter
On August 27-28, 2018, the CMSA will be hosting a Kickoff workshop on Topology and Quantum Phases of Matter. New ideas rooted in topology have recently had a big impact on condensed matter physics, and have highlighted new connections with high energy physics, mathematics and quantum information theory. Additionally, these ideas have found applications in the design of photonic systems and of materials with novel mechanical properties. The aim of this program will be to deepen these connections by fostering discussion and seeding new collaborations within and across disciplines.
This workshop is a part of the CMSA’s program on Program on Topological Aspects of Condensed Matter, and will be the first of two workshops, in addition to a visitor program and seminars.
The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Please register here
Speakers:
- Zhen Bi, MIT
- Meng Cheng, Yale
- Dima Feldman, Brown
- Dominic Else, UCSB
- Liang Fu, MIT
- Fabian Grusdt, Harvard
- Ying Fei Gu, Harvard
- Bert Halperin, Harvard
- Anton Kapustin, Caltech
- Patrick Lee, MIT
- L. Mahadevan, Harvard
- Brad Marston, Brown
- Max Metlitski, MIT
- Emil V. Prodan, Yeshiva
- Achim Rosch, University of Cologne
- Mathias Scheurer, Harvard
- Marin Soljacic, MIT
- X. G. Wen, MIT
- Cenke Xu, UCSB
- Frank Zhang, Cornell
Static vacuum extensions of Bartnik boundary data near flat domains
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on a joint work with Lan-Hsuan Huang.
Geometry and Physics Seminar
During the summer of 2020, the CMSA will be hosting a new Geometry Seminar. Talks will be scheduled on Mondays at 9:30pm or Tuesdays at 9:30am, depending on the location of the speaker. This seminar is organized by Tsung-Ju Lee, Yoosik Kim, and Du Pei.
To learn how to attend this seminar, please contact Tsung-Ju Lee (tjlee@cmsa.fas.harvard.edu).
Date Speaker Title/Abstract 6/2/2020
9:30am ETSiu-Cheong Lau
Boston UniversityThis meeting will be taking place virtually on Zoom. Speaker: Equivariant Floer theory and SYZ mirror symmetry
Abstract: In this talk, we will first review a symplectic realization of the SYZ program and some of its applications. Then I will explain some recent works on equivariant Lagrangian Floer theory and disc potentials of immersed SYZ fibers. They are joint works with Hansol Hong, Yoosik Kim and Xiao Zheng.
6/8/2020
9:30pm ETYoungjin Bae (KIAS) This meeting will be taking place virtually on Zoom. Title: Legendrian graphs and their invariants
Abstract: Legendrian graphs naturally appear in the study of Weinstein manifolds with a singular Lagrangian skeleton, and a tangle decomposition of Legendrian submanifolds. I will introduce various invariant of Legendrian graphs including DGA type, polynomial type, sheaf theoretic one, and their relationship. This is joint work with Byunghee An, and partially with Tamas Kalman and Tao Su.
6/16/2020
9:30am ETMichael McBreen (CMSA) This meeting will be taking place virtually on Zoom. Title: Loops in hypertoric varieties and symplectic duality
Abstract: Hypertoric varieties are algebraic symplectic varieties associated to graphs, or more generally certain hyperplane arrangements. They make many appearances in modern geometric representation theory. I will discuss certain infinite dimensional or infinite type generalizations of hypertoric varieties which occur in the study of enumerative invariants, focusing on some elementary examples. Joint work with Artan Sheshmani and Shing-Tung Yau.
6/22/2020
9:30pm ETZiming Ma (CUHK) This meeting will be taking place virtually on Zoom. Title: The geometry of Maurer–Cartan equation near degenerate Calabi–Yau varieties
Abstract: In this talk, we construct a \(dgBV algebra PV*(X)\) associated to a possibly degenerate Calabi–Yau variety X equipped with local thickening data. This gives a version of the Kodaira–Spencer dgLa which is applicable to degenerated spaces including both log smooth or maximally degenerated Calabi–Yau. We use this to prove an unobstructedness result about the smoothing of degenerated Log Calabi–Yau varieties X satisfying Hodge–deRham degeneracy property for cohomology of X, in the spirit of Kontsevich–Katzarkov–Pantev. This is a joint work with Kwokwai Chan and Naichung Conan Leung.
6/30/2020
9:30pm ETSunghyuk Park (Caltech) This meeting will be taking place virtually on Zoom. Title: 3-manifolds, q-series, and topological strings
Abstract: \(\hat{Z}\) is an invariant of 3-manifolds valued in q-series (i.e. power series in q with integer coefficients), which has interesting modular properties. While originally from physics, this invariant has been mathematically constructed for a big class of 3-manifolds, and conjecturally it can be extended to all 3-manifolds. In this talk, I will give a gentle introduction to \(\hat{Z}\) and what is known about it, as well as highlighting some recent developments, including the use of R-matrix, generalization to higher rank, large N-limit and interpretation as open topological string partition functions.
7/7/2020
9:30am ETJeremy Lane (McMaster University) This meeting will be taking place virtually on Zoom. Title: Collective integrable systems and global action-angle coordinates
Abstract: A “collective integrable system” on a symplectic manifold is a commutative integrable system constructed from a Hamiltonian action of a non-commutative Lie group. Motivated by the example of Gelfand-Zeitlin systems, we give a construction of collective integrable systems that generate a Hamiltonian torus action on a dense subset of any Hamiltonian K-manifold, where K is any compact connected Lie group. In the case where the Hamiltonian K-manifold is compact and multiplicity free, the resulting Hamiltonian torus action is completely integrable and yields global action angle coordinates. Moreover, the image of the moment map is a (non-simple) convex polytope.
7/13/2020
9:30pm ETPo-Shen Hsin (Caltech) This meeting will be taking place virtually on Zoom. Title: Berry phase in quantum field theory
Abstract: We will discuss Berry phase in family of quantum field theories using effective field theory. The family is labelled by parameters which we promote to be spacetime-dependent sigma model background fields. The Berry phase is equivalent to Wess-Zumino-Witten action for the sigma model. We use Berry phase to study diabolic points in the phase diagram of the quantum field theory and discuss applications to deconfined quantum criticality and new tests for boson/fermion dualities in \((2+1)d\).
7/20/2020
9:30pm ETSangwook Lee (KIAS) This meeting will be taking place virtually on Zoom. Title: A geometric construction of orbifold Jacobian algebras
Abstract: We review the definition of a twisted Jacobian algebra of a Landau-Ginzburg orbifold due to Kaufmann et al. Then we construct an A-infinity algebra of a weakly unobstructed Lagrangian submanifold in a symplectic orbifold. We work on an elliptic orbifold sphere and see that above two algebras are isomorphic, and furthermore their structure constants are related by a modular identity which was used to prove the mirror symmetry of closed string pairings. This is a joint work with Cheol-Hyun Cho.
7/27/2020 9:30pm ET Mao Sheng (USTC) This meeting will be taking place virtually on Zoom. Title: Parabolic de Rham bundles: motivic vs periodic
Abstract: Let \($C$\) be a complex smooth projective curve. We consider the set of parabolic de Rham bundles over \($C$\) (with rational weights in parabolic structure). Many examples arise from geometry: let \($f: X\to U$\) be a smooth projective morphism over some nonempty Zariski open subset \($U\subset C$\). Then the Deligne–Iyer–Simpson canonical parabolic extension of the Gauss–Manin systems associated to \($f$\) provides such examples. We call a parabolic de Rham bundle \emph{motivic}, if it appears as a direct summand of such an example of geometric origin. It is a deep question in the theory of linear ordinary differential equations and in Hodge theory, to get a characterization of motivic parabolic de Rham bundles. In this talk, I introduce another subcategory of parabolic de Rham bundles, the so-called \emph{periodic} parabolic de Rham bundles. It is based on the work of Lan–Sheng–Zuo on Higgs-de Rham flows, with aim towards linking the Simpson correspondence over the field of complex numbers and the Ogus–Vologodsky correspondence over the finite fields. We show that motivic parabolic de Rham bundles are periodic, and conjecture that they are all periodic parabolic de Rham bundles. The conjecture for rank one case follows from the solution of Grothendieck–Katz p-curvature conjecture, and for some versions of rigid cases should follow from Katz’s work on rigid local systems. The conjecture implies that in a spread-out of any complex elliptic curve, there will be infinitely many supersingular primes, a result of N. Elkies for rational elliptic curves. Among other implications of the conjecture, we would like to single out the conjectural arithmetic Simpson correspondence, which asserts that the grading functor is an equivalence of categories from the category of periodic parabolic de Rham bundles to the category of periodic parabolic Higgs bundles. This is a joint work in progress with R. Krishnamoorthy.
8/4/2020
9:30am EtPavel Safronov (University of Zurich) This meeting will be taking place virtually on Zoom. Title: Kapustin–Witten TFT on 3-manifolds and skein modules
Abstract: Kapustin and Witten have studied a one-parameter family of topological twists of \(4d N=4\) super Yang–Mills. They have shown that the categories of boundary conditions on a surface are exactly the categories participating in the geometric Langlands program of Beilinson and Drinfeld. Moreover, S-duality is manifested as a quantum geometric Langlands duality after the topological twist. In this talk I will describe some mathematical formalizations of Hilbert spaces of states on a 3-manifold. I will outline an equivalence between two such possible formalizations: complexified Floer homology of Abouzaid–Manolescu and skein modules. This is a report on work in progress joint with Sam Gunningham.8/11/2020
9:30amXujia Chen (Stonybrook) This meeting will be taking place virtually on Zoom. Title: Lifting cobordisms and Kontsevich-type recursions for counts of real curves
Abstract: Kontsevich’s recursion, proved in the early 90s, is a recursion formula for the counts of rational holomorphic curves in complex manifolds. For complex fourfolds and sixfolds with a real structure (i.e. a conjugation), signed invariant counts of real rational holomorphic curves were defined by Welschinger in 2003. Solomon interpreted Welschinger’s invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline of a potential approach of proving them. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon’s recursions by re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne-Mumford moduli spaces of stable real curves (which is different from Solomon’s approach).
8/18/2020
9:30am ETDongmin Gang (Asia Pacific Center for Theoretical Physics) This meeting will be taking place virtually on Zoom. Title: M-theoretic genesis of topological phases
Abstract: I will talk about a novel way of constructing \((2+1)d\) topological phases using M-theory. They emerge as macroscopic world-volume theories of M5-branes wrapped on non-hyperbolic 3-manifolds. After explaining the algorithm of extracting modular structures of the topological phase from topological data of the 3-manifold, I will discuss the possibility of full classification of topological orders via the geometrical construction.
8/25/2020
9:30pm ETMykola Dedushenko (Caltech) This meeting will be taking place virtually on Zoom. Title: Algebras and traces at the boundary of \(4d N=4\) SYM
Abstract: I will describe how the structure of supersymmetric boundary correlators in \(4d N=4\) SYM can be encoded in a class of associative algebras equipped with twisted traces. In the case of interfaces, this yields a new connection to integrability.
C-P-T Fractionalization, and Quantum Criticality Beyond the Standard Model
Member Seminar
Speaker: Juven Wang
Title: C-P-T Fractionalization, and Quantum Criticality Beyond the Standard Model
Abstract: Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as a Z2-involution on the spacetime coordinates; but together with a charge conjugation C and the fermion parity (−1)^F, these symmetries can be further fractionalized forming nonabelian C-P-R-T-(−1)^F group structures, in various examples such as relativistic Lorentz invariant Dirac spinor quantum field theories (QFT), or nonrelativistic quantum many-body systems (involving Majorana zero modes). This result answers Prof. Shing-Tung Yau’s question on “Can C-P-T symmetries be fractionalized more than involutions?” based on arxiv:2109.15320.
In the second part of my talk, I will sketch to explain how can we modify the so(10) Grand Unified Theory (GUT) by adding a new topological term such that two GUTs of Georgi-Glashow and Pati-Salam can smoother into each other in a quantum phase transition, where the Standard Model and new dark sector physics can occur naturally near the critical region. The new modified so(10) GUT requires a double Spin structure that we name DSpin. This phenomenon is inspired by the “deconfined quantum criticality” in condensed matter. Based on arxiv:2106.16248.
Peeling properties of the spinor fields and the solutions to nonlinear Dirac equations
Abstract: The Dirac equation is a relativistic equation that describes the spin-1/2 particles. We talk about Dirac equations in Minkowski spacetime. In a geometric viewpoint, we can see that the spinor fields satisfying the Dirac equations enjoy the so-called peeling properties. It means the null components of the solution will decay at different rates along the null hypersurface. Based on this decay mechanism, we can obtain a fresh insight to the spinor null forms which is used to prove a small data global existence result especially for some quadratic Dirac models.
Wall-crossing from Higgs bundles to vortices
Speaker: Du Pei
Title: Wall-crossing from Higgs bundles to vortices
Abstract: Quantum field theories can often be used to uncover hidden algebraic structures in geometry and hidden geometric structures in algebra. In this talk, I will demonstrate how such “wall-crossing” can relate the moduli space of Higgs bundles with the moduli space of vortices.
The Large D Limit of Einstein’s Equations
Abstract: Taking the large dimension limit of Einstein’s equations is a useful strategy for solving and understanding the dynamics that these equations encode. I will introduce the underlying ideas and the progress that has resulted in recent years from this line of research. Most of the discussion will be classical in nature and will concern situations where there is a black hole horizon. A main highlight of this approach is the formulation of effective membrane theories of black hole dynamics. These have made possible to efficiently study, with relatively simple techniques, some of the thorniest problems in black hole physics, such as the non-linear evolution of the instabilities of black strings and black branes, and the collisions and mergers of higher-dimensional black holes. Open directions and opportunities will also be discussed. To get a flavor of what this is about, you may read the first few pages of the review (with C.P. Herzog) e-Print: 2003.11394.
Instability of naked singularities in general relativity
Member Seminar
Speaker: Jue Liu
Title: Instability of naked singularities in general relativity
Abstract: One of the fundamental problems in mathematical relativity is the weak cosmic censorship conjecture, proposed by Penrose, which roughly states that for generic physical spacetime, the singularities (if existed) must be hidden behind the black holes. Unfortunately, the singularities visible to faraway observers, which are called by naked singularities, indeed exist. The first example constructed by Christodoulou in 1994 is a family of self-similar spherically symmetric spacetime, in which the naked singularity forms due to a self-gravitating scalar field. Therefore the suitable censorship conjecture should be reduced to prove the instability of the naked singularities. In 1999 Christodoulou succeeded to prove the weak cosmic censorship conjecture in spherically symmetric cases, and recently the co-author and I found that the corresponding results have a big probability to be extended to spacetime without symmetries. In this talk I will discuss how to prove the instability of naked singularities using the energy method, and it is this wild method that helps us to extend some results to the asymmetric cases.
The classical interior of charged black holes with AdS asymptotics
Abstract: The gravitational dual to the grand canonical ensemble of a large N holographic theory is a charged black hole. These spacetimes can have Cauchy horizons that render the classical gravitational dynamics of the black hole interior incomplete. We show that a (spatially uniform) deformation of the CFT by a neutral scalar operator generically leads to a black hole with no inner horizon. There is instead a spacelike Kasner singularity in the interior. For relevant deformations, Cauchy horizons never form. We then consider charged scalars, which are known to condense at low temperatures, thus providing a holographic realization of superconductivity. We look inside the horizon of these holographic superconductors and find intricate dynamical behavior. The spacetime ends at a spacelike Kasner singularity, and there is no Cauchy horizon. Before reaching the singularity, there are several intermediate regimes which we study both analytically and numerically. These include strong Josephson oscillations in the condensate and possible `Kasner inversions’ in which after many e-folds of expansion, the Einstein-Rosen bridge contracts towards the singularity. Due to the Josephson oscillations, the number of Kasner inversions depends very sensitively on temperature, and diverges at a discrete set of temperatures that accumulate at the critical temperature. Near this discrete set of temperatures, the final Kasner exponent exhibits fractal-like behavior.
The complex Monge-Ampere equation in K\”ahler geometry
Speaker: Freid Tong
Title: The complex Monge-Ampere equation in Kahler geometry
Abstract: The complex Monge-Ampere equations occupies an central role in K\”ahler geometry, beginning with Yau’s famous solutions of the Calabi conjecture. Later developments has led to many interesting geometric applications and opening of new fields. In this talk, I will introduce the complex Monge-Ampere equation and discuss the interplay between their analysis and geometry, with a particular focus on the a priori C^0 estimates and their various applications. In the end, I will also try to discuss some recent work with B. Guo and D.H. Phong on a new approach for proving sharp C^0 estimates for complex Monge-Ampere equations, this new approach avoids the machinery of pluripotential theory that was previously necessary and has the advantage of generalizing to a large class of fully nonlinear equations.
Knowledge Graph Embeddings and Inference
Member Seminar
Speaker: Michael Douglas
Title: Knowledge Graph Embeddings and Inference
Abstract: A knowledge graph (KG) is a data structure which represents entities and relations as the vertices and edges of a directed graph. Two examples are Wikidata for general knowledge and SemMedDB for biomedical data.
A popular KG representation method is graph embedding, which facilitates question answering, inferring missing edges, and logical reasoning tasks. In this talk we introduce the topic and explain relevant mathematical results on graph embedding. We then analyze KG inference into several mechanisms: motif learning, network learning, and unstructured statistical inference, and describe experiments to measure the contributions of each mechanism.Joint work with M. Simkin, O. Ben-Eliezer, T. Wu, S. P. Chin, T. V. Dang and A. Wood.
Derived categories of nodal quintic del Pezzo threefolds
Abstract: Conifold transitions are important algebraic geometric constructions that have been of special interests in mirror symmetry, transforming Calabi-Yau 3-folds between A- and B-models. In this talk, I will discuss the change of the quintic del Pezzo 3-fold (Fano 3-fold of index 2 and degree 5) under the conifold transition at the level of the bounded derived category of coherent sheaves. The nodal quintic del Pezzo 3-fold X has at most 3 nodes. I will construct a semiorthogonal decomposition for D^b(X) and in the case of 1-nodal X, detail the change of derived categories from its smoothing to its small resolution.
Causality Comparison and Postive Mass
Abstract: Penrose et al. investigated the physical incoherence of the space-time with negative mass via the bending of light. Precise estimates of the time-delay of null geodesics were needed and played a pivotal role in their proof. In this paper, we construct an intermediate diagonal metric and reduce this problem to a causality comparison in the compactified space-time regarding time-like connectedness near conformal infinities. This different approach allows us to avoid encountering the difficulties and subtle issues that Penrose et al. met. It provides a new, substantially simple, and physically natural non-partial differential equation viewpoint to understand the positive mass theorem. This elementary argument modestly applies to asymptotically flat solutions that are vacuum and stationary near infinity
Growth and zero sets of eigenfunctions and of solutions to elliptic partial differential equations
From February 25 to March 1, the CMSA will be hosting a workshop on Growth and zero sets of eigenfunctions and of solutions to elliptic partial differential equations.
Key participants of this workshop include David Jerison (MIT), Alexander Logunov (IAS), and Eugenia Malinnikova (IAS). This workshop will have morning sessions on Monday-Friday of this week from 9:30-11:30am, and afternoon sessions on Monday, Tuesday, and Thursday from 3:00-5:00pm.
The sessions will be held in \(G02\) (downstairs) at 20 Garden, except for Tuesday afternoon, when the talk will be in \(G10\).9/24/2021 General Relativity Seminar
Title: On the Observable Shape of Black Hole Photon Rings
Abstract: The photon ring is a narrow ring-shaped feature, predicted by General Relativity but not yet observed, that appears on images of sources near a black hole. It is caused by extreme bending of light within a few Schwarzschild radii of the event horizon and provides a direct probe of the unstable bound photon orbits of the Kerr geometry. I will argue that the precise shape of the observable photon ring is remarkably insensitive to the astronomical source profile and can therefore be used as a stringent test of strong-field General Relativity. In practice, near-term interferometric observations may be limited to the visibility amplitude alone, which contains incomplete shape information: for convex curves, the amplitude only encodes the set of projected diameters (or “widths”) of the shape. I will describe the freedom in reconstructing a convex curve from its widths, giving insight into the photon ring shape information probed by technically plausible future astronomical measurements.
9/17/2021 General Relativity Seminar
Title: Stable Big Bang formation for the Einstein equations
Abstract: I will discuss recent work concerning stability of cosmological singularities described by the generalized Kasner solutions. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be stable under perturbations of the Kasner initial data, as long as the Kasner exponents are “sub-critical”. We prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the full regime where stable monotonic-type curvature blowup is expected. We treat the 3+1-dimensional Einstein-scalar field system and the D+1-dimensional Einstein-vacuum equations for D≥10. This is joint work with Speck and Fournodavlos.
9/10/2021 General Relativity Seminar
Title: Asymptotic localization, massive fields, and gravitational singularities
Abstract: I will review three recent developments on Einstein’s field equations under low decay or low regularity conditions. First, the Seed-to-Solution Method for Einstein’s constraint equations, introduced in collaboration with T.-C. Nguyen generates asymptotically Euclidean manifolds with the weakest or strongest possible decay (infinite ADM mass, Schwarzschild decay, etc.). The ‘asymptotic localization problem’ is also proposed an alternative to the ‘optimal localization problem’ by Carlotto and Schoen. We solve this new problem at the harmonic level of decay. Second, the Euclidian-Hyperboloidal Foliation Method, introduced in collaboration with Yue Ma, applies to nonlinear wave systems which need not be asymptotically invariant under Minkowski’s scaling field and to solutions with low decay in space. We established the global nonlinear stability of self-gravitating massive matter field in the regime near Minkowski spacetime. Third, in collaboration with Bruno Le Floch and Gabriele Veneziano, I studied spacetimes in the vicinity of singularity hypersurfaces and constructed bouncing cosmological spacetimes of big bang-big crunch type. The notion of singularity scattering map provides a flexible tool for formulating junction conditions and, by analyzing Einstein’s constraint equations, we established a surprising classification of all gravitational bouncing laws. Blog: philippelefloch.org
Threshold phenomena in random graphs and hypergraphs
Member Seminar
Speaker: Michael Simkin
Title: Threshold phenomena in random graphs and hypergraphs
Abstract: In 1959 Paul Erdos and Alfred Renyi introduced a model of random graphs that is the cornerstone of modern probabilistic combinatorics. Now known as the “Erdos-Renyi” model of random graphs it has far-reaching applications in combinatorics, computer science, and other fields.
The model is defined as follows: Given a natural number $n$ and a parameter $p \in [0,1]$, let $G(n;p)$ be the distribution on graphs with $n$ vertices in which each of the $\binom{n}{2}$ possible edges is present with probability $p$, independent of all others. Despite their apparent simplicity, the study of Erdos-Renyi random graphs has revealed many deep and non-trivial phenomena.
A central feature is the appearance of threshold phenomena: For all monotone properties (e.g., connectivity and Hamiltonicity) there is a critical probability $p_c$ such that if $p >> p_c$ then $G(n;p)$ possesses the property with high probability (i.e., with probability tending to 1 as $n \to \infty$) whereas if $p << p_c$ then with high probability $G(n;p)$ does not possess the property. In this talk we will focus on basic properties such as connectivity and containing a perfect matching. We will see an intriguing connection between these global properties and the local property of having no isolated vertices. We will then generalize the Erdos-Renyi model to higher dimensions where many open problems remain.
Stability and convergence issues in mathematical cosmology
Member Seminar
Speaker: Puskar Mondal
Title: Stability and convergence issues in mathematical cosmology
Abstract: The standard model of cosmology is built on the fact that while viewed on a sufficiently coarse-grained scale the portion of our universe that is accessible to observation appears to be spatially homogeneous and isotropic. Therefore this observed `homogeneity and isotropy’ of our universe is not known to be dynamically derived. In this talk, I will present an interesting dynamical mechanism within the framework of the Einstein flow (including physically reasonable matter sources) which suggests that many closed manifolds that do not support homogeneous and isotropic metrics at all will nevertheless evolve to be asymptotically compatible with the observed approximate homogeneity and isotropy of the physical universe. This asymptotic spacetime is naturally isometric to the standard FLRW models of cosmology. In order to conclude to what extent the asymptotic state is physically realized, one needs to study its stability properties. Therefore, I will briefly discuss the stability issue and its consequences (e.g., structure formation, etc).
Geometry, Entanglement and Quasi Local Data
Member Seminar
Speaker: Itamar Shamir
Title: Geometry, Entanglement and Quasi Local Data
Abstract: I will review some general ideas about gravity as motivation for an approach based on quasi local quantities.
Surfacehedra and the Binary Positive Geometry of Particle and “String” Amplitudes
Speaker: Nima Arkani-Hamed, IAS
Title: Surfacehedra and the Binary Positive Geometry of Particle and “String” Amplitudes
Geometric Analysis Seminar, Tuesdays at 9:50am
The seminar on geometric analysis will be held on Tuesdays from 9:50am to 10:50am with time for questions afterwards in CMSA Building, 20 Garden Street, Room G10. The tentative schedule can be found below. Titles will be added as they are provided.
Evolution Equations Seminar, Thursdays at 9:50am
The seminar for evolution equations, hyperbolic equations, and fluid dynamics will be held on Thursdays from 9:50am to 10:50am with time for questions afterwards in CMSA Building, 20 Garden Street, Room G10. The tentative schedule of speakers is below. Titles for the talks will be added as they are received.
Social Science Applications Forum
During the Summer of 2020, the CMSA will be hosting a periodic Social Science Applications Seminar.
The list of speakers is below and will be updated as details are confirmed.
For a list of past Social Science Applications talks, please click here.
Date Speaker Title/Abstract 7/13/2020 10:00-11:00am ET Ludovic Tangpi (Princeton) Please note, this seminar will take place online using Zoom. Title: Convergence of Large Population Games to Mean Field Games with Interaction Through the Controls
Abstract: This work considers stochastic differential games with a large number of players, whose costs and dynamics interact through the empirical distribution of both their states and their controls. We develop a framework to prove convergence of finite-player games to the asymptotic mean field game. Our approach is based on the concept of propagation of chaos for forward and backward weakly interacting particles which we investigate by fully probabilistic methods, and which appear to be of independent interest. These propagation of chaos arguments allow to derive moment and concentration bounds for the convergence of both Nash equilibria and social optima in non-cooperative and cooperative games, respectively. Incidentally, we also obtain convergence of a system of second order parabolic partial differential equations on finite dimensional spaces to a second order parabolic partial differential equation on the Wasserstein space.
For security reasons, you will have to show your full name to join the meeting.7/27/2020
10:00pmMichael Ewens (Caltech) Please note, this seminar will take place online using Zoom. Title: Measuring Intangible Capital with Market Prices
Abstract: Despite the importance of intangibles in today’s economy, current standards prohibit the capitalization of internally created knowledge and organizational capital, resulting in a downward bias of reported assets. As a result, researchers estimate this value by capitalizing prior flows of R&D and SG&A. In doing so, a set of capitalization parameters, i.e. the R&D depreciation rate and the fraction of SG&A that represents a long-lived asset, must be assumed. Parameters now in use are derived from models with strong assumptions or are ad hoc. We develop a capitalization model that motivates the use of market prices of intangibles to estimate these parameters. Two settings provide intangible asset values: (1) publicly traded equity prices and (2) acquisition prices. We use these parameters to estimate intangible capital stocks and subject them to an extensive set of diagnostic analyses that compare them with stocks estimated using existing parameters. Intangible stocks developed from exit price parameters outperform both stocks developed by publicly traded parameters and those stocks developed with existing estimates. (Joint work with Ryan Peters and Sean Wang.)
Small Cosmological Constants in String Theory
Abstract: We construct supersymmetric AdS4 vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by Kachru, Kallosh, Linde, and Trivedi. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the α 0 expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude < 10^{-123} in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol.
Mathematical supergravity and its applications to differential geometry
Virtual and in 20 Garden Street, Room G10Speaker: Carlos S. Shahbazi (Hamburg University)
Title: Mathematical supergravity and its applications to differential geometry
Abstract: I will discuss the recent developments in the mathematical theory of supergravity that lay the mathematical foundations of the universal bosonic sector of four-dimensional ungauged supergravity and its Killing spinor equations in a differential-geometric framework. I will provide the necessary context and background. explaining the results pedagogically from scratch and highlighting several open mathematical problems which arise in the mathematical theory of supergravity, as well as some of its potential mathematical applications. Work in collaboration with Vicente Cortés and Calin Lazaroiu.
Frontiers in Applied Mathematics and Computation
Together with the School of Engineering and Applied Sciences, the CMSA will be hosting a lecture series on the Frontiers in Applied Mathematics and Computation. Talks in this series will aim to highlight current research trends at the interface of applied math and computation and will explore the application of these trends to challenging scientific, engineering, and societal problems.
Lectures will take place on March 25, April 1, and April 29, 2021.
Speakers:
- George Biros (U.T. Austin)
- Laura Grigori (INRIA Paris)
- Samory K. Kpotufe (Columbia)
- Jonas Martin Peters (University of Copenhagen)
- Joseph M. Teran (UCLA)
The schedule below will be updated as talks are confirmed.
Date/Time Speaker Title/Abstract 3/25/2021
10:00 – 11:00am ETJoseph M. Teran Title: Affine-Particle-In-Cell with Conservative Resampling and Implicit Time Stepping for Surface Tension Forces Abstract: The Particle-In-Cell (PIC) method of Harlow is one of the first and most widely used numerical methods for Partial Differential Equations (PDE) in computational physics. Its relative efficiency, versatility and intuitive implementation have made it particularly popular in computational incompressible flow, plasma physics and large strain elastoplasticity. PIC is characterized by its dual particle/grid (Lagrangian/Eulerian) representation of material where particles are generally used to track material transport in a Lagrangian way and a structured Eulerian grid is used to discretize remaining spatial derivatives in the PDE. I will discuss the importance of conserving linear and angular momentum when switching between these two representations and the recent Affine-Particle-In-Cell (APIC) extension to PIC designed for this conservation. I will also discuss a recent APIC technique for discretizing surface tension forces and their linearizations needed for implicit time stepping. This technique is characterized by a novel surface resampling strategy and I will discuss a generalization of the APIC conservation to this setting.
4/1/2021
9:00 – 10:00am ETGeorge Biros Title: Inverse biophysical modeling and its application to neurooncology Abstract: A predictive, patient-specific, biophysical model of tumor growth would be an invaluable tool for causally connecting diagnostics with predictive medicine. For example, it could be used for tumor grading, characterization of the tumor microenvironment, recurrence prediction, and treatment planning, e.g., chemotherapy protocol or enrollment eligibility for clinical trials. Such a model also would provide an important bridge between molecular drivers of tumor growth and imaging-based phenotypic signatures, and thus, help identify and quantify mechanism-based associations between these two. Unfortunately, such a predictive biophysical model does not exist. Existing models undergoing clinical evaluation are too simple–they do not even capture the MRI phenotype. Although many highly complex models have been proposed, the major hurdle in deploying them clinically is their calibration and validation.
In this talk, I will discuss the challenges related to the calibration and validation of biophysical models, and in particular the mathematical structure of the underlying inverse problems. I will also present a new algorithm that localizes the tumor origin within a few millimeters.
4/1/2021
10:00 – 11:00am ETSamory K. Kpotufe Title: From Theory to Clustering Abstract: Clustering is a basic problem in data analysis, consisting of partitioning data into meaningful groups called clusters. Practical clustering procedures tend to meet two criteria: flexibility in the shapes and number of clusters estimated, and efficient processing. While many practical procedures might meet either of these criteria in different applications, general guarantees often only hold for theoretical procedures that are hard if not impossible to implement. A main aim is to address this gap.
We will discuss two recent approaches that compete with state-of-the-art procedures, while at the same time relying on rigorous analysis of clustering. The first approach fits within the framework of density-based clustering, a family of flexible clustering approaches. It builds primarily on theoretical insights on nearest-neighbor graphs, a geometric data structure shown to encode local information on the data density. The second approach speeds up kernel k-means, a popular Hilbert space embedding and clustering method. This more efficient approach relies on a new interpretation – and alternative use – of kernel-sketching as a geometry-preserving random projection in Hilbert space.
Finally, we will present recent experimental results combining the benefits of both approaches in the IoT application domain.
The talk is based on various works with collaborators Sanjoy Dasgupta, Kamalika Chaudhuri, Ulrike von Luxburg, Heinrich Jiang, Bharath Sriperumbudur, Kun Yang, and Nick Feamster.4/29/2021
12:00 – 1:00pm ETJonas Martin Peters Title: Causality and Distribution Generalization Abstract: Purely predictive methods do not perform well when the test distribution changes too much from the training distribution. Causal models are known to be stable with respect to distributional shifts such as arbitrarily strong interventions on the covariates, but do not perform well when the test distribution differs only mildly from the training distribution. We discuss anchor regression, a framework that provides a trade-off between causal and predictive models. The method poses different (convex and non-convex) optimization problems and relates to methods that are tailored for instrumental variable settings. We show how similar principles can be used for inferring metabolic networks. If time allows, we discuss extensions to nonlinear models and theoretical limitations of such methodology.
4/29/2021
1:00 – 2:00pm ETLaura Grigori Title: Randomization and communication avoiding techniques for large scale linear algebra Abstract: In this talk we will discuss recent developments of randomization and communication avoiding techniques for solving large scale linear algebra operations. We will focus in particular on solving linear systems of equations and we will discuss a randomized process for orthogonalizing a set of vectors and its usage in GMRES, while also exploiting mixed precision. We will also discuss a robust multilevel preconditioner that allows to further accelerate solving large scale linear systems on parallel computers.
Decoding Divergent Distances
Abstract: Motivated by a relationship between the Zamolodchikov and NLSM metrics to the so-called quantum information metric, I will discuss recent work (2106.11313) on understanding infinite distance limits within the context of information theory. I will describe how infinite distance points represent theories that are hyper-distinguishable, in the sense that they can be distinguished from “nearby” theories with certainty in relatively few measurements. I will then discuss necessary and sufficient ingredients for the appearance of these infinite distance points, illustrate these in simple examples, and describe how this perspective can help the swampland program.
2020 Big Data Conference (Virtual)
On August 24-25, 2020 the CMSA hosted our sixth annual Conference on Big Data. The Conference featured many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics. The 2020 Big Data Conference took place virtually.
Organizers:
- Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University
- Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business
- Horng-Tzer Yau, Professor of Mathematics, Harvard University
- Sergiy Verstyuk, CMSA, Harvard University
Speakers:
- Sanjeev Arora, Princeton University
- Juan Camilo Castillo, University of Pennsylvania
- Joseph Dexter, Dartmouth College
- Nicole Immorlica, Microsoft
- Amin Saberi, Stanford University
- Vira Semenova, University of California, Berkeley
- Varda Shalev, Tel Aviv University
Schedule:
Computational Biology Symposium
On May 3, 2021 the CMSA will be hosting a Computational Biology Symposium virtually on Zoom. This symposium will be organized by Vijay Kuchroo.
The symposium will begin at 10:00am ET. There will be a morning and afternoon session, with an hour break for lunch.
Videos of the talks can be found in this Youtube playlist. Links are also available in the schedule below.
Confirmed participants:
- Uri Alon, Weizmann Institute
- Elana Fertig, Johns Hopkins
- Martin Hemberg, Brigham and Women’s Hospital
- Peter Kharchenko, Harvard University
- Smita Krishnaswamy, Yale University
- John Marioni, EMBL-EBI
- Eran Segal, Weizmann Institute
- Meromit Singer, Harvard Medical School
Schedule:
Mathematical Physics Seminar, Mondays
The seminar on mathematical physics will be held on Mondays from 10:00 – 11:00am ET on Zoom. Please email the seminar organizers to learn how toattend. This year’s Seminar will be organized by Yoosik Kim (yoosik@cmsa.fas.harvard.edu), Tsung-Ju Lee (tjlee@cmsa.fas.harvard.edu), and Yang Zhou (yangzhou@cmsa.fas.harvard.edu).
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The list of speakers for the upcoming academic year will be posted below and updated as details are confirmed. Titles and abstracts for the talks will be added as they are received.
Spring 2021:
Date Speaker Title/Abstract 2/1/2021 Choa Dongwook
(KIAS)VideoTitle: Fukaya category of Landau-Ginzburg orbifolds. Abstract: Landau-Ginzburg orbifold is just another name for a holomorphic function W with its abelian symmetry G. Its Fukaya category can be viewed as a categorification of a homology group of its Milnor fiber. In this introductory talk, we will start with some classical results on the topology of isolated singularities and its Fukaya-Seidel category. Then I will explain a new construction for such category to deal with a non-trivial symmetry group G. The main ingredients are classical variation map and the Reeb dynamics at the contact boundary. If time permits, I will show its application to mirror symmetry of LG orbifolds and its Milnor fiber. This is a joint work with C.-H. Cho and W. Jeong
2/8/2021 Jérémy Guéré (Fourier Institute) Title: Congruences on K-theoretic Gromov-Witten invariants Abstract: K-theoretic Gromov-Witten invariants of smooth projective varieties have been introduced by YP Lee, using the Euler characteristic of a virtual structure sheaf. In particular, they are integers. In this talk, I look at these invariants for the quintic threefold and I will explain how to compute them modulo 41, using the virtual localization formula under a finite group action, up to genus 19 and degree 40.
2/15/2021 Zhiwei Zheng (Max Planck Institute) Title: Some new results on automorphisms of hypersurfaces Abstract: It is natural to study automorphisms of hypersurfaces in projective spaces. In this talk, I will discuss a new approach to determine all possible orders of automorphisms of smooth hypersurfaces with fixed degree and dimension. Then we consider the specific case of cubic fourfolds, and discuss the relation with Hodge theory.
2/22/2021 Yu-Shen Lin (Boston University) Title: Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces Abstract: Strominger–Yau–Zaslow conjecture predicts the existence of special Lagrangian fibrations on Calabi–Yau manifolds. The conjecture inspires the development of mirror symmetry while the original conjecture has little progress. In this talk, I will confirm the conjecture for the complement of a smooth anti-canonical divisor in del Pezzo surfaces. Moreover, I will also construct the dual torus fibration on its mirror. As a consequence, the special Lagrangian fibrations detect a non-standard semi-flat metric and some Ricci-flat metrics that don’t obviously appear in the literature. This is based on a joint work with T. Collins and A. Jacob.
3/1/2021 Carlos S. Shahbazi (Hamburg University) Title: Mathematical supergravity and its applications to differential geometry. Abstract: I will discuss the recent developments in the mathematical theory of supergravity that lay the mathematical foundations of the universal bosonic sector of four-dimensional ungauged supergravity and its Killing spinor equations in a differential-geometric framework. I will provide the necessary context and background. explaining the results pedagogically from scratch and highlighting several open mathematical problems which arise in the mathematical theory of supergravity, as well as some of its potential mathematical applications. Work in collaboration with Vicente Cortés and Calin Lazaroiu.
3/8/2021 Miguel Moreira (ETH) Title: Virasoro constraints for stable pairs. Abstract: The theory of stable pairs (PT) with descendents, defined on a 3-fold X, is a sheaf theoretical curve counting theory. Conjecturally, it is equivalent to the Gromov-Witten (GW) theory of X via a universal (but intricate) transformation, so we can expect that the Virasoro conjecture on the GW side should have a parallel in the PT world. In joint work with A. Oblomkov, A. Okounkov, and R. Pandharipande, we formulated such a conjecture and proved it for toric 3-folds in the stationary case. The Hilbert scheme of points on a surface S might be regarded as a component of the moduli space of stable pairs on S x P1, and the Virasoro conjecture predicts a new set of relations satisfied by tautological classes on S[n] which can be proven by reduction to the toric case.
3/15/2021 Spring break 3/22/2021 Ying Xie (Shanghai Center for Mathematical Sciences) Title: Derived categories for Grassmannian flips Abstract: Flip is a fundamental surgery operation for constructing minimal models in higher-dimensional birational geometry. In this talk, I will introduce a series of flips from Lie theory and investigate their derived categories. This is a joint program with Conan Leung.
3/29/2021 Emanuel Scheidegger (Peking University) Title: On the quantum K-theory of the quintic.
Abstract: Quantum cohomology is a deformation of the cohomology of a projective variety governed by counts of stable maps from a curve into this variety. Quantum K-theory is in a similar way a deformation of K-theory but also of quantum cohomology, It has recently attracted attention in physics since a realization in a physical theory has been found. Currently, both the structure and examples in quantum K-theory are far less understood than in quantum cohomology.
We will explain the properties of quantum K-theory in comparison with quantum cohomology, and we will discuss the examples of projective space and the quintic hypersurface in P^4.4/5/2021 Gaëtan Borot (HU Berlin) Title: Topological recursion in 4d N = 2 supersymmetric gauge theories Abstract: According to the Alday-Gaiotto-Tachikawa conjecture (proved in this case by Schiffman and Vasserot), the instanton partition function in 4d N = 2 SU(r) supersymmetric gauge theory on P^2 with equivariant parameters \epsilon_1,\epsilon_2 is the norm of a Whittaker vector for W(gl_r) algebra. I will explain how these Whittaker vectors can be computed (at least perturbatively in the energy scale) by topological recursion for \epsilon_1 +\epsilon_2 = 0, and by a non-commutation version of the topological recursion in the Nekrasov-Shatashvili regime where \epsilon_1/\epsilon_2 is fixed. This is a joint work to appear with Bouchard, Chidambaram and Creutzig.
4/12/2021 Fei Yan (Rutgers) Title: Networks and quantization Abstract: I will describe two quantization scenarios. The first scenario involves the construction of a quantum trace map computing a link “invariant” (with possible wall-crossing behavior) for links L in a 3-manifold M, where M is a Riemann surface C times a real line. This construction unifies the computation of familiar link invariant with the refined counting of framed BPS states for line defects in 4d N=2 theories of class S. Certain networks on C play an important role in the construction. The second scenario concerns the study of Schroedinger equations and their higher order analogues, which could arise in the quantization of Seiberg-Witten curves in 4d N=2 theories. Here similarly certain networks play an important part in the exact WKB analysis for these Schroedinger-like equations. At the end of my talk I will also try to sketch a possibility to bridge these two scenarios.
4/19/2021 Hazel Mak (Brown University) Title: Branching Rules and Young Tableaux Methods: 10D & 11D Supergravity Abstract: In this talk, I will review 4D, N = 1 off-shell supergravity. Then I present explorations to construct 10D and 11D supergravity theories in two steps. The first step is to decompose scalar superfield into Lorentz group representations which involves branching rules and related methods. Interpretations of component fields by Young tableaux methods will be presented. The second step is to implement an analogue of Breitenlohner’s approach for 4D supergravity to 10D and 11D theories.
4/26/2021 Owen Gwilliam (UMass. Amherst) Title: Topological-holomorphic field theories and their BV quantizations Abstract: Topological field theories and holomorphic field theories have each had a substantial impact in both physics and mathematics, so it is natural to consider theories that are hybrids of the two, which we call topological-holomorphic and denote as THFTs. Examples include Kapustin’s twist of N=2, D=4 supersymmetric Yang-Mills theory and Costello’s 4-dimensional Chern-Simons theory. In this talk about joint work with Rabinovich and Williams, I will define THFTs, describe several examples, and then explain how to quantize them rigorously and explicitly, by building on techniques of Si Li. Time permitting, I will indicate how these results offer a novel perspective on the Gaudin model via 3-dimensional field theories.
Fall 2020:
Date Speaker Title/Abstract 9/14/2020 Lino Amorim (Kansas State University) Title: Non-commutative Gromov-Witten invariants Abstract: I will describe an analogue of Saito’s theory of primitive forms for Calabi-Yau A-infinity categories. Under some conditions on the Hochschild cohomology of the category, this construction recovers the (genus zero) Gromov-Witten invariants of a symplectic manifold from its Fukaya category. This includes many compact toric manifolds, in particular projective spaces.
9/21/2020 Yuhan Sun (Rutgers) Title: Displacement energy of Lagrangian 3-spheres Abstract: We study local and global Hamiltonian dynamical behaviors of some Lagrangian submanifolds near a Lagrangian sphere S in a symplectic manifold X. When dim S = 2, we show that there is a one-parameter family of Lagrangian tori near S, which are nondisplaceable in X. When dim S = 3, we obtain a new estimate of the displacement energy of S, by estimating the displacement energy of a one-parameter family of Lagrangian tori near S.
9/28/2020 Shota Komatsu (CERN) Title: Wilson loops as matrix product states Abstract: In this talk, I will discuss a reformulation of the Wilson loop in large N gauge theories in terms of matrix product states. The construction is motivated by the analysis of supersymmetric Wilson loops in the maximally super Yang–Mills theory in four dimensions, but can be applied to any other large N gauge theories and matrix models, although less effective. For the maximally super Yang–Mills theory, one can further perform the computation exactly as a function of ‘t Hooft coupling by combining our formulation with the relation to integrable spin chains.
10/5/2020 Ming Zhang (UBC) Title: Verlinde/Grassmannian correspondence and applications. Abstract: In the 90s’, Witten gave a physical derivation of an isomorphism between the Verlinde algebra of $GL(n)$ of level $l$ and the quantum cohomology ring of the Grassmannian $\text{Gr}(n,n+l)$. In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten’s work by relating the $\text{GL}_{n}$ Verlinde numbers to the level $l$ quantum K-invariants of the Grassmannian $\text{Gr}(n,n+l)$, and refer to it as the Verlinde/Grassmannian correspondence.
The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there. In this talk, I will discuss the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner. At the end of the talk, I will describe some applications of this correspondence.
10/12/2020 Cancelled -Columbus Day 10/19/2020 Ben Gammage (Harvard) Title: 3d mirror symmetry for abelian gauge groups Abstract: 3d mirror symmetry is a proposed duality relating a pair of 3-dimensional supersymmetric gauge theories. Various consequences of this duality have been heavily explored by representation theorists in recent years, under the name of “symplectic duality”. In joint work in progress with Justin Hilburn, for the case of abelian gauge groups, we provide a fully mathematical explanation of this duality in the form of an equivalence of 2-categories of boundary conditions for topological twists of these theories. We will also discuss some applications to homological mirror symmetry and geometric Langlands duality.
10/26/2020 Cancelled 11/2/2020 Haoyu Sun (Berkeley) Title: Double-Janus linear sigma models and generalized quadratic reciprocity
Abstract: We study the supersymmetric partition function of a 2d linear sigma-model whose target space is a torus with a complex structure that varies along one worldsheet direction and a Kähler modulus that varies along the other. This setup is inspired by the dimensional reduction of a Janus configuration of 4d N=4 U(1) Super-Yang-Mills theory compactified on a mapping torus (T^2 fibered over S^1) times a circle with an SL(2,Z) duality wall inserted on S^1, but our setup has minimal supersymmetry. The partition function depends on two independent elements of SL(2,Z), one describing the duality twist, and the other describing the geometry of the mapping torus. It is topological and can be written as a multivariate quadratic Gauss sum. By calculating the partition function in two different ways, we obtain identities relating different quadratic Gauss sums, generalizing the Landsberg-Schaar relation. These identities are a subset of a collection of identities discovered by F. Deloup. Each identity contains a phase which is an eighth root of unity, and we show how it arises as a Berry phase in the supersymmetric Janus-like configuration. Supersymmetry requires the complex structure to vary along a semicircle in the upper half-plane, as shown by Gaiotto and Witten in a related context, and that semicircle plays an important role in reproducing the correct Berry phase.11/9/2020 An Huang (Brandeis) Title: p-adic strings, Einstein equations, Green’s functions, and Tate’s thesis
Abstract: I shall discuss a recent work on how p-adic strings can produce perturbative quantum gravity, and an adelic physics interpretation of Tate’s thesis.11/16/2020
10:00am ETMatt Kerr (WUSTL) Title: Differential equations and mixed Hodge structures Abstract: We report on a new development in asymptotic Hodge theory, arising from work of Golyshev–Zagier and Bloch–Vlasenko, and connected to the Gamma Conjectures in Fano/LG-model mirror symmetry. The talk will focus exclusively on the Hodge/period-theoretic aspects through two main examples.
Given a variation of Hodge structure M on a Zariski open in P^1, the periods of the limiting mixed Hodge structures at the punctures are interesting invariants of M. More generally, one can try to compute these asymptotic invariants for iterated extensions of M by “Tate objects”, which may arise for example from normal functions associated to algebraic cycles. The main point of the talk will be that (with suitable assumptions on M) these invariants are encoded in an entire function called the motivic Gamma function, which is determined by the Picard-Fuchs operator L underlying M. In particular, when L is hypergeometric, this is easy to compute and we get a closed-form answer (and a limiting motive). In the non-hypergeometric setting, it yields predictions for special values of normal functions; this part of the story is joint with V. Golyshev and T. Sasaki.11/23/2020 11:30am ET
Kyoung-Seog Lee (U of Miami) Title: Derived categories and motives of moduli spaces of vector bundles on curves Abstract: Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. On the other hand, moduli spaces of vector bundles on curves provide attractive and important examples of algebraic varieties and there have been intensive works studying them. In this talk, I will discuss derived categories and motives of moduli spaces of vector bundles on curves. This talk is based on joint works with I. Biswas and T. Gomez.
11/30/2020 Zijun Zhou (IPMU) Title: 3d N=2 toric mirror symmetry and quantum K-theory Abstract: In this talk, I will introduce a new construction for the K-theoretic mirror symmetry of toric varieties/stacks, based on the 3d N=2 mirror symmetry introduced by Dorey-Tong. Given the toric datum, i.e. a short exact sequence 0 -> Z^k -> Z^n -> Z^{n-k} -> 0, we consider the toric Artin stack of the form [C^n / (C^*)^k]. Its mirror is constructed by taking the Gale dual of the defining short exact sequence. As an analogue of the 3d N=4 case, we consider the K-theoretic I-function, with a suitable level structure, defined by counting parameterized quasimaps from P^1. Under mirror symmetry, the I-functions of a mirror pair are related to each other under the mirror map, which exchanges the K\”ahler and equivariant parameters, and maps q to q^{-1}. This is joint work with Yongbin Ruan and Yaoxiong Wen.
12/7/2020 Thomas Grimm (Utrecht) Title: Moduli Space Holography and the Finiteness of Flux Vacua Abstract: In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using asymptotic Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The bulk reconstruction from the boundary data is provided by the sl(2)-orbit theorem of Schmid and Cattani, Kaplan, Schmid, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us, in work with C. Schnell, to prove that the number of self-dual flux vacua is finite
For a listing of previous Mathematical Physics Seminars, please click here.
Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces
Speaker: Yu-Shen Lin (Boston University)
Title: Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts the existence of special Lagrangian fibrations on Calabi-Yau manifolds. The conjecture inspires the development of mirror symmetry while the original conjecture has little progress. In this talk, I will confirm the conjecture for the complement of a smooth anti-canonical divisor in del Pezzo surfaces. Moreover, I will also construct the dual torus fibration on its mirror. As a consequence, the special Lagrangian fibrations detect a non-standard semi-flat metric and some Ricci-flat metrics that don’t obviously appear in the literature. This is based on a joint work with T. Collins and A. Jacob.
CMSA Math-Science Literature Lecture: Subfactors–in Memory of Vaughan Jones
Zhengwei Liu (Tsinghua University)
Title: Subfactors–in Memory of Vaughan Jones
Abstract: Jones initiated modern subfactor theory in early 1980s and investigated this area for his whole academic life. Subfactor theory has both deep and broad connections with various areas in mathematics and physics. One well-known peak in the development of subfactor theory is the discovery of the Jones polynomial, for which Jones won the Fields Metal in 1990. Let us travel back to the dark room at the beginning of the story, to appreciate how radically our viewpoint has changed.
Talk chair: Arthur Jaffe
Previous Random Matrix & Probability Theory Seminars
Spring 2020:
Date Speaker Title/Abstract 2/26/2020 Louigi Addario-Berry (McGill University) Title: Hipster random walks and their ilk Abstract: I will describe how certain recursive distributional equations can be solved by importing rigorous results on the convergence of approximation schemes for degenerate PDEs, from numerical analysis. This project is joint work with Luc Devroye, Hannah Cairns, Celine Kerriou, and Rivka Maclaine Mitchell.
4/1/2020 Ian Jauslin (Princeton) This meeting will be taking place virtually on Zoom. Title: A simplified approach to interacting Bose gases
Abstract: I will discuss some new results about an effective theory introduced by Lieb in 1963 to approximate the ground state energy of interacting Bosons at low density. In this regime, it agrees with the predictions of Bogolyubov. At high densities, Hartree theory provides a good approximation. In this talk, I will show that the ’63 effective theory is actually exact at both low and high densities, and numerically accurate to within a few percents in between, thus providing a new approach to the quantum many body problem that bridges the gap between low and high density.4/22/2020 Martin Gebert (UC Davis) This meeting will be taking place virtually on Zoom. Title: Lieb-Robinson bounds for a class of continuum many-body fermion systems
Abstract: We introduce a class of UV-regularized two-body interactions for
fermions in $\R^d$ and prove a Lieb-Robinson estimate for the dynamics
of this class of many-body systems. As a step towards this result, we
also prove a propagation bound of Lieb-Robinson type for continuum
one-particle Schr\“odinger operators. We apply the propagation bound to
prove the existence of a strongly continuous infinite-volume dynamics on
the CAR algebra.4/29/2020 Marcin Napiórkowski (University of Warsaw) This meeting will be taking place virtually on Zoom. Title: Free energy asymptotics of the quantum Heisenberg spin chain
Abstract: Spin wave theory suggests that low temperature properties of the Heisenberg model can be described in terms of noninteracting quasiparticles called magnons. In my talk I will review the basic concepts and predictions of spin wave approximation and report on recent rigorous results in that direction. Based on joint work with Robert Seiringer.
5/6/2020 Antti Knowles (University of Geneva) Title: Field theory as a limit of interacting quantum Bose gases
Abstract: We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d = 1,2,3. For d > 1 the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. The proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials. Joint work with Jürg Fröhlich, Benjamin Schlein, and Vedran Sohinger.5/13/2020 Sven Bachmann (University of British Columbia) Title: Quantized quantum transport and Abelian anyons Abstract: I’ll discuss recent developments in the study of quantized quantum transport, focussing on the quantum Hall effect. Beyond presenting an index taking rational values, and which is the Hall conductance in the adapted setting, I will explain how the index is intimately paired with the existence of quasi-particle excitations having non-trivial braiding properties.
5/20/2020 Kristina Schubert (TU Dortmund) Title: Fluctuation Results for General Ising Models — Block Spin Ising Models and Random Interactions Abstract: Starting from the classical Curie-Weiss model in statistical mechanics, we will consider more general Ising models. On the one hand, we introduce a block structure, i.e. a model of spins in which the vertices are divided into a finite number of blocks and where pair interactions are given according to their blocks. The magnetization is then the vector of magnetizations within each block, and we are interested in its behaviour and in particular in its fluctuations. On the other hand, we consider Ising models on Erdős-Rényi random graphs. Here, I will also present results on the fluctuations of the magnetization.
Fall 2019:
Date Speaker Title/Abstract 9/11/2019 Subhabrata Sen Title: Sampling convergence for random graphs: graphexes and multigraphexes Abstract: We will look at structural properties of large, sparse random graphs through the lens of sampling convergence (Borgs, Chayes, Cohn and Veitch ’17). Sam- pling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a graphex. We will introduce this framework and motivate the components of a graphex. Subsequently, we will discuss the graphex limit for several well-known sparse random (multi)graph models. This is based on joint work with Christian Borgs, Jennifer Chayes, and Souvik Dhara.
9/25/2019 Jeff Schenker (Michigan State) Title: An ergodic theorem for homogeneously distributed quantum channels with applications to matrix product states Abstract: Quantum channels represent the most general physical evolution of a quantum system through unitary evolution and a measurement process. Mathematically, a quantum channel is a completely positive and trace preserving linear map on the space of $D\times D$ matrices. We consider ergodic sequences of channels, obtained by sampling channel valued maps along the trajectories of an ergodic dynamical system. The repeated composition of these maps along such a sequence could represent the result of repeated application of a given quantum channel subject to arbitrary correlated noise. It is physically natural to assume that such repeated compositions are eventually strictly positive, since this is true whenever any amount of decoherence is present in the quantum evolution. Under such an hypothesis, we obtain a general ergodic theorem showing that the composition of maps converges exponentially fast to a rank-one — “entanglement breaking’’ – channel. We apply this result to describe the thermodynamic limit of ergodic matrix product states and prove that correlations of observables in such states decay exponentially in the bulk. (Joint work with Ramis Movassagh)
10/3/2019 Thursday
4:30pm
Jian Ding (UPenn) Title: Distances associated with Liouville quantum gravity Abstract: I will review some recent progresses on distances associated with Liouville quantum gravity, which is a random measure obtained from exponentiating a planar Gaussian free field.
The talk is based on works with Julien Dubédat, Alexander Dunlap, Hugo Falconet, Subhajit Goswami, Ewain Gwynne, Ofer Zeitouni and Fuxi Zhang in various combinations.
10/9/2019 Ruth Williams (UCSD) Title: Stability of a Fluid Model for Fair Bandwidth Sharing with General File Size Distributions Abstract: Massoulie and Roberts introduced a stochastic model for a data communication network where file sizes are generally distributed and the network operates under a fair bandwidth sharing policy. It has been a standing problem to prove stability of this general model when the average load on the system is less than the network’s capacity. A crucial step in an approach to this problem is to prove stability of an associated measure-valued fluid model. We shall describe prior work on this question done under various strong assumptions and indicate how to prove stability of the fluid model under mild conditions.
This talk is based on joint work with Yingjia Fu.
10/11/2019 Cancelled 10/16/2019 Wei-Kuo Chen (University of Minnesota) Title: The generalized TAP free energy Abstract: Spin glasses are disordered spin systems initially invented by theoretical physicists with the aim of understanding some strange magnetic properties of certain alloys. In particular, over the past decades, the study of the Sherrington-Kirkpatrick (SK) mean-field model via the replica method has received great attention. In this talk, I will discuss another approach to studying the SK model proposed by Thouless-Anderson-Palmer (TAP). I will explain how the generalized TAP correction appears naturally and give the corresponding generalized TAP representation for the free energy. Based on a joint work with D. Panchenko and E. Subag.
10/23/2019 Souvik Dhara (MIT) Title: A new universality class for critical percolation on networks with heavy-tailed degrees Abstract: The talk concerns critical behavior of percolation on finite random networks with heavy-tailed degree distribution. In a seminal paper, Aldous (1997) identified the scaling limit for the component sizes in the critical window of phase transition for the Erdős-Rényi random graph. Subsequently, there has been a surge in the literature identifying two universality classes for the critical behavior depending on whether the asymptotic degree distribution has a finite or infinite third moment.
In this talk, we will present a completely new universality class that arises in the context of degrees having infinite second moment. Specifically, the scaling limit of the rescaled component sizes is different from the general description of multiplicative coalescent given by Aldous and Limic (1998). Moreover, the study of critical behavior in this regime exhibits several surprising features that have never been observed in any other universality classes so far.
This is based on joint works with Shankar Bhamidi, Remco van der Hofstad, Johan van Leeuwaarden.
10/30/2019 Aram Harrow (MIT) Title: Random quantum circuits, phase transitions and complexity Abstract: Random unitary dynamics are a toy model for chaotic quantum dynamics and also have applications to quantum information theory and computing. Recently, random quantum circuits were the basis of Google’s announcement of “quantum computational supremacy,” meaning performing a task on a programmable quantum computer that would difficult or infeasible for any classical computer. Google’s approach is based on the conjecture that random circuits are as hard to classical computers to simulate as a worst-case quantum computation would be. I will describe evidence in favor of this conjecture for deep random circuits and against this conjecture for shallow random circuits. (Deep/shallow refers to the number of time steps of the quantum circuit.) For deep random circuits in Euclidean geometries, we show that quantum dynamics match the first few moments of the Haar measure after roughly the amount of time needed for a signal to propagate from one side of the system to the other. In non-Euclidean geometries, such as the Schwarzschild metric in the vicinity of a black hole, this turns out not to be always true. I will also explain how shallow quantum circuits are easier to simulate when the gates are randomly chosen than in the worst case. This uses a simulation algorithm based on tensor contraction which is analyzed in terms of an associated stat mech model.
This is based on joint work with Saeed Mehraban (1809.06957) and with John Napp, Rolando La Placa, Alex Dalzell and Fernando Brandao (to appear).
11/6/2019 Bruno Nachtergaele (UC Davis) Title: The transmission time and local integrals of motion for disordered spin chains Abstract: We investigate the relationship between zero-velocity Lieb-Robinson bounds and the existence of local integrals of motion (LIOMs) for disordered quantum spin chains. We also study the effect of dilute random perturbations on the dynamics of many-body localized spin chains. Using a notion of transmission time for propagation in quantum lattice systems we demonstrate slow propagation by proving a lower bound for the transmission time. This result can be interpreted as a robustness property of slow transport in one dimension. (Joint work with Jake Reschke)
11/13/2019 Gourab Ray (University of Victoria) Title: Logarithmic variance of height function of square-iceAbstract: A homomorphism height function on a finite graph is a integer-valued function on the set of vertices constrained to have adjacent vertices take adjacent integer values. We consider the uniform distribution over all such functions defined on a finite subgraph of Z^2 with predetermined values at some fixed boundary vertices. This model is equivalent to the height function of the six-vertex model with a = b = c = 1, i.e. to the height function of square-ice. Our main result is that in a subgraph of Z^2 with zero boundary conditions, the variance grows logarithmically in the distance to the boundary. This establishes a strong form of roughness of the planar uniform homomorphisms. Joint work with: Hugo Duminil Copin, Matan Harel, Benoit Laslier and Aran Raoufi.
11/20/2019 Vishesh Jain (MIT) Title: A combinatorial approach to the quantitative invertibility of random matrices. Abstract: Abstract: Let $s_n(M_n)$ denote the smallest singular value of an $n\times n$ random matrix $M_n$. We will discuss a novel combinatorial approach (in particular, not using either inverse Littlewood–Offord theory or net arguments) for obtaining upper bounds on the probability that $s_n(M_n)$ is smaller than $\eta \geq 0$ for quite general random matrix models. Such estimates are a fundamental part of the non-asymptotic theory of random matrices and have applications to the strong circular law, numerical linear algebra etc. In several cases of interest, our approach provides stronger bounds than those obtained by Tao and Vu using inverse Littlewood–Offord theory.
2018-2019
Date Speaker Title/Abstract 9/28/2018 *Friday, 10:00am*
Yash Deshpande (MIT) Title: Estimating low-rank matrices in noise: phase transitions from spin glass theory Abstract: Estimating low-rank matrices from noisy observations is a common task in statistical and engineering applications. Following the seminal work of Johnstone, Baik, Ben-Arous and Peche, versions of this problem have been extensively studied using random matrix theory. In this talk, we will consider an alternative viewpoint based on tools from mean field spin glasses. We will present two examples that illustrate how these tools yield information beyond those from classical random matrix theory. The first example is the two-groups stochastic block model (SBM), where we will obtain a full information-theoretic understanding of the estimation phase transition. In the second example, we will augment the SBM with covariate information at nodes, and obtain results on the altered phase transition.
This is based on joint works with Emmanuel Abbe, Andrea Montanari, Elchanan Mossel and Subhabrata Sen.
10/3/2018 Ian Jauslin (IAS) Title: Liquid Crystals and the Heilmann-Lieb model Abstract: In 1979, O.Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of a nematic liquid crystal phase in it. In such a phase, dimers spontaneously align, but there is no long range translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. I will discuss a recent proof of this conjecture. This is joint work with Elliott H. Lieb.
10/10/2018 Afonso Bandeira (NYU Title: Statistical estimation under group actions: The Sample Complexity of Multi-Reference Alignment Abstract: Many problems in signal/image processing, and computer vision amount to estimating a signal, image, or tri-dimensional structure/scene from corrupted measurements. A particularly challenging form of measurement corruption are latent transformations of the underlying signal to be recovered. Many such transformations can be described as a group acting on the object to be recovered. Examples include the Simulatenous Localization and Mapping (SLaM) problem in Robotics and Computer Vision, where pictures of a scene are obtained from different positions and orientations; Cryo-Electron Microscopy (Cryo-EM) imaging where projections of a molecule density are taken from unknown rotations, and several others.
One fundamental example of this type of problems is Multi-Reference Alignment: Given a group acting in a space, the goal is to estimate an orbit of the group action from noisy samples. For example, in one of its simplest forms, one is tasked with estimating a signal from noisy cyclically shifted copies. We will show that the number of observations needed by any method has a surprising dependency on the signal-to-noise ratio (SNR), and algebraic properties of the underlying group action. Remarkably, in some important cases, this sample complexity is achieved with computationally efficient methods based on computing invariants under the group of transformations.
10/17/2018 3:30pm
Thomas Chen (UT Austin) Title: Dynamics of a heavy quantum tracer particle in a Bose gas Abstract: We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in R^3. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to 1/N where N is the expected particle number. Assuming that the mass of the tracer particle is proportional to N, we derive generalized Hartree equations in the limit where N tends to infinity. Moreover, we prove the global well-posedness of the associated Cauchy problem for sufficiently weak interaction potentials. This is joint work with Avy Soffer (Rutgers University).
10/24/2018 *Room G02*
Tselil Schramm (Harvard/MIT) Title: (Nearly) Efficient Algorithms for the Graph Matching Problem in Correlated Random Graphs Abstract: The Graph Matching problem is a robust version of the Graph Isomorphism problem: given two not-necessarily-isomorphic graphs, the goal is to find a permutation of the vertices which maximizes the number of common edges. We study a popular average-case variant; we deviate from the common heuristic strategy and give the first quasi-polynomial time algorithm, where previously only sub-exponential time algorithms were known.
Based on joint work with Boaz Barak, Chi-Ning Chou, Zhixian Lei, and Yueqi Sheng.
10/30/2018 *Tuesday
10:30am
SC 507*
Lauren Williams (Harvard) Title: Introduction to the asymmetric simple exclusion process (from a combinatorialist’s point of view) Abstract: The asymmetric simple exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice, subject to the condition that there is at most one particle per site. This model was introduced in 1970 by biologists (as a model for translation in protein synthesis) but has since been shown to display a rich mathematical structure. There are many variants of the model — e.g. the lattice could be a ring, or a line with open boundaries. One can also allow multiple species of particles with different “weights.” I will explain how one can give combinatorial formulas for the stationary distribution using various kinds of tableaux. I will also explain how the ASEP is related to interesting families of orthogonal polynomials, including Askey-Wilson polynomials, Koornwinder polynomials, and Macdonald polynomials.
11/7/2018 Willhelm Schlag (Yale) Title: on the Bourgain-Dyatlov fractal uncertainty principle Abstract: We will present the Bourgain-Dyatlov theorem on the line, it’s connection with other uncertainty principles in harmonic analysis, and my recent partial progress with Rui Han on the problem of higher dimensions.
11/14/2018 David Gamarnik (MIT) Title: Two Algorithmic Hardness Results in Spin Glasses and Compressive Sensing. Abstract: I will discuss two computational problems in the area of random combinatorial structures. The first one is the problem of computing the partition function of a Sherrington-Kirkpatrick spin glass model. While the the problem of computing the partition functions associated with arbitrary instances is known to belong to the #P complexity class, the complexity of the problem for random instances is open. We show that the problem of computing the partition function exactly (in an appropriate sense) for the case of instances involving Gaussian couplings is #P-hard on average. The proof uses Lipton’s trick of computation modulo large prime number, reduction of the average case to the worst case instances, and the near uniformity of the ”stretched” log-normal distribution.
In the second part we will discuss the problem of explicit construction of matrices satisfying the Restricted Isometry Property (RIP). This challenge arises in the field of compressive sensing. While random matrices are known to satisfy the RIP with high probability, the problem of explicit (deterministic) construction of RIP matrices eluded efforts and hits the so-called ”square root” barrier which I will discuss in the talk. Overcoming this barrier is an open problem explored widely in the literature. We essentially resolve this problem by showing that an explicit construction of RIP matrices implies an explicit construction of graphs satisfying a very strong form of Ramsey property, which has been open since the seminal work of Erdos in 1947.
11/28/2018 Sean O’ Rourke (UC Boulder) Title: Universality and least singular values of random matrix products Abstract: We consider the product of m independent iid random matrices as m is fixed and the sizes of the matrices tend to infinity. In the case when the factor matrices are drawn from the complex Ginibre ensemble, Akemann and Burda computed the limiting microscopic correlation functions. In particular, away from the origin, they showed that the limiting correlation functions do not depend on m, the number of factor matrices. We show that this behavior is universal for products of iid random matrices under a moment matching hypothesis. In addition, we establish universality results for the linear statistics for these product models, which show that the limiting variance does not depend on the number of factor matrices either. The proofs of these universality results require a near-optimal lower bound on the least singular value for these product ensembles.
12/5/2018 *Room G02*
Omer Angel (UBC) Title: balanced excited random walks Abstract: I will present results on the scaling limit and asymptotics of the balanced excited random walk and related processes. This is a walk the that moves vertically on the first visit to a vertex, and horizontally on every subsequent visit. We also analyze certain versions of “clairvoyant scheduling” of random walks.
Joint work with Mark Holmes and Alejandro Ramirez.
2/7/2019 Science Center 530
Ramis Movassagh (IMB Research) Title: Generic Gaplessness, and Hamiltonian density of states from free probability theory Abstract: Quantum many-body systems usually reside in their lowest energy states. This among other things, motives understanding the gap, which is generally an undecidable problem. Nevertheless, we prove that generically local quantum Hamiltonians are gapless in any dimension and on any graph with bounded maximum degree.
We then provide an applied and approximate answer to an old problem in pure mathematics. Suppose the eigenvalue distributions of two matrices M_1 and M_2 are known. What is the eigenvalue distribution of the sum M_1+M_2? This problem has a rich pure mathematics history dating back to H. Weyl (1912) with many applications in various fields. Free probability theory (FPT) answers this question under certain conditions. We will describe FPT and show examples of its powers for approximating physical quantities such as the density of states of the Anderson model, quantum spin chains, and gapped vs. gapless phases of some Floquet systems. These physical quantities are often hard to compute exactly (provably NP-hard). Nevertheless, using FPT and other ideas from random matrix theory excellent approximations can be obtained. Besides the applications presented, we believe the techniques will find new applications in fresh new contexts.
2/14/2019 Nike Sun (MIT) Title: Capacity lower bound for the Ising perceptron Abstract: The perceptron is a toy model of a simple neural network that stores a collection of given patterns. Its analysis reduces to a simple problem in high-dimensional geometry, namely, understanding the intersection of the cube (or sphere) with a collection of random half-spaces. Despite the simplicity of this model, its high-dimensional asymptotics are not well understood. I will describe what is known and present recent results.
2/21/2019 Michael Loss (Georgia Tech) Title: Some results for functionals of Aharonov-Bohm type Abstract: In this talk I present some variational problems of Aharonov-Bohm type, i.e., they include a magnetic flux that is entirely concentrated at a point. This is maybe the simplest example of a variational problems for systems, the wave function being necessarily complex. The functional is rotationally invariant and the issue to be discussed is whether the optimizer have this symmetry or whether it is broken.
3/6/2019 4:15pm
Science Center 411
Ilya Kachkovskiy (Michigan State University) Title: Localization and delocalization for interacting 1D quasiperiodic particles. Abstract: We consider a system of two interacting one-dimensional quasiperiodic particles as an operator on $\ell^2(\mathbb Z^2)$. The fact that particle frequencies are identical, implies a new effect compared to generic 2D potentials: the presence of large coupling localization depends on symmetries of the single-particle potential. If the potential has no cosine-type symmetries, then we are able to show large coupling localization at all energies, even if the interaction is not small (with some assumptions on its complexity). If symmetries are present, we can show localization away from finitely many energies, thus removing a fraction of spectrum from consideration. We also demonstrate that, in the symmetric case, delocalization can indeed happen if the interaction is strong, at the energies away from the bulk spectrum. The result is based on joint works with Jean Bourgain and Svetlana Jitomirskaya.
3/14/2019 5:45pm
Science Center 232
Anna Vershynina (University of Houston) Title: How fast can entanglement be generated in quantum systems? Abstract: We investigate the maximal rate at which entanglement can be generated in bipartite quantum systems. The goal is to upper bound this rate. All previous results in closed systems considered entanglement entropy as a measure of entanglement. I will present recent results, where entanglement measure can be chosen from a large class of measures. The result is derived from a general bound on the trace-norm of a commutator, and can, for example, be applied to bound the entanglement rate for Renyi and Tsallis entanglement entropies.
3/28/2019 Room G02
Xuwen Chen (University of Rochester) Title: The Derivation of the Energy-critical NLS from Quantum Many-body Dynamics Abstract: We derive the 3D energy-critical quintic NLS from quantum many-body dynamics with 3-body interaction in the T^3 (periodic) setting. Due to the known complexity of the energy critical setting, previous progress was limited in comparison to the 2-body interaction case yielding energy subcritical cubic NLS. We develop methods to prove the convergence of the BBGKY hierarchy to the infinite Gross-Pitaevskii (GP) hierarchy, and separately, the uniqueness of large GP solutions. Since the trace estimate used in the previous proofs of convergence is the false sharp trace estimate in our setting, we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de Finetti theorem. For the large solution uniqueness argument, we discover the new HUFL (hierarchical uniform frequency localization) property for the GP hierarchy and use it to prove a new type of uniqueness theorem.
4/4/2019 Paul Bourgade (NYU) Title: Log-correlations and branching structures in analytic number theory Abstract: Fyodorov, Hiary and Keating have predicted the size of local maxima of L-function along the critical axis, based on analogous random matrix statistics. I will explain this prediction in the context of the log-correlated universality class and branching structures. In particular I will explain why the Riemann zeta function exhibits log-correlations, and outline the proof for the leading order of the maximum in the Fyodorov, Hiary and Keating prediction. Joint work with Arguin, Belius, Radziwill and Soundararajan.
4/9/2019 Tuesday
12:00pm
Room G02
Giulio Biroli (ENS Paris) Title: Large deviations for the largest eigenvalues and eigenvectors of spiked random matrices Abstract: I consider matrices formed by a random $N\times N$ matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength $\theta$, and focus on the largest eigenvalue, $x$, and the component, $u$, of the corresponding eigenvector in the direction associated to the rank-one perturbation. I will show how to obtain the large deviation principle governing the atypical joint fluctuations of $x$ and $u$. Interestingly, for $\theta>1$, in large deviations characterized by a small value of $u$, i.e. $u<1-1/\theta$, the second-largest eigenvalue pops out from the Wigner semi-circle and the associated eigenvector orients in the direction corresponding to the rank-one perturbation. These results can be generalized to the Wishart Ensemble, and extended to the first $n$ eigenvalues and the associated eigenvectors.
Finally, I will discuss motivations and applications of these results to the study of the geometric properties of random high-dimensional functions—a topic that is currently attracting a lot of attention in physics and computer science.
4/11/2019 Rui Han (Georgia Tech) Title: Spectral gaps in graphene structures Abstract: We present a full analysis of the spectrum of graphene in magnetic fields with constant flux through every hexagonal comb. In particular, we provide a rigorous foundation for self-similarity by showing that for irrational flux, the spectrum of graphene is a zero measure Cantor set. We also show that for vanishing flux, the spectral bands have nontrivial overlap, which proves the discrete Bethe-Sommerfeld conjecture for the graphene structure. This is based on joint works with S. Becker, J. Fillman and S. Jitomirskaya.
4/25/2019 Benjamin Fehrman (Oxford) Title: Pathwise well-posedness of nonlinear diffusion equations with nonlinear, conservative noise Abstract: We present a pathwise well-posedness theory for stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, approximate the Dean-Kawasaki equation in fluctuating fluid dynamics, describe the fluctuating hydrodynamics of the zero range process, and model the evolution of a thin film in the regime of negligible surface tension. Motivated by the theory of stochastic viscosity solutions, we pass to the equation’s kinetic formulation, where the noise enters linearly and can be inverted using the theory of rough paths. The talk is based on joint work with Benjamin Gess.
4/30/2019 TBA TBA 5/2/2019 Jian Ding (UPenn) TBA 2017-2018
Date………… Name……………. Title/Abstract 2-16-20183:30pm G02
Reza Gheissari (NYU) Dynamics of Critical 2D Potts ModelsAbstract: The Potts model is a generalization of the Ising model to $q\geq 3$ states with inverse temperature $\beta$. The Gibbs measure on $\mathbb Z^2$ has a sharp transition between a disordered regime when $\beta<\beta_c(q)$ and an ordered regime when $\beta>\beta_c(q)$. At $\beta=\beta_c(q)$, when $q\leq 4$, the phase transition is continuous while when $q>4$, the phase transition is discontinuous and the disordered and ordered phases coexist. We will discuss recent progress, joint with E. Lubetzky, in analyzing the time to equilibrium (mixing time) of natural Markov chains (e.g., heat bath/Metropolis) for the 2D Potts model, where the mixing time on an $n \times n$ torus should transition from $O(\log n)$ at high temperatures to $\exp(c_\beta n)$ at low temperatures, via a critical slowdown at $\beta_c(q)$ that is polynomial in $n$ when $q \leq 4$ and exponential in $n$ when $q>4$.
2-23-20183:30pm G02
Mustazee Rahman (MIT) On shocks in the TASEPAbstract: The TASEP particle system runs into traffic jams when the particle density to the left is smaller than the density to the right. Macroscopically, the particle density solves Burgers’ equation and traffic jams correspond to its shocks. I will describe work with Jeremy Quastel on a specialization of the TASEP shock whereby we identify the microscopic fluctuations around the shock by using exact formulas for the correlation functions of TASEP and its KPZ scaling limit. The resulting laws are related to universal laws of random matrix theory. For the curious, here is a video of the shock forming in Burgers’ equation:
4-20-20182:00-3:00pm Carlo Lucibello(Microsoft Research NE) The Random Perceptron Problem: thresholds, phase transitions, and geometryAbstract: The perceptron is the simplest feedforward neural network model, the building block of the deep architectures used in modern machine learning practice. In this talk, I will review some old and new results, mostly focusing on the case of binary weights and random examples. Despite its simplicity, this model provides an extremely rich phenomenology: as the number of examples per synapses is increased, the system undergoes different phase transitions, which can be directly linked to solvers’ performances and to information theoretic bounds. A geometrical analysis of the solution space shows how two different types of solutions, akin to wide and sharp minima, have different generalization capabilities when presented with new examples. Solutions in dense clusters generalize remarkably better, partially closing the gap with Bayesian optimal estimators. Most of the results I will present were first obtained using non rigorous techniques from spin glass theory and many of them haven’t been rigorously established yet, although some big steps forward have been taken in recent years. 4-20-20183:00-4:00pm Yash Despande(MIT) Phase transitions in estimating low-rank matricesAbstract: Low-rank perturbations of Wigner matrices have been extensively studied in random matrix theory; much information about the corresponding spectral phase transition can be gleaned using these tools. In this talk, I will consider an alternative viewpoint based on tools from spin glass theory, and two examples that illustrate how these they yield information beyond traditional spectral tools. The first example is the stochastic block model,where we obtain a full information-theoretic picture of estimation. The second example demonstrates how side information alters the spectral threshold. It involves a new phase transition that interpolates between the Wigner and Wishart ensembles. Date Name Title/Abstract 9-27-17 Herbert Spohn, Technische Universität München Hydrodynamics of integrable classical and quantum systems Abstract: In the cold atoms community there is great interest in developing Euler-type hydrodynamics for one-dimensional integrable quantum systems, in particular with application to domain wall initial states. I provide some mathematical physics background and also compare with integrable classical systems.
10-23-17 *12:00-1:00pm, Science Center 232*
Madhu Sudan, Harvard SEAS General Strong Polarization A recent discovery (circa 2008) in information theory called Polar Coding has led to a remarkable construction of error-correcting codes and decoding algorithms, resolving one of the fundamental algorithmic challenges in the field. The underlying phenomenon studies the “polarization” of a “bounded” martingale. A bounded martingale, X_0,…,X_t,… is one where X_t in [0,1]. This martingale is said to polarize if Pr[lim_{t\to infty} X_t \in {0,1}] = 1. The questions of interest to the results in coding are the rate of convergence and proximity: Specifically, given epsilon and tau > 0 what is the smallest t after which it is the case that Pr[X_t in (tau,1-tau)] < epsilon? For the main theorem, it was crucial that t <= min{O(log(1/epsilon)), o(log(1/tau))}. We say that a martingale polarizes strongly if it satisfies this requirement. We give a simple local criterion on the evolution of the martingale that suffices for strong polarization. A consequence to coding theory is that a broad class of constructions of polar codes can be used to resolve the afore-mentioned algorithmic challenge.
In this talk I will introduce the concepts of polarization and strong polarization. Depending on the audience interest I can explain why this concept is useful to construct codes and decoding algorithms, or explain the local criteria that help establish strong polarization (and the proof of why it does so).
Based on joint work with Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, and Atri Rudra.
10-25-17 *2:00-4:00pm*
Subhabrata Sen (Microsoft and MIT) Noga Alon,(Tel Aviv University)
Subhabrata Sen, “Partitioning sparse random graphs: connections with mean-field spin glasses” Abstract: The study of graph-partition problems such as Maxcut, max-bisection and min-bisection have a long and rich history in combinatorics and theoretical computer science. A recent line of work studies these problems on sparse random graphs, via a connection with mean field spin glasses. In this talk, we will look at this general direction, and derive sharp comparison inequalities between cut-sizes on sparse Erd\ ̋{o}s-R\'{e}nyi and random regular graphs.
Based on joint work with Aukosh Jagannath.
Noga Alon, “Random Cayley Graphs”
Abstract: The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators. Several intriguing questions that remain open will be mentioned as well.
11-1-17 *2:00-4:00pm*
Kay Kirkpatrick (Illinois) and
Wei-Ming Wang (CNRS)
Kay Kirkpatrick, Quantum groups, Free Araki-Woods Factors, and a Calculus for Moments Abstract: We will discuss a central limit theorem for quantum groups: that the joint distributions with respect to the Haar state of the generators of free orthogonal quantum groups converge to free families of generalized circular elements in the large (quantum) dimension limit. We also discuss a connection to free Araki-Woods factors, and cases where we have surprisingly good rates of convergence. This is joint work with Michael Brannan. Time permitting, we’ll mention another quantum central limit theorem for Bose-Einstein condensation and work in progress.
Wei-Min Wang, Quasi-periodic solutions to nonlinear PDE’s
Abstract: We present a new approach to the existence of time quasi-periodic solutions to nonlinear PDE’s. It is based on the method of Anderson localization, harmonic analysis and algebraic analysis. This can be viewed as an infinite dimensional analogue of a Lagrangian approach to KAM theory, as suggested by J. Moser.
11-8-17 Elchanan Mossel Optimal Gaussian Partitions. Abstract: How should we partition the Gaussian space into k parts in a way that minimizes Gaussian surface area, maximize correlation or simulate a specific distribution.
The problem of Gaussian partitions was studied since the 70s first as a generalization of the isoperimetric problem in the context of the heat equation. It found a renewed interest in context of the double bubble theorem proven in geometric measure theory and due to connection to problems in theoretical computer science and social choice theory.
I will survey the little we know about this problem and the major open problems in the area.
11-10-17 *12pm SC 232*
Zhe Wang (NYU) A Driven Tagged Particle in One-dimensional Simple Exclusion Process Abstract: We study the long-time behavior of a driven tagged particle in a one-dimensional non-nearest- neighbor simple exclusion process. We will discuss two scenarios when the tagged particle has a speed. Particularly, for the ASEP, the tagged particle can have a positive speed even when it has a drift with negative mean; for the SSEP with removals, we can compute the speed explicitly. We will characterize some nontrivial invariant measures of the environment process by using coupling arguments and color schemes.
11-15-17 *4:00-5:00pm*
*G02*
Daniel Sussman (BU) Multiple Network Inference: From Joint Embeddings to Graph Matching Abstract: Statistical theory, computational methods, and empirical evidence abound for the study of individual networks. However, extending these ideas to the multiple-network framework remains a relatively under-explored area. Individuals today interact with each other through numerous modalities including online social networks, telecommunications, face-to-face interactions, financial transactions, and the sharing and distribution of goods and services. Individually these networks may hide important activities that are only revealed when the networks are studied jointly. In this talk, we’ll explore statistical and computational methods to study multiple networks, including a tool to borrow strength across networks via joint embeddings and a tool to confront the challenges of entity resolution across networks via graph matching.
11-20-17 *Monday
12:00-1:00pm*
Yue M. Lu (Harvard)
Asymptotic Methods for High-Dimensional Inference: Precise Analysis, Fundamental Limits, and Optimal DesignsAbstract: Extracting meaningful information from the large datasets being compiled by our society presents challenges and opportunities to signal and information processing research. On the one hand, many classical methods, and the assumptions they are based on, are simply not designed to handle the explosive growth of the dimensionality of the modern datasets. On the other hand, the increasing dimensionality offers many benefits: in particular, the very high-dimensional settings allow one to apply powerful asymptotic methods from probability theory and statistical physics to obtain precise characterizations that would otherwise be too complicated in moderate dimensions. I will mention recent work on exploiting such blessings of dimensionality via sharp asymptotic methods. In particular, I will show (1) the exact characterization of a widely-used spectral method for nonconvex signal recoveries; (2) the fundamental limits of solving the phase retrieval problem via linear programming; and (3) how to use scaling and mean-field limits to analyze nonconvex optimization algorithms for high-dimensional inference and learning. In these problems, asymptotic methods not only clarify some of the fascinating phenomena that emerge with high-dimensional data, they also lead to optimal designs that significantly outperform commonly used heuristic choices.11-29-17 David Gamarink (MIT) (Arguably) Hard on Average Constraint Satisfaction Problems Abstract: Many combinatorial optimization problems defined on random instances such as random graphs, exhibit an apparent gap between the optimal value, which can be estimated by non-constructive means, and the best values achievable by fast (polynomial time) algorithms. Through a combined effort of mathematicians, computer scientists and statistical physicists, it became apparent that a potential and in some cases a provable obstruction for designing algorithms bridging this gap is an intricate geometry of nearly optimal solutions, in particular the presence of chaos and a certain Overlap Gap Property (OGP), which we will introduce in this talk. We will demonstrate how for many such problems, the onset of the OGP phase transition indeed nearly coincides with algorithmically hard regimes. Our examples will include the problem of finding a largest independent set of a graph, finding a largest cut in a random hypergrah, random NAE-K-SAT problem, the problem of finding a largest submatrix of a random matrix, and a high-dimensional sparse linear regression problem in statistics.
Joint work with Wei-Kuo Chen, Quan Li, Dmitry Panchenko, Mustazee Rahman, Madhu Sudan and Ilias Zadik.
12-6-17 *2:00-4:00pm*
Philippe Rigollet (MIT) 2-3 pm
&
Ankur Moitra (MIT)
3-4 pm
Philippe Rigollet (MIT), Exact Recovery in the Ising Block Model Abstract: Over the past fifteen years, the problem of learning Ising models from independent samples has been of significant interest in the statistics, machine learning, and statistical physics communities. Much of the effort has been directed towards finding algorithms with low computational cost for various restricted classes of models, primarily in the case where the interaction graph is sparse. In parallel, stochastic blockmodels have played a more and more preponderant role in community detection and clustering as an average case model for the minimum bisection model. In this talk, we introduce a new model, called Ising blockmodel for the community structure in an Ising model. It imposes a block structure on the interactions of a dense Ising model and can be viewed as a structured perturbation of the celebrated Curie-Weiss model. We show that interesting phase transitions arise in this model and leverage this probabilistic analysis to develop an algorithm based on semidefinite programming that recovers exactly the community structure when the sample size is large enough. We also prove that exact recovery of the block structure is actually impossible with fewer samples.
This is joint work with Quentin Berthet (University of Cambridge) and Piyush Srivastava (Tata Institute).
Ankur Moitra (MIT), A New Approach to Approximate Counting and Sampling
Abstract: Over the past sixty years, many remarkable connections have been made between statistical physics, probability, analysis and theoretical computer science through the study of approximate counting. While tight phase transitions are known for many problems with pairwise constraints, much less is known about problems with higher-order constraints.
Here we introduce a new approach for approximately counting and sampling in bounded degree systems. Our main result is an algorithm to approximately count the number of solutions to a CNF formula where the degree is exponential in the number of variables per clause. Our algorithm extends straightforwardly to approximate sampling, which shows that under Lovasz Local Lemma-like conditions, it is possible to generate a satisfying assignment approximately uniformly at random. In our setting, the solution space is not even connected and we introduce alternatives to the usual Markov chain paradigm.12-14-17 TBD Strongly Correlated Quantum Materials and High-Temperature Superconductors Series
In the 2020-2021 academic year, the CMSA will be hosting a lecture series on Strongly Correlated Materials and High Tc Superconductor. All talks will take place from 10:30-12:00pm ET virtually on Zoom.
Cuprate high-temperature superconductors are a classic quantum material system to demonstrate the beauty of “Emergence and Entanglement” in the quantum phases of matter. Merely by adding more holes into an antiferromagnetic insulator, several fascinating phases emerge, including a d-wave superconductor, a pseudo-gap metal, and strange metal. After intensive studies from experimental, theoretical, and numerical communities for more than three decades, remarkable progress has been made, but basic questions remain:
- What is the origin of the superconductivity? What are the relative contributions of electron-phonon coupling, spin fluctuations, or resonating-valence-bonds?
- How do we explain the pseudo-gap and the Fermi arc in the underdoped region above the critical temperature? Are they from some symmetry breaking order parameters, or do we need an unconventional picture involving fractionalization?
- Is the strange metal at optimal doping associated with a quantum critical point? And if so, what is the driving force of this phase transition?
The cuprate quantum materials have been a major source for many new concepts in modern condensed matter physics, such as quantum spin liquids, topological order, and non-Fermi liquids. In the coming years, it is clear that the study of the cuprates will continually motivate new concepts and development of new techniques. In this seminar series, we hope to accelerate this process by bringing together deeper conversations between experimental, theoretical, and numerical experts with different backgrounds and perspectives.
The Strongly Correlated Quantum Materials and High-Temperature Superconductors series is a part of the Quantum Matter in Mathematics and Physics seminar.
Seminar organizers: Juven Wang (Harvard CMSA) and Yahui Zhang (Harvard).
Scientific program advisors: Professor Subir Sachdev (Harvard), Professor Patrick Lee (MIT).
In order to learn how to attend this series, please fill out this form.
For more information, please contact Juven Wang (jw@cmsa.fas.harvard.edu) and Yahui Zhang (yahui_zhang@g.harvard.edu)
Spring 2022
April 20, 2022 | 11:30 – 1:00 pm ET
Harold Y. Hwang (Stanford University & SLAC National Accelerator Laboratory)
Title: Superconductivity in infinite-layer nickelates
Abstract: Since its discovery, unconventional superconductivity in cuprates has motivated the search for materials with analogous electronic or atomic structure. We have used soft chemistry approaches to synthesize superconducting infinite layer nickelates from their perovskite precursor phase. We will present the synthesis and transport properties of the nickelates, observation of a doping-dependent superconducting dome, and our current understanding of their electronic and magnetic structure.
February 3, 2022 | 11:30 – 1:00 pm ET
Lu Li (U Michigan)
Title: Quantum Oscillations of Electrical Resistivity in an Insulator
Abstract: In metals, orbital motions of conduction electrons are quantized in magnetic fields, which is manifested by quantum oscillations in electrical resistivity. This Landau quantization is generally absent in insulators, in which all the electrons are localized. Here we report a notable exception in an insulator — ytterbium dodecaboride (YbB12). The resistivity of YbB12, despite much larger than that of usual metals, exhibits profound quantum oscillations under intense magnetic fields. This unconventional oscillation is shown to arise from the insulating bulk instead of conducting surface states. The large effective masses indicate strong correlation effects between electrons. Our result is the first discovery of quantum oscillations in the electrical resistivity of a strongly correlated insulator and will bring crucial insight into understanding the ground state in gapped Kondo systems.
2020 – 2021
September 2, 2020 | 10:30am ET
Subir Sachdev (Harvard) Title: Metal-to-metal quantum phase transitions not described by symmetry-breaking orders
Abstract: Numerous experiments have explored the phases of the cuprates with increasing doping density p from the antiferromagnetic insulator. There is now strong evidence that the small p region is a novel phase of matter, often called the pseudogap metal, separated from conventional Fermi liquid at larger p by a quantum phase transition. Symmetry-breaking orders play a spectator role, at best, at this quantum phase transition. I will describe trial wavefunctions across this metal-metal transition employing hidden layers of ancilla qubits (proposed by Ya-Hui Zhang). Quantum fluctuations are described by a gauge theory of ghost fermions that carry neither spin nor charge. I will also
describe a separate approach to this transition in a t-J model with random exchange interactions in the limit of large dimensions. This approach leads to a partly solvable SYK-like critical theory of holons and spinons, and a linear in temperature resistivity from time reparameterization fluctuations. Near criticality, both approaches have in common emergent fractionalized excitations, and a significantly larger entropy than naively expected.September 23, 2020 | 10:30am ET
Subir Sachdev (Harvard) Title: Metal-to-metal quantum phase transitions not described by symmetry-breaking orders II
Abstract: In this second talk, I will focus on (nearly) solvable models of metal-metal transition in random systems. The t-J model with random and all-to-all hopping and exchange can be mapped onto a quantum impurity model coupled self-consistently to an environment (the mapping also applies to a t-J model in a large dimension lattice, with random nearest-neighbor exchange). Such models will be argued to exhibit metal-metal quantum phase transitions in the universality class of the SYK model, accompanied by a linear-in-T resistivity from time reparameterization fluctuations. I will also present the results of exact diagonalization of random t-J clusters, obtained recently with Henry Shackleton, Alexander Wietek, and Antoine Georges.
September 24, 2020 | 12:00pm ET
Inna Vishik (University of California, Davis)
Title: Universality vs materials-dependence in cuprates: ARPES studies of the model cuprate Hg1201Abstract: The cuprate superconductors exhibit the highest ambient-pressure superconducting transition temperatures (T c ), and after more than three decades of extraordinary research activity, continue to pose formidable scientific challenges. A major experimental obstacle has been to distinguish universal phenomena from materials- or technique-dependent ones. Angle-resolved photoemission spectroscopy (ARPES) measures momentum-dependent single-particle electronic excitations and has been invaluable in the endeavor to determine the anisotropic momentum-space properties of the cuprates. HgBa 2 CuO 4+d (Hg1201) is a single-layer cuprate with a particularly high optimal T c and a simple crystal structure; yet there exists little information from ARPES about the electronic properties of this model system. I will present recent ARPES studies of doping-, temperature-, and momentum-dependent systematics of near-nodal dispersion anomalies in Hg1201. The data reveal a hierarchy of three distinct energy scales which establish several universal phenomena, both in terms of connecting multiple experimental techniques for a single material, and in terms of connecting comparable spectral features in multiple structurally similar cuprates.VideoOctober 15, 2020 | 10:30am ET
Louis Taillefer (Université de Sherbrooke) Title: New signatures of the pseudogap phase of cuprate superconductors
Abstract: The pseudogap phase of cuprate superconductors is arguably the most enigmatic phase of quantum matter. We aim to shed new light on this phase by investigating the non- superconducting ground state of several cuprate materials at low temperature across a wide doping range, suppressing superconductivity with a magnetic field. Hall effect measurements across the pseudogap critical doping p* reveal a sharp drop in carrier density n from n = 1 + p above p* to n = p below p, signaling a major transformation of the Fermi surface. Angle-dependent magneto-resistance (ADMR) directly reveals a change in Fermi surface topology across p. From specific heat measurements, we observe the classic thermodynamic signatures of quantum criticality: the electronic specific heat C el shows a sharp peak at p, where it varies in temperature as C el ~ – T logT. At p and just above, the electrical resistivity is linear in T at low T, with an inelastic scattering rate that obeys the Planckian limit. Finally, the pseudogap phase is found to have a large negative thermal Hall conductivity, which extends to zero doping. We show that the pseudogap phase makes phonons become chiral. Understanding the mechanisms responsible for these various new signatures will help elucidate the nature of the pseudogap phase.
October 28, 2020 | 10:30am ET
Patrick Lee (MIT) Title: The not-so-normal normal state of underdoped Cuprate
Abstract: The underdoped Cuprate exhibits a rich variety of unusual properties that have been exposed after years of experimental investigations. They include a pseudo-gap near the anti-nodal points and “Fermi arcs” of gapless excitations, together with a variety of order such as charge order, nematicity and possibly loop currents and time reversal and inversion breaking. I shall argue that by making a single assumption of strong pair fluctuations at finite momentum (Pair density wave), a unified description of this phenomenology is possible. As an example, I will focus on a description of the ground state that emerges when superconductivity is suppressed by a magnetic field which supports small electron pockets. [Dai, Senthil, Lee, Phys Rev B101, 064502 (2020)] There is some support for the pair density wave hypothesis from STM data that found charge order at double the usual wave-vector in the vicinity of vortices, as well as evidence for a fragile form of superconductivity persisting to fields much above Hc2. I shall suggest a more direct experimental probe of the proposed fluctuating pair density wave.
November 6, 2020 |12:30pm ET
Zhi-Xun Shen (Stanford University) Title: Essential Ingredients for Superconductivity in Cupper Oxide Superconductors
Abstract: High‐temperature superconductivity in cupper oxides, with critical temperature well above what wasanticipated by the BCS theory, remains a major unsolved physics problem. The problem is fascinating because it is simultaneously simple ‐ being a single band and 1⁄2 spin system, yet extremely rich ‐ boasting d‐wave superconductivity, pseudogap, spin and charge orders, and strange metal phenomenology. For this reason, cuprates emerge as the most important model system for correlated electrons – stimulating conversations on the physics of Hubbard model, quantum critical point, Planckian metal and beyond.
Central to this debate is whether the Hubbard model, which is the natural starting point for the undoped
magnetic insulator, contains the essential ingredients for key physics in cuprates. In this talk, I will discuss our photoemission evidence for a multifaceted answer to this question [1‐3]. First, we show results that naturally points to the importance of Coulomb and magnetic interactions, including d‐wave superconducting gap structure [4], exchange energy (J) control of bandwidth in single‐hole dynamics [5]. Second, we evidence effects beyond the Hubbard model, including band dispersion anomalies at known phonon frequencies [6, 7], polaronic spectral lineshape and the emergence of quasiparticle with doping [8]. Third, we show properties likely of hybrid electronic and phononic origin, including the pseudogap [9‐11], and the almost vertical phase boundary near the critical 19% doping [12]. Fourth, we show examples of small q phononic coupling that cooperates with d‐wave superconductivity [13‐15]. Finally, we discuss recent experimental advance in synthesizing and investigating doped one‐dimensional (1D) cuprates [16]. As theoretical calculations of the 1D Hubbard model are reliable, a robust comparison can be carried out. The experiment reveals a near‐neighbor attractive interaction that is an order of magnitude larger than the attraction generated by spin‐superexchange in the Hubbard model. Addition of such an attractive term, likely of phononic origin, into the Hubbard model with canonical parameters provides a quantitative explanation for all important experimental observable: spinon and holon dispersions, and holon‐ holon attraction. Given the structural similarity of the materials, It is likely that an extended two‐dimensional
(2D) Hubbard model with such an attractive term, will connect the dots of the above four classes of
experimental observables and provide a holistic understanding of cuprates, including the elusive d‐wave superconductivity in 2D Hubbard model.[1] A. Damascelli, Z. Hussain, and Z.‐X. Shen, Review of Modern Physics, 75, 473 (2003)
[2] M. Hashimoto et al., Nature Physics 10, 483 (2014)
[3] JA Sobota, Y He, ZX Shen ‐ arXiv preprint arXiv:2008.02378, 2020; submitted to Rev. of Mod. Phys.
[4] Z.‐X. Shen et al., Phys. Rev. Lett. 70, 1553 (1993)
[5] B.O. Wells et al., Phys. Rev. Lett. 74, 964 (1995)
[6] A. Lanzara et al., Nature 412, 510 (2001)
[7] T. Cuk et al., Phys. Rev. Lett., 93, 117003 (2004)
[8] K.M. Shen et al., Phys. Rev. Lett., 93, 267002 (2004)
[9] D.M. King et al., J. of Phys. & Chem of Solids 56, 1865 (1995)
[10] D.S. Marshall et al., Phy. Rev. Lett. 76, 484 (1996)
[11] A.G. Loeser et al., Science 273, 325 (1996)
[12] S. Chen et al., Science, 366, 6469 (2019)
[13] T.P. Devereaux, T. Cuk, Z.X. Shen, N. Nagaosa, Phys. Rev. Lett., 93, 117004 (2004)
[14] S. Johnston et al., Phys. Rev. Lett. 108, 166404 (2012)
[15] Yu He et al., Science, 362, 62 (Oct. 2018)
[16] Z. Chen, Y. Wang et al., preprint, 2020November 12, 2020 |10:30am ET
Chandra Varma (Visting Professor, University of California, Berkeley.
Emeritus Distinguished Professor, University of California, Riverside.)Title: Loop-Current Order and Quantum-Criticality in CupratesThis talk is organized as follows:
1. Physical Principles leading to Loop-current order and quantum criticality as the central feature in the physics of Cuprates.
2. Summary of the essentially exact solution of the dissipative xy model for Loop-current fluctuations.
3. Quantitative comparison of theory for the quantum-criticality with a variety of experiments.
4. Topological decoration of loop-current order to understand ”Fermi-arcs” and small Fermi-surface magneto-oscillations.Time permitting,
(i) Quantitative theory and experiment for fluctuations leading to d-wave superconductivity.
(ii) Extensions to understand AFM quantum-criticality in heavy-fermions and Fe-based superconductors.
(iii) Problems.VideoNovember 18, 2020 |10:30am ET
Antoine Georges (Collège de France, Paris and Flatiron Institute, New York) Abstract: Simplified as it is, the Hubbard model embodies much of the complexity of the `strong correlation problem’ and has established itself as a paradigmatic model in the field. In this talk, I will argue that several key aspects of its physics in two dimensions can now be established beyond doubt, thanks to the development of controlled and accurate computational methods. These methods implement different and complementary points of view on the quantum many-body problem. Along with pushing forward each method, the community has recently embarked into a major effort to combine and critically compare these approaches, and in several instances a consistent picture of the physics has emerged as a result. I will review in this perspective our current understanding of the emergence of a pseudogap in both the weak and strong coupling regimes. I will present recent progress in understanding how the pseudogap phase may evolve into a stripe-dominated regime at low temperature, and briefly address the delicate question of the competition between stripes and superconductivity. I will also emphasize outstanding questions which are still open, such as the possibility of a Fermi surface reconstruction without symmetry breaking. Whenever possible, connections to the physics of cuprate superconductors will be made. If time permits, I may also address the question of Planckian transport and bad metallic transport at high temperature.
November 19, 2020 |10:30am ET
Eduardo Fradkin (University of Illinois at Urbana-Champaign) Title: Pair Density Waves and Intertwined Orders in High Tc Superconductors
Abstract: I will argue that the orders that are present in high temperature superconductors naturally arise with the same strength and are better regarded as intertwined rather than competing. I illustrate this concept in the context of the orders that are present in the pair-density-wave state and the phase diagrams that result from this analysis.November 25, 2020 |10:30am ET
Qimiao Si (Rice University) Title: Bad Metals and Electronic Orders – Nematicity from Iron Pnictides to Graphene Moiré Systems
Abstract: Strongly correlated electron systems often show bad-metal behavior, as operationally specified in terms of a resistivity at room temperature that reaches or exceeds the Mott-Ioffe-Regel limit. They display a rich landscape of electronic orders, which provide clues to the underlying microscopic physics. Iron-based superconductors present a striking case study, and have been the subject of extensive efforts during the past decade or so. They are well established to be bad metals, and their phase diagrams prominently feature various types of electronic orders that are essentially always accompanied by nematicity. In this talk, I will summarize these characteristic features and discuss our own efforts towards understanding the normal state through the lens of the electronic orders and their fluctuations. Implications for superconductivity will be briefly discussed. In the second part of the talk, I will consider the nematic correlations that have been observed in the graphene-based moiré narrow-band systems. I will present a theoretical study which demonstrates nematicity in a “fragile insulator”, predicts its persistence in the bad metal regime and provides an overall perspective on the phase diagram of these correlated systems.
December 2, 2020 |10:30am ET
Andrey Chubukov (University of Minnesota) Title: Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal
Abstract: I discuss the interplay between non-Fermi liquid behaviour and pairing near a quantum-critical point (QCP) in a metal. These tendencies are intertwined in the sense that both originate from the same interaction mediated by gapless fluctuations of a critical order parameter. The two tendencies compete because fermionic incoherence destroys the Cooper logarithm, while the pairing eliminates scattering at low energies and restores fermionic coherence. I discuss this physics for a class of models with an effective dynamical interaction V (Ω) ~1/|Ω|^γ (the γ-model). This model describes, in particular, the pairing at a 2D Ising-nematic critical point in (γ=1/3), a 2D antiferromagnetic critical point (γ=1/2) and the pairing by an Einstein phonon with vanishing dressed Debye frequency (γ=2). I argue the pairing wins, unless the pairing component of the interaction is artificially reduced, but because of fermionic incoherence in the normal state, the system develops a pseudogap, preformed pairs behaviour in the temperature range between the onset of the pairing at Tp and the onset of phase coherence at the actual superconducting Tc. The ratio Tc/Tp decreases with γ and vanishes at γ =2. I present two complementary arguments of why this happens. One is the softening of longitudinal gap fluctuations, which become gapless at γ =2. Another is the emergence of a 1D array of dynamical vortices, whose number diverges at γ =2. I argue that once the number of vortices becomes infinite, quasiparticle energies effectively get quantized and do not get re-arranged in the presence of a small phase variation. I show that a new non-superconducting ground state emerges at γ >2.December 9, 2020 |10:30am ET
David Hsieh (Caltech) Title: Signatures of anomalous symmetry breaking in the cuprates
Abstract: The temperature versus doping phase diagram of the cuprate high-Tc superconductors features an enigmatic pseudogap region whose microscopic origin remains a subject of intensive study. Experimentally resolving its symmetry properties is imperative for narrowing down the list of possible explanations. In this talk I will give an overview of how optical second harmonic generation (SHG) can be used as a sensitive probe of symmetry breaking, and recap the ways it has been used to solve outstanding problems in condensed matter physics. I will then describe how we have been applying SHG polarimetry and spectroscopy to interrogate the cuprate pseudogap. In particular, I will discuss our data on YBa2Cu3Oy [1], which show an order parameter-like increase in SHG intensity below the pseudogap temperature T* across a broad range of doping levels. I will then focus on our more recent results on a model parent cuprate Sr2CuO2Cl2 [2], where evidence of anomalous broken symmetries surprisingly also exists. Possible connections between these observations will be speculated upon.
[1] L. Zhao, C. A. Belvin, R. Liang, D. A. Bonn, W. N. Hardy, N. P. Armitage and D. Hsieh, “A global inversion-symmetry-broken phase inside the pseudogap region of YBa2Cu3Oy,” Nature Phys. 13, 250 (2017).[2] A. de la Torre, K. L. Seyler, L. Zhao, S. Di Matteo, M. S. Scheurer, Y. Li, B. Yu, M. Greven, S. Sachdev, M. R. Norman and D. Hsieh. “Anomalous mirror symmetry breaking in a model insulating cuprate Sr2CuO2Cl2,” Preprint at https://arxiv.org/abs/2008.06516
December 16, 2020 |10:30am ET
Zheng-Yu Weng (Tsinghua University) Title: Organizing Principle of Mottness and Complex Phenomenon in High Temperature Superconductors
Abstract: The complex phenomenon in the high-Tc cuprate calls for a microscopic understanding based on general principles. In this Lecture, an exact organizing principle for a typical doped Mott insulator will be presented, in which the fermion sign structure is drastically reduced to a mutual statistics. Its nature as a long-range spin-charge entanglement of many-body quantum mechanics will be exemplified by exact numerical calculations. The phase diagram of the cuprate may be unified in a “bottom-up” fashion by a “parent” ground state ansatz with hidden orders constructed based on the organizing principle. Here the pairing mechanism will go beyond the “RVB” picture and the superconducting state is of non-BCS nature with modified London equation and novel elementary excitations. In particular, the Bogoliubov/Landau quasiparticle excitation are emerging with a two-gap structure in the superconducting state and the Fermi arc in a pseudogap regime. A mathematic framework of fractionalization and duality transformation guided by the organizing principle will be introduced to describe the above emergent phenomenon.December 17, 2020 |10:30am ET
Steven Kivelson (Stanford University) Abstract: Despite the fact that papers submitted to glossy journals universally start by bemoaning the absence of theoretical understanding, I will argue that the answer to the title question is “quite a lot.” To focus the discussion, I will take the late P.W. Anderson’s “Last Words on the Cuprates” (arXiv:1612.03919) as a point of departure, although from a perspective that differs from his in many key points.
January 20, 2021 |10:30am ET
Thomas Peter Devereaux (Stanford University) Title: Numerical investigations of models of the cuprates
Abstract: Richard Feynman once said “Anyone who wants to analyze the properties of matter in a real problem might want to start by writing down the fundamental equations and then try to solve them mathematically. Although there are people who try to use such an approach, these people are the failures in this field. . . ”I will summarize efforts to solve microscopic models of the cuprates using quantum Monte Carlo and density matrix renormalization group computational methods, with emphasis on how far one can get before failing to describe the real materials. I will start with an overview of the quantum chemistry of the cuprates that guides our choices of models, and then I will discuss “phases” of these models, both realized and not. I will lastly discuss the transport properties of the models in the “not-so-normal” regions of the phase diagram.
February 3, 2021 |10:30am ET
Philip Phillips (University of Illinois Urbana-Champaign) Title: Beyond BCS: An Exact Model for Superconductivity and Mottness
Abstract: High-temperature superconductivity in the cuprates remains an unsolved problem because the cuprates start off their lives as Mott insulators in which no organizing principle such a Fermi surface can be invoked to treat the electron interactions. Consequently, it would be advantageous to solve even a toy model that exhibits both Mottness and superconductivity. Part of the problem is that the basic model for a Mott insulator, namely the Hubbard model is unsolvable in any dimension we really care about. To address this problem, I will start by focusing on the overlooked Z_2 emergent symmetry of a Fermi surface first noted by Anderson and Haldane. Mott insulators break this emergent symmetry. The simplest model of this type is due to Hatsugai/Kohmoto. I will argue that this model can be thought of a fixed point for Mottness. I will then show exactly[1] that this model when appended with a weak pairing interaction exhibits not only the analogue of Cooper’s instability but also a superconducting ground state, thereby demonstrating that a model for a doped Mott insulator can exhibit superconductivity. The properties of the superconducting state differ drastically from that of the standard BCS theory. The elementary excitations of this superconductor are not linear combinations of particle and hole states but rather are superpositions of doublons and holons, composite excitations signaling that the superconducting ground state of the doped Mott insulator inherits the non-Fermi liquid character of the normal state. Additional unexpected features of this model are that it exhibits a superconductivity-induced transfer of spectral weight from high to low energies and a suppression of the superfluid density as seen in the cuprates.
[1] PWP, L. Yeo, E. Huang, Nature Physics, 16, 1175-1180 (2020).February 10, 2021 |10:30am ET
Senthil Todadri (MIT) Abstract: The strange metal regime is one of the most prominent features of the cuprate phase diagram but yet has remained amongst the most mysterious. Seemingly similar metallic behavior is seen in a few other metals. In this talk, I will discuss, in great generality, some properties of `strange metals’ in an ideal clean system. I will discuss general constraints[1] on the emergent low energy symmetries of any such strange metal. These constraints may be viewed as a generalization of the Luttinger theorem of ordinary Fermi liquids. Many, if not all, non-Fermi liquids will have the same realization of emergent symmetry as a Fermi liquid (even though they could have very different dynamics). Such phases – dubbed ersatz Fermi liquids – share some (but not all) universal properties with Fermi liquids. I will discuss the implications for understanding the strange metal physics observed in experiments . Combined with a few experimental observations, I will show that these general model-independent considerations lead to concrete predictions[2] about a class of strange metals. The most striking of these is a divergent susceptibility of an observable that has the same symmetries as the loop current order parameter.
[1]. Dominic Else, Ryan Thorngren, T. Senthil, https://arxiv.org/abs/2007.07896
[2]. Dominic Else, T. Senthil, https://arxiv.org/abs/2010.10523April 1, 2021 |9:00am ET
Naoto Nagaosa (University of Tokyo) Title: Applied physics of high-Tc theories
Abstract: Since the discovery of high temperature superconductors in cuprates in 1986, many theoretical ideas have been proposed which have enriched condensed matter theory. Especially, the resonating valence bond (RVB) state for (doped) spin liquids is one of the most fruitful idea. In this talk, I would like to describe the development of RVB idea to broader class of materials, especially more conventional magnets. It is related to the noncollinear spin structures with spin chirality and associated quantal Berry phase applied to many phenomena and spintronics applications. It includes the (quantum) anomalous Hall effect, spin Hall effect, topological insulator, multiferroics, various topological spin textures, e.g., skyrmions, and nonlinear optics. I will show that even though the phenomena are extensive, the basic idea is rather simple and common in all of these topics.
April 22, 2021 |10:30am ET
Dung-Hai Lee (UC Berkeley) Title: “Non-abelian bosonization in two and three spatial dimensions and some applications”
Abstract: In this talk, we generalize Witten’s non-abelian bosonization in $(1+1)$-D to two and three spatial dimensions. Our theory applies to fermions with relativistic dispersion. The bosonized theories are non-linear sigma models with level-1 Wess-Zumino-Witten terms. As applications, we apply the bosonization results to the $SU(2)$ gauge theory of the $\pi$ flux mean-field theory of half-filled Hubbard model, critical spin liquids of “bipartite-Mott insulators” in 1,2,3 spatial dimensions, and twisted bilayer graphene.
May 12, 2021 |10:30am ET
André-Marie Tremblay (Université de Sherbrooke) Title: A unified theoretical perspective on the cuprate phase diagram
Abstract: Many features of the cuprate phase diagram are a challenge for the usual tools of solid state physics. I will show how a perspective that takes into account both the localized and delocalized aspects of conduction electrons can explain, at least qualitatively, many of these features. More specifically, I will show that the work of several groups using cluster extensions of dynamical mean-field theory sheds light on the pseudogap, on the quantum-critical point and on d-wave superconductivity. I will argue that the charge transfer gap and oxygen hole content are the best indicators of strong superconductivity and that many observations are a signature of the influence of Mott physics away from half-filling. I will also briefly comment on what information theoretic measures tell us about this problem.
August 11, 2021 |10:30am ET
Piers Coleman (Rutgers) Title: Order Fractionalization*
Abstract: I will discuss the interplay of spin fractionalization with broken
symmetry. When a spin fractionalizes into a fermion, the resulting particle
can hybridize or pair with the mobile electrons to develop a new kind of
fractional order parameter. The concept of “order fractionalization” enables
us to extend the concept of off-diagonal order to encompass the formation of
such order parameters with fractional quantum numbers, such as spinorial
order[1].
A beautiful illustration of this phenomenon is provided by a model
which incorporates the Yao-Lee-Kitaev model into a Kondo lattice[2]. This
model explicitly exhibits order fractionalization and is expected to undergo a
discrete Ising phase transition at finite temperature into an
order-fractionalized phase with gapless Majorana excitations.
The broader implications of these considerations for Quantum
Materials and Quantum Field Theory will be discussed.
Work done in collaboration with Yashar Komijani, Anna Toth and Alexei
Tsvelik.
[1] Order Fractionalization, Yashar Komijani, Anna Toth, Premala Chandra, Piers Coleman, (2018).
[2] Order Fractionalization in a Kitaev Kondo model, Alexei Tsvelik and Piers Coleman, (2021).September 15, 2021 |10:30am ET
Liang Fu (MIT) Title: Three-particle mechanism for pairing and superconductivity
Abstract: I will present a new mechanism and an exact theory of electron pairing due to repulsive interaction in doped insulators. When the kinetic energy is small, the dynamics of adjacent electrons on the lattice is strongly correlated. By developing a controlled kinetic energy expansion, I will show that two doped charges can attract and form a bound state, despite and because of the underlying repulsion. This attraction by repulsion is enabled by the virtual excitation of a third electron in the filled band. This three-particle pairing mechanism leads to a variety of novel phenomena at finite doping, including spin-triplet superconductivity, pair density wave, BCS-BEC crossover and Feshbach resonance involving “trimers”. Possible realizations in moire materials, ZrNCl and WTe2 will be discussed.
[1] V. Crepel and L. Fu, Science Advances 7, eabh2233 (2021)
[2] V. Crepel and L. Fu, arXiv:2103.12060
[3] K. Slagle and L. Fu, Phys. Rev. B 102, 235423 (2020)September 29, 2021 |11:30am ET (special time)
Nai Phuan Ong (Princeton University)
Title:.Abstract: The layered honeycomb magnet alpha-RuCl3 orders below 7 K in a zigzag phase in zero field. An in-plane magnetic field H||a suppresses the zigzag order at 7 Tesla, leaving a spin-disordered phase widely believed to be a quantum spin liquid (QSL) that extends to ~12 T. We have observed oscillations in the longitudinal thermal conductivity Kxx vs. H from 0.4 to 4 K. The oscillations are periodic in 1/H (with a break-in-slope at 7 T). The amplitude function is maximal in the QSL phase (7 –11.5 T). I will describe a benchmark for crystalline disorder, the reproducibility and intrinsic nature of the oscillations, and discuss implications for the QSL state. I will also show detailed data on the thermal Hall conductivity Kxy measured from 0.4 K to 10 K and comment on recent half-quantization results.*Czajka et al., Nature Physics 17, 915 (2021).Collaborators: Czajka, Gao, Hirschberger, Lampen Kelley, Banerjee, Yan, Mandrus and Nagler.Date TBA |10:30am ET
Suchitra Sebastian (University of Cambridge) Title: TBA
Date TBA |10:30am ET
Jenny Hoffman (Harvard University) Title: TBA
Exact symmetries and threshold states in two-dimensional models for QCD
Speaker: Silviu Pufu (Princeton University)
Title: Exact symmetries and threshold states in two-dimensional models for QCD
Abstract: Two-dimensional QCD models form an interesting playground for studying phenomena such as confinement and screening. In this talk I will describe one such model, namely a 2d SU(N) gauge theory with an adjoint and a fundamental fermion, and explain how to compute the spectrum of bound states using discretized light-cone quantization at large N. Surprisingly, the spectrum of the discretized theory exhibits a large number of exact degeneracies, for which I will provide two different explanations. I will also discuss how these degeneracies provide a physical picture of confinement in 2d QCD with just a massless adjoint fermion. This talk is based on joint work with R. Dempsey and I. Klebanov.
Swampland Seminar Series
During the 2021-22 academic year, the CMSA will be co-hosting a seminar on Swampland, with the Harvard Physics Department, organized by Miguel Montero, Cumrun Vafa, Irene Valenzuela. This seminar is a part of the Swampland Program. This seminar will take place on Mondays at 10:00 am – 11:30 am (Boston time). To learn how to attend, please subscribe here.
Talks will be posted on the Swampland Seminars YouTube channel. The schedule below will be updated as talks are confirmed.
Spring 2022
Date Speaker Title/Abstract 1/31/2022 Rafael Álvarez-García (DESY Hamburg) Title: Membrane Limits in Quantum Gravity 2/7/2022 Du Pei (Harvard CMSA) Title: Holomorphic CFTs and topological modular forms Abstract: The theory of topological modular forms leads to many interesting constraints and predictions for two-dimensional quantum field theories, and some of them might have interesting implications for the swampland program. In this talk, I will show that a conjecture by Segal, Stolz and Teichner requires the constant term of the partition function of a bosonic holomorphic CFTs to be divisible by specific integers determined by the central charge. We verify this constraint in large classes of physical examples, and rule out the existence of an infinite set of “extremal CFTs”, including those with central charges c = 48, 72, 96 and 120.
2/28/2022 Tom Rudelius (UC, Berkeley) Title: Generalized Global Symmetries and the Weak Gravity Conjecture 3/7/2022 Fernando Marchesano (UAM-CSIC, Madrid) and Max Wiesner (Harvard CMSA) Title: 4d strings at strong coupling 3/21/2022 Patrick Draper (Univ. of Illinois) and Alvaro Herraez (IPhT Saclay). Open Mic Discussion
Topic: Entropy bounds (species bound, Bekenstein bound, CKN bound, and the like)3/28/2022 Fernando Quevedo (Cambridge) Title: On renormalisation group induced moduli stabilisation and brane-antibrane inflation Abstract: A proposal to use the renormalisation group to address moduli stabilisation in IIB string perturbation theory will be described. We revisit brane-antibrane inflation combining this proposal with non-linearly realised supersymmetry.
4/5/2022 Simon Caron-Huot (McGill University) and Julio Parra (Caltech) Title: Causality constraints on corrections to Einstein gravity Abstract: We study constraints from causality and unitarity on 2→2 graviton scattering in four-dimensional weakly-coupled effective field theories. Together, causality and unitarity imply dispersion relations that connect low-energy observables to high-energy data. Using such dispersion relations, we derive two-sided bounds on gravitational Wilson coefficients in terms of the mass M of new higher-spin states. Our bounds imply that gravitational interactions must shut off uniformly in the limit G→0, and prove the scaling with M expected from dimensional analysis (up to an infrared logarithm). We speculate that causality, together with the non-observation of gravitationally-coupled higher-spin states at colliders, severely restricts modifications to Einstein gravity that could be probed by experiments in the near future.
4/11/2022 Timm Wrase and Eduardo Gonzalo (Lehigh) Title: Type IIB flux compactifications with $h^{1,1}=0$ Abstract: We revisit type IIB flux compactification that are mirror dual to type IIA on rigid Calabi-Yau manifolds. We find a variety of interesting new solutions, like fully stabilized Minkowski vacua and infinite families of AdS$_4$ solutions with arbitrarily large numbers of spacetime filling D3 branes. We discuss how these solutions fit into the web of swampland conjectures.
4/18/2022 José Calderón (IFT Madrid) Open mic Swampland Discussion Topic: Cobordism
5/9/2022 Georges Obie (Harvard) Title: Inflation and light Dark Matter constraints from the Swampland Abstract: I will explore the interplay between Swampland conjectures and models of inflation and light Dark Matter. To that end, I will briefly review the weak gravity conjecture (WGC) and the related Festina Lente (FL) bound. These have implications for light darkly and milli-charged particles and can disfavor a large portion of parameter space. The FL bound also implies strong restrictions on the field content of our universe during inflation and presents an opportunity for inflationary model building. At the same time, it rules out some popular models like chromo-natural inflation and gauge-flation. Finally, I will review another Swampland conjecture related to Stückelberg photon masses and discuss its implications for astro-particle physics.
Fall 2021
Date Speaker Title/Abstract 9/13/2021 John Stout (Harvard) Title: Decoding Divergent Distances Abstract: Motivated by a relationship between the Zamolodchikov and NLSM metrics to the so-called quantum information metric, I will discuss recent work (2106.11313) on understanding infinite distance limits within the context of information theory. I will describe how infinite distance points represent theories that are hyper-distinguishable, in the sense that they can be distinguished from “nearby” theories with certainty in relatively few measurements. I will then discuss necessary and sufficient ingredients for the appearance of these infinite distance points, illustrate these in simple examples, and describe how this perspective can help the swampland program.
9/20/2021 Manki Kim (MIT) Title: Small Cosmological Constants in String Theory Abstract: We construct supersymmetric AdS4 vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by Kachru, Kallosh, Linde, and Trivedi. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the α 0 expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude < 10^{-123} in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol.
9/27/2021 Eran Palti (Ben Gurion) Title: Convexity of Charged Operators in CFTs and the Weak Gravity Conjecture Abstract: In this talk I will introduce a particular formulation of the Weak Gravity Conjecture in AdS space in terms of the self-binding energy of a particle. The holographic CFT dual of this formulation corresponds to a certain convex-like structure for operators charged under continuous global symmetries. Motivated by this, we propose a conjecture that this convexity is a general property of all CFTs, not just those with weakly-curved gravitational duals. It is possible to test this in simple CFTs, the conjecture passes all the tests performed so far.
10/18/2021 Thomas Van Riet (KU Leuven) Title: The Festina Lente Bound Abstract: I will explain what the Festina Lente bound means and where it comes from. Then I discuss its possible implications for phenomenology, both top-down and bottom-up.
10/25/2021 Joe Conlon (Oxford) Title: Exploring the Holographic Swampland Abstract: I describe our work looking at `traditional’ scenarios of moduli stabilisation from a holographic perspective. This reveals some interesting structure that is not apparent from the top-down perspective. For vacua in the extreme regions of moduli space, such as LVS in type IIB or the DGKT flux vacua in type IIA, the dual moduli conformal dimensions reduce to fixed values – in a certain sense, the low-conformal dimension part of the CFT is unique and independent of the large number of flux choices. For the DGKT flux vacua these conformal dimensions are also integer, for reasons we do not understand.
11/01/2021 Pieter Bomans (Princeton) Title: Bubble instability of mIIA on AdS_4 x S^6 Abstract: Recently, a set of non-supersymmetric AdS_4 vacua of massive type IIA string theory has been constructed. These vacua are perturbatively stable with respect to the full KK spectrum of type mIIA supergravity and furthermore, they are stable against a variety of non-perturbative decay channels. Hence, at this point, they represent a serious challenge to the AdS swampland conjecture. In my talk, I will review in detail the construction of these vacua as well as introduce a new decay channel, ultimately sealing their fate as being unstable.
11/15/2021 Nima Arkani-Hamed (IAS), and Gary Shiu (UW-Madison) This week’s seminar will be an open mic discussion which will be led by Nima Arkani-Hamed (IAS), and by Gary Shiu (UW-Madison), and the topic will be “Swampland constraints, Unitarity and Causality”. They will start with a brief introduction sharing their thoughts about the topic and moderate a discussion afterwards. 11/22/2021 Thomas Grimm (Utrecht University) Title: Taming the Landscape Abstract: In this talk I will introduce a generalized notion of finiteness that provides a structural principle for the set of effective theories that can be consistently coupled to quantum gravity. More concretely, I will propose a ‘tameness conjecture’ that states that all scalar field spaces and coupling functions that appear in such an effective theory must be definable in an o-minimal structure. The fascinating field of tame geometry has seen much recent progress and I will argue that the results can be used to support the above swampland conjecture. The strongest evidence arises from a new finiteness theorem for the flux landscape which is shown using the tameness of the period map.
11/29/2021 Timm Wrase (Lehigh University) Title: Scale separated AdS vacua? Abstract: In this talk I will review massive type IIA flux compactifications that seem to give rise to infinite families of supersymmetric 4d AdS vacua. These vacua provide an interesting testing ground for the swampland program. After reviewing potential shortcomings of this setup, I will discuss recent progress on overcoming them and getting a better understanding of these solutions.
12/6/2021 Lars Aalsma (University of Wisconsin-Madison) Title: Extremal Black Hole Corrections from Iyer-Wald Abstract: Extremal black holes play a key role in our understanding of various swampland conjectures and in particular the WGC. The mild form of the WGC states that higher-derivative corrections should decrease the mass of extremal black holes at fixed charge. Whether or not this conjecture is satisfied depends on the sign of the combination of Wilson coefficients that control corrections to extremality. Typically, corrections to extremality need to be computed on a case-by-case basis, but in this talk I will present a universal derivation of extremal black hole corrections using the Iyer-Wald formalism. This leads to a formula that expresses general corrections to the extremality bound in terms of the stress tensor of the perturbations under consideration, clarifying the relation between the WGC and energy conditions. This shows that a necessary condition for the mild form of the WGC to be satisfied is a violation of the Dominant Energy Condition. This talk is based on 2111.04201.
Global Anomalies on the Hilbert Space
Speaker: Jaume Gomis (Perimeter PI)
Title: Global Anomalies on the Hilbert Space
Abstract: We will discuss an elementary way of detecting some global anomalies from the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory, give a physical description of the imprint of the “layers” that enter in the cobordism classification of anomalies and discuss applications, including how anomalies can imply a supersymmetric spectrum in strongly coupled (nonsupersymmetric) gauge theories.
2/18/2021 Quantum Matter Seminar
Speaker: Xiao-Gang Wen (MIT)
Title: A solution to the chiral fermion problem
Abstract: Motivated by the relation between anomaly and topological/SPT order in one higher dimension, we propose a solution to the chiral fermion problem. In particular, we find several sufficient conditions, such that a chiral fermion field theory can be regularized by an interacting lattice model in the same dimension. We also discuss some related issues, such as mass without mass term, and why ‘topological’ phase transitions are usually not “topological” phase transitions.
Hybrid Fracton Orders
Nathanan Tantivasadakarn (Harvard) Title: Hybrid Fracton Orders Abstract: I will introduce a family of gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders called “Hybrid Fracton Orders”. First, I will present the simplest example of such an order: the “Hybrid X-cube” model, where excitations can be labeled identically to those of the Z2 toric code tensored with the Z2 X-cube model, but exhibit fusion and braiding properties between the two sets of excitations. Next, I will provide a general construction of hybrid fracton orders which inputs a finite group G and an abelian normal subgroup N and produces an exactly solvable model. Such order can host non-abelian fracton excitations when G is non-abelian. Furthermore, the mobilities of a general excitation is dictated by the choice of N, from which by varying, one can view as “interpolating” between a pure 3D topological order and a pure fracton order.
Based on 2102.09555 and 2106.03842
The nu=5/2 enigma: Recent insights from theory and experiment
peaker: Ady Stern & David Mross (Weizmann)
Speaker: Ady Stern & David Mross (Weizmann
Title: The nu=5/2 enigma: Recent insights from theory and experiment
Abstract: Non-Abelian phases of matter have long inspired quantum physicists across various disciplines. The strongest experimental evidence of such a phase arises in quantum Hall systems at the filling factor 5/2 but conflicts with decades of numerical works. We will briefly introduce the 5/2 plateau and explain some of the key obstacles to identifying its topological order. We will then describe recent experimental and theoretical progress, including a proposal for resolving the 5/2 enigma based on electrical conductance measurements.
General Relativity 2021-22
During the 2021–22 academic year, the CMSA will be hosting a seminar on General Relativity, organized by Aghil Alaee, Jue Liu, Daniel Kapec, and Puskar Mondal. This seminar will take place on Thursdays at 9:30am – 10:30am (Eastern time). The meetings will take place virtually on Zoom. To learn how to attend, please fill out this form.
The schedule below will be updated as talks are confirmed.
Spring 2022
Date Speaker Title/Abstract 2/10/2022 Tin Yau Tsang (UC Irvine) Title: Dihedral ridigity and mass Abstract: To characterise scalar curvature, Gromov proposed the dihedral rigidity conjecture which states that a positively curved polyhedron having dihedral angles less than those of a corresponding flat polyhedron should be isometric to a flat one. In this talk, we will discuss some recent progress on this conjecture and its connection with general relativity (ADM mass and quasilocal mass).
2/17/2022 Shiraz Minwalla
(Tata Institute of Fundamental Research, Mumbai)Title: Black Hole dynamics at Large D Abstract: I demonstrate that black hole dynamics simplifies – without trivializing – in the limit in which the number of spacetime dimensions D in which the black holes live is taken to infinity. In the strict large D limit and under certain conditions I show the equations that govern black hole dynamics reduce to the equations describing the dynamics of a non gravitational membrane propagating in an unperturbed spacetime (e.g. flat space). In the stationary limit black hole thermodynamics maps to membrane thermodynamics, which we formulate in a precise manner. We also demonstrate that the large D black hole membrane agrees with the fluid gravity map in the appropriate regime.
2/24/2022 Achilleas Porfyriadis
(Harvard Black Hole Initiative)Title: Extreme Black Holes: Anabasis and Accidental Symmetry Abstract: The near-horizon region of black holes near extremality is universally AdS_2-like. In this talk I will concentrate on the simplest example of AdS_2 x S^2 as the near-horizon of (near-)extreme Reissner-Nordstrom. I will first explain the SL(2) transformation properties of the spherically symmetric linear perturbations of
AdS_2 x S^2 and show how their backreaction leads to the Reissner-Nordstrom black hole. This backreaction with boundary condition change is called an anabasis. I will then show that the linear Einstein equation near AdS_2 x S^2, with or without additional matter, enjoys an accidental symmetry that may be thought of as an on-shell large diffeomorphism of AdS_2.3/10/2022 David Fajman (University of Vienna) Title: The Einstein-flow on manifolds of negative curvature
Abstract: We consider the Cauchy problem for the Einstein equations for cosmological spacetimes, i.e. spacetimes with compact spatial hypersurfaces. Various classes of those dynamical spacetimes have been constructed and analyzed using CMC foliations or equivalently the CMC-Einstein flow. We will briefly review the Andersson-Moncrief stability result of negative Einstein metrics under the vacuum Einstein flow and then present various recent generalizations to the nonvacuum case. We will emphasize what difficulties arise in those generalizations, how they can be handled depending on the matter model at hand, and what implications we can draw from these results for cosmology. We then turn to a scenario where the CMC Einstein flow leads to a large data result in 2+1-dimensions.3/21/2022 Prof. Arick Shao (Queen Mary University of London) Title: Bulk-boundary correspondence for vacuum asymptotically Anti-de Sitter spacetimes Abstract: The AdS/CFT conjecture in physics posits the existence of a correspondence between gravitational theories in asymptotically Anti-de Sitter (aAdS) spacetimes and field theories on their conformal boundary. In this presentation, we prove rigorous mathematical statements toward this conjecture.
In particular, we show there is a one-to-one correspondence between aAdS solutions of the Einstein-vacuum equations and a suitable space of data on the conformal boundary (consisting of the boundary metric and the boundary stress-energy tensor). We also discuss consequences of this result, as well as the main ingredient behind its proof: a unique continuation property for wave equations on aAdS spacetimes.
This is joint work with Gustav Holzegel (and makes use of joint works with Alex McGill and Athanasios Chatzikaleas).
3/24/2022 Qian Wang, University of Oxford Title: Rough solutions of the $3$-D compressible Euler equations Abstract: I will talk about my work on the compressible Euler equations. We prove the local-in-time existence the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, mega) \in H^s\times H^s\times H^{s’}$, $2<s'<s$. The result extends the sharp result of Smith-Tataru and Wang, established in the irrotational case, i.e $mega=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e. with a constant density, the result is known to hold for $mega\in H^s$, $s>3/2$ and fails for $s\le 3/2$, see the work of Bourgain-Li. It is thus natural to conjecture that the optimal result should be $(v,\varrho, mega) \in H^s\times H^s\times H^{s’}$, $s>2, \, s’>\frac{3}{2}$. We view our work as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of a sound wave from the transported part and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.
3/28/2022 Emanuele Berti, Johns Hopkins University Title: Black Hole Spectroscopy Abstract: According to general relativity, the remnant of a binary black hole merger should be a perturbed Kerr black hole. Perturbed Kerr black holes emit “ringdown” radiation which is well described by a superposition of quasinormal modes, with frequencies and damping times that depend only on the mass and spin of the remnant. Therefore the observation of gravitational radiation emitted by black hole mergers might finally provide direct evidence of black holes with the same certainty as, say, the 21 cm line identifies interstellar hydrogen. I will review the current status of this “black hole spectroscopy” program. I will focus on two important open issues: (1) When is the waveform well described by linear black hole perturbation theory? (2) What is the current observational status of black hole spectroscopy?
4/7/2022 CMSA General Relativity Conference 4/14/2022 Chao Liu, Huazhong University of Science and Technology Title: Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n≥4 Abstract: In this talk, we briefly introduce our recent work on establishing the global existence and stability to the future of non-linear perturbation of de Sitter-like solutions to the Einstein-Yang-Mills system in n≥4 spacetime dimension. This generalizes Friedrich’s (1991) Einstein-Yang-Mills stability results in dimension n=4 to all higher dimensions. This is a joint work with Todd A. Oliynyk and Jinhua Wang.
4/21/2022 Jinhua Wang,
Xiamen UniversityTitle: Future stability of the $1+3$ Milne model for the Einstein-Klein-Gordon system Abstract: We study the small perturbations of the $1+3$-dimensional Milne model for the Einstein-Klein-Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on the $1+3$ splitting of the Bianchi-Klein-Gordon equations. Moreover, we treat the Bianchi-Klein-Gordon equations as evolution equations and establish the energy scheme in the sense that we only commute the Bianchi-Klein-Gordon equations with spatially covariant derivatives while normal derivative is not allowed. We propose some refined estimates for lapse and the hierarchies of energy estimates to close the energy argument.
4/28/2022 Allen Fang, Sorbonne University Title: A new proof for the nonlinear stability of slowly-rotating Kerr-de Sitter Abstract: The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vectorfield techniques to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.
Fall 2021
Date Speaker Title/Abstract 9/10/2021 (10:30am – 11:30am (Boston time)
Philippe G. LeFloch, Sorbonne University and CNRS Title: Asymptotic localization, massive fields, and gravitational singularities Abstract: I will review three recent developments on Einstein’s field equations under low decay or low regularity conditions. First, the Seed-to-Solution Method for Einstein’s constraint equations, introduced in collaboration with T.-C. Nguyen generates asymptotically Euclidean manifolds with the weakest or strongest possible decay (infinite ADM mass, Schwarzschild decay, etc.). The ‘asymptotic localization problem’ is also proposed an alternative to the ‘optimal localization problem’ by Carlotto and Schoen. We solve this new problem at the harmonic level of decay. Second, the Euclidian-Hyperboloidal Foliation Method, introduced in collaboration with Yue Ma, applies to nonlinear wave systems which need not be asymptotically invariant under Minkowski’s scaling field and to solutions with low decay in space. We established the global nonlinear stability of self-gravitating massive matter field in the regime near Minkowski spacetime. Third, in collaboration with Bruno Le Floch and Gabriele Veneziano, I studied spacetimes in the vicinity of singularity hypersurfaces and constructed bouncing cosmological spacetimes of big bang-big crunch type. The notion of singularity scattering map provides a flexible tool for formulating junction conditions and, by analyzing Einstein’s constraint equations, we established a surprising classification of all gravitational bouncing laws. Blog: philippelefloch.org
9/17/2021 (10:30am – 11:30am (Boston time)
Igor Rodnianski, Princeton University Title: Stable Big Bang formation for the Einstein equations Abstract: I will discuss recent work concerning stability of cosmological singularities described by the generalized Kasner solutions. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be stable under perturbations of the Kasner initial data, as long as the Kasner exponents are “sub-critical”. We prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the full regime where stable monotonic-type curvature blowup is expected. We treat the 3+1-dimensional Einstein-scalar field system and the D+1-dimensional Einstein-vacuum equations for D≥10. This is joint work with Speck and Fournodavlos.
9/24/2021 (10:30am – 11:30am (Boston time)
Alex Lupsasca Title: On the Observable Shape of Black Hole Photon Rings Abstract: The photon ring is a narrow ring-shaped feature, predicted by General Relativity but not yet observed, that appears on images of sources near a black hole. It is caused by extreme bending of light within a few Schwarzschild radii of the event horizon and provides a direct probe of the unstable bound photon orbits of the Kerr geometry. I will argue that the precise shape of the observable photon ring is remarkably insensitive to the astronomical source profile and can therefore be used as a stringent test of strong-field General Relativity. In practice, near-term interferometric observations may be limited to the visibility amplitude alone, which contains incomplete shape information: for convex curves, the amplitude only encodes the set of projected diameters (or “widths”) of the shape. I will describe the freedom in reconstructing a convex curve from its widths, giving insight into the photon ring shape information probed by technically plausible future astronomical measurements.
10/1/2021 (10:30am – 11:30am (Boston time)
Zhongshan An, University of Connecticut Title: Static vacuum extensions of Bartnik boundary data near flat domains Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on a joint work with Lan-Hsuan Huang.
10/8/2021 (10:30am – 11:30am (Boston time)
Xiaoning Wu, Chinese Academy of Sciences Title: Causality Comparison and Postive Mass Abstract: Penrose et al. investigated the physical incoherence of the space-time with negative mass via the bending of light. Precise estimates of the time-delay of null geodesics were needed and played a pivotal role in their proof. In this paper, we construct an intermediate diagonal metric and reduce this problem to a causality comparison in the compactified space-time regarding time-like connectedness near conformal infinities. This different approach allows us to avoid encountering the difficulties and subtle issues that Penrose et al. met. It provides a new, substantially simple, and physically natural non-partial differential equation viewpoint to understand the positive mass theorem. This elementary argument modestly applies to asymptotically flat solutions that are vacuum and stationary near infinity
10/15/2021 (10:30am – 11:30am (Boston time)
Jiong-Yue Li, Sun Yat-Sen University Title: Peeling properties of the spinor fields and the solutions to nonlinear Dirac equations Abstract: The Dirac equation is a relativistic equation that describes the spin-1/2 particles. We talk about Dirac equations in Minkowski spacetime. In a geometric viewpoint, we can see that the spinor fields satisfying the Dirac equations enjoy the so-called peeling properties. It means the null components of the solution will decay at different rates along the null hypersurface. Based on this decay mechanism, we can obtain a fresh insight to the spinor null forms which is used to prove a small data global existence result especially for some quadratic Dirac models.
10/22/2021 (11:00am – 12:30pm (Boston time)
Roberto Emparan, University of Barcelona Title: The Large D Limit of Einstein’s Equations Abstract: Taking the large dimension limit of Einstein’s equations is a useful strategy for solving and understanding the dynamics that these equations encode. I will introduce the underlying ideas and the progress that has resulted in recent years from this line of research. Most of the discussion will be classical in nature and will concern situations where there is a black hole horizon. A main highlight of this approach is the formulation of effective membrane theories of black hole dynamics. These have made possible to efficiently study, with relatively simple techniques, some of the thorniest problems in black hole physics, such as the non-linear evolution of the instabilities of black strings and black branes, and the collisions and mergers of higher-dimensional black holes. Open directions and opportunities will also be discussed. To get a flavor of what this is about, you may read the first few pages of the review (with C.P. Herzog) e-Print: 2003.11394.
10/28/2021 Jorge Santos, University of Cambridge Title: The classical interior of charged black holes with AdS asymptotics Abstract: The gravitational dual to the grand canonical ensemble of a large N holographic theory is a charged black hole. These spacetimes can have Cauchy horizons that render the classical gravitational dynamics of the black hole interior incomplete. We show that a (spatially uniform) deformation of the CFT by a neutral scalar operator generically leads to a black hole with no inner horizon. There is instead a spacelike Kasner singularity in the interior. For relevant deformations, Cauchy horizons never form. We then consider charged scalars, which are known to condense at low temperatures, thus providing a holographic realization of superconductivity. We look inside the horizon of these holographic superconductors and find intricate dynamical behavior. The spacetime ends at a spacelike Kasner singularity, and there is no Cauchy horizon. Before reaching the singularity, there are several intermediate regimes which we study both analytically and numerically. These include strong Josephson oscillations in the condensate and possible `Kasner inversions’ in which after many e-folds of expansion, the Einstein-Rosen bridge contracts towards the singularity. Due to the Josephson oscillations, the number of Kasner inversions depends very sensitively on temperature, and diverges at a discrete set of temperatures that accumulate at the critical temperature. Near this discrete set of temperatures, the final Kasner exponent exhibits fractal-like behavior.
11/4/2021
at 10 am ETElena Giorgi, Columbia University Title: The stability of charged black holes Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
11/11/2021
*9:30 am ET*Siyuan Ma, Sorbonne University Title: Sharp decay for Teukolsky equation in Kerr spacetimes Abstract: Teukolsky equation in Kerr spacetimes governs the dynamics of the spin $s$ components, $s=0, \pm 1, \pm 2$ corresponding to the scalar field, the Maxwell field, and the linearized gravity, respectively. I will discuss recent joint work with L. Zhang on proving the precise asymptotic profiles for these spin $s$ components in Schwarzschild and Kerr spacetimes.
11/19/2021 (10:30–11:30 am ET)
Nishanth Gudapati, Clark University Title: On Curvature Propagation and ‘Breakdown’ of the Einstein Equations on U(1) Symmetric Spacetimes Abstract: The analysis of global structure of the Einstein equations for general relativity, in the context of the initial value problem, is a difficult and intricate mathematical subject. Any additional structure in their formulation is welcome, in order to alleviate the problem. It is expected that the initial value problem of the Einstein equations on spacetimes admitting a translational, fixed-point free, spatial U(1) isometry group are globally well-posed. In our previous works, we discussed the special structure provided by the dimensional reduction of 3+1 dimensional U(1) symmetric Einstein equations to 2+1 Einstein-wave map system and demonstrated global existence in the equivariant case for large data. In this talk, after discussing some preliminaries and background, we shall discuss about yet another structure of the U(1) symmetric Einstein equations, namely the analogy with Yang-Mills theory via the Cartan formalism and reconcile with the dimensionally reduced field equations. We shall also discuss implications for ‘breakdown’ criteria of U(1) symmetric Einstein equations.
12/2/2021 Professor Geoffrey Comp
ére, Université Libre de BruxellesTitle: Kerr Geodesics and Self-consistent match between Inspiral and Transition-to-merger Abstract: The two-body motion in General Relativity can be solved perturbatively in the small mass ratio expansion. Kerr geodesics describe the leading order motion. After a short summary of the classification of polar and radial Kerr geodesic motion, I will consider the inspiral motion of a point particle around the Kerr black hole subjected to the self-force. I will describe its quasi-circular inspiral motion in the radiation timescale expansion. I will describe in parallel the transition-to-merger motion around the last stable circular orbit and prove that it is controlled by the Painlevé transcendental equation of the first kind. I will then prove that one can consistently match the two motions using the method of asymptotically matched expansions.
12/16/2021 Xinliang An, University of Singapore Title: Low regularity ill-posedness for 3D elastic waves and for 3D ideal compressible MHD driven by shock formation Abstract: We construct counterexamples to the local existence of low-regularity solutions to elastic wave equations and to the ideal compressible magnetohydrodynamics (MHD) system in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad’s classic results on the scalar wave equation by showing that the Cauchy problems for 3D elastic waves and for 3D MHD system are ill-posed in $H^3(R^3)$ and $H^2(R^3)$, respectively. Both elastic waves and MHD are physical systems with multiple wave speeds. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. In particular, when the magnetic field is absent in MHD, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proofs for elastic waves and for MHD are based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D elastic waves and the 3D ideal MHD equations into $6\times 6$ and $7\times 7$ non-strictly hyperbolic systems. Via detailed calculations, we reveal their hidden subtle structures. With them, we give a complete description of solutions’ dynamics up to the earliest singular event, when a shock forms. This talk is based on joint works with Haoyang Chen and Silu Yin.
2021 Summer Introduction to Mathematical Research
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithm, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited to 10 people, so don’t wait too long to apply.
CMSA Math-Science Literature Lecture: Birational geometry
Vyacheslav V. Shokurov (Johns Hopkins University)
Title: Birational geometry
Abstract: About main achievements in birational geometry during the last fifty years.
Talk chair: Caucher Birkar
CMSA Math-Science Literature Lecture: Nonlinear stability of Kerr black holes for small angular momentum
Sergiu Klainerman (Princeton University)
Title: Nonlinear stability of Kerr black holes for small angular momentum
Abstract: According to a well-known conjecture, initial data sets, for the Einstein vacuum equations, sufficiently close to a Kerr solution with parameters $a, m$, $|a|/m <1$, have maximal developments with complete future null infinity and with domain of outer communication (i.e complement of a future event horizon) which approaches (globally) a nearby Kerr solution. I will describe the main ideas in my recent joint work with Jeremie Szeftel concerning the resolution of the conjecture for small angular momentum, i.e. $, $|a|/m $ sufficiently small. The work, ArXiv:2104.11857v1, also depends on forthcoming work on solutions of nonlinear wave equations in realistic perturbations of Kerr, with Szeftel and Elena Giorgi, which I will also describe.
Talk chair: Lydia Bieri
CMSA Math-Science Literature Lecture: Black Hole Formation
Lydia Bieri (University of Michigan)
Title: Black Hole Formation
Abstract: Can black holes form through the focusing of gravitational waves? This was an outstanding question since the early days of general relativity. In his breakthrough result of 2008, Demetrios Chrstodoulou answered this question with “Yes!” In order to investigate this result, we will delve deeper into the dynamical mathematical structures of the Einstein equations. Black holes are related to the presence of trapped surfaces in the spacetime manifold. Christodoulou proved that in the regime of pure general relativity and for arbitrarily dispersed initial data, trapped surfaces form through the focusing of gravitational waves provided the incoming energy is large enough in a precisely defined way. The proof combines new ideas from geometric analysis and nonlinear partial differential equations as well as it introduces new methods to solve large data problems. These methods have many applications beyond general relativity. D. Christodoulou’s result was generalized in various directions by many authors. It launched mathematical activities going into multiple fields in mathematics and physics. In this talk, we will discuss the mathematical framework of the above question. Then we will outline the main ideas of Christodoulou’s result and its generalizations, show relations to other questions and give an overview of implications in other fields.
FRG Workshop on Geometric Methods for Analyzing Discrete Shapes
This workshop will take place May 7-9 (Friday-Sunday), 2021 virtually on Zoom
The aim of the workshop is to bring together a community of researchers in mathematics, computer science, and data science who develop theoretical and computational models to characterize shapes and analysis of image data.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The first half of the workshop will feature talks aimed at graduate students, newcomers, and a broad spectrum of audiences. Christopher Bishop (Stony Brook) and Keenan Crane (Carnegie Mellon) will each give two featured talks. The remaining part will have both background and research talks. There will also be organized discussions of open problems and potential applications.
For the discussions, we are soliciting open problems in mathematical theory and applications of shape analysis. You are encouraged to post problems by sending an email to geometricproblemsfrg@gmail.com.
We invite junior researchers to present a short talk in the workshop. The session will be held on Friday, May 7th or Saturday, May 8th at 4pm and are expected to be 15-20 minutes in length. It is a great opportunity to share your work and get to know others at the workshop. Depending on the number of contributed talks, the organizers will review the submissions and let you know if you have been selected. If you are interested please send your title and abstract to tianqi@cmsa.fas.harvard.edu by the end of May 2nd.
Organizers:
- David Glickenstein, University of Arizona
- Joel Hass, University of California, Davis
- Patrice Koehl, University of California, Davis
- Feng Luo, Rutgers University, New Brunswick
- Tianqi Wu, Harvard University
- Shing-Tung Yau, Harvard University
Featured lectures:
- Christopher Bishop, Stony Brook
- Keenan Crane, Carnegie Mellon
Speakers include:
- Miri Ben-Chen, Technion – Israel Institute of Technology
- Alexander Bobenko, Technische Universität Berlin, Germany
- Ulrike Buecking, Free University, Germany
- Nadav Dym, Duke University
- Ivan Izmestiev, Vienna University of Technology
- Yanwen Luo, Rutgers
- Stephan Tillmann, The University of Sydney
- Max Wardetzky, University of Goettingen
- Xu Xu, Wuhan University
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Simplices in the Calabi–Yau web
Abstract: Calabi–Yau manifolds of a given dimension are connected by an intricate web of birational maps. This web has deep consequences for the derived categories of coherent sheaves on such manifolds, and for the associated string theories. In particular, for 4-folds and beyond, I will highlight certain simplices appearing in the web, and identify corresponding derived category structures.
Previous Colloquia
The CMSA Colloquium will take place every Wednesday from 4:30-5:30pm in CMSA Building, 20 Garden Street, G10.
Spring 2020
Date Speaker Title/Abstract 1/29/2020 David Yang (Harvard) Abstract: Data-intensive technologies such as AI may reshape the modern world. We propose that two features of data interact to shape innovation in data-intensive economies: first, states are key collectors and repositories of data; second, data is a non-rival input in innovation. We document the importance of state-collected data for innovation using comprehensive data on Chinese facial recognition AI firms and government contracts. Firms produce more commercial software and patents, particularly data-intensive ones, after receiving government public security contracts. Moreover, effects are largest when contracts provide more data. We then build a directed technical change model to study the state’s role in three applications: autocracies demanding AI for surveillance purposes, data-driven industrial policy, and data regulation due to privacy concerns. When the degree of non-rivalry is as strong as our empirical evidence suggests, the state’s collection and processing of data can shape the direction of innovation and growth of data-intensive economies.
2/5/2020 Scott Aaronson (UT Austin) Title: Gentle Measurement of Quantum States and Differential Privacy Abstract: I’ll discuss a recent connection between two seemingly unrelated problems: how to measure a collection of quantum states without damaging them too much (“gentle measurement”), and how to provide statistical data without leaking too much about individuals (“differential privacy,” an area of classical CS). This connection leads, among other things, to a new protocol for “shadow tomography”
of quantum states (that is, answering a large number of questions about a quantum state given few copies of it).Based on joint work with Guy Rothblum (arXiv:1904.08747)
2/12/2020 Scott Kominers (Harvard) Title: A Compact, Logical Approach to Large-Market Analysis Abstract: In game theory, we often use infinite models to represent “limit” settings, such as markets with a large number of agents or games with a long time horizon. Yet many game-theoretic models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. Here, we show how to extend key results from (finite) models of matching, games on graphs, and trading networks to infinite models by way of Logical Compactness, a core result from Propositional Logic. Using Compactness, we prove the existence of man-optimal stable matchings in infinite economies, as well as strategy-proofness of the man-optimal stable matching mechanism. We then use Compactness to eliminate the need for a finite start time in a dynamic matching model. Finally, we use Compactness to prove the existence of both Nash equilibria in infinite games on graphs and Walrasian equilibria in infinite trading networks.
2/19/2020 Peter Shor (MIT) Title: Quantum Money from Lattices Abstract: Quantum money is a cryptographic protocol for quantum computers. A quantum money protocol consists of a quantum state which can be created (by the mint) and verified (by anybody with a quantum computer who knows what the “serial number” of the money is), but which cannot be duplicated, even by somebody with a copy of the quantum state who knows the verification protocol. Several previous proposals have been made for quantum money protocols. We will discuss the history of quantum money and give a protocol which cannot be broken unless lattice cryptosystems are insecure.
2/26/2020 Daneil Wise (McGill) Title: The Cubical Route to Understanding Groups Abstract: Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that culminated in the resolution of the virtual Haken conjecture for 3-manifolds and simultaneously dramatically extended our understanding of many infinite groups.3/4/2020 4:45 – 5:45pm
Salil Vadhan (Harvard) Title: Derandomizing Algorithms via Spectral Graph Theory Abstract: Randomization is a powerful tool for algorithms; it is often easier to design efficient algorithms if we allow the algorithms to “toss coins” and output a correct answer with high probability. However, a longstanding conjecture in theoretical computer science is that every randomized algorithm can be efficiently “derandomized” — converted into a deterministic algorithm (which always outputs the correct answer) with only a polynomial increase in running time and only a constant-factor increase in space (i.e. memory usage).
In this talk, I will describe an approach to proving the space (as opposed to time) version of this conjecture via spectral graph theory. Specifically, I will explain how randomized space-bounded algorithms are described by random walks on directed graphs, and techniques in algorithmic spectral graph theory (e.g. solving Laplacian systems) have yielded deterministic space-efficient algorithms for approximating the behavior of such random walks on undirected graphs and Eulerian directed graphs (where every vertex has the same in-degree as out-degree). If these algorithms can be extended to general directed graphs, then the aforementioned conjecture about derandomizing space-efficient algorithms will be resolved.
3/11/2020 Postponed
Jose Scheinkman (Columbia)
This colloquium will be rescheduled at a later date. Title: Menu Costs and the Volatility of Inflation
Abstract: We present a state-dependent equilibrium pricing model that generates inflation rate fluctuations from idiosyncratic shocks to the cost of price changes of individual firms. A firm’s nominal price increase lowers other firms’ relative prices, thereby inducing further nominal price increases. We first study a mean-field limit where the equilibrium is characterized by a variational inequality and exhibits a constant rate of inflation. We use the limit model to show that in the presence of a large but finite number n of firms the snowball effect of repricing causes fluctuations to the aggregate price level and these fluctuations converge to zero slowly as n grows. The fluctuations caused by this mechanism are larger when the density of firms at the repricing threshold is high, and the density at the threshold is high when the trend inflation level is high. However a calibration to US data shows that this mechanism is quantitatively important even at modest levels of trend inflation and can account for the positive relationship between inflation level and volatility that has been observed empirically.
3/12/2020 4:00 – 5:00pm
Daniel Forger (University of Michigan) This meeting will be taking place virtually on Zoom. Title: Math, Music and the Mind; Mathematical analysis of the performed Trio Sonatas of J. S. Bach
Abstract: I will describe a collaborative project with the University of Michigan Organ Department to perfectly digitize many performances of difficult organ works (the Trio Sonatas by J.S. Bach) by students and faculty at many skill levels. We use these digitizations, and direct representations of the score to ask how music should encoded in the mind. Our results challenge the modern mathematical theory of music encoding, e.g., based on orbifolds, and reveal surprising new mathematical patterns in Bach’s music. We also discover ways in which biophysical limits of neuronal computation may limit performance.
Daniel Forger is the Robert W. and Lynn H. Browne Professor of Science, Professor of Mathematics and Research Professor of Computational Medicine and Bioinformatics at the University of Michigan. He is also a visiting scholar at Harvard’s NSF-Simons Center and an Associate of the American Guild of Organists.
3/25/2020 Cancelled 4/1/2020 Mauricio Santillana (Harvard) This meeting will be taking place virtually on Zoom. Title: Data-driven machine learning approaches to monitor and predict events in healthcare. From population-level disease outbreaks to patient-level monitoring
Abstract: I will describe data-driven machine learning methodologies that leverage Internet-based information from search engines, Twitter microblogs, crowd-sourced disease surveillance systems, electronic medical records, and weather information to successfully monitor and forecast disease outbreaks in multiple locations around the globe in near real-time. I will also present data-driven machine learning methodologies that leverage continuous-in-time information coming from bedside monitors in Intensive Care Units (ICU) to help improve patients’ health outcomes and reduce hospital costs.
4/8/2020 Juven Wang (CMSA) This meeting will be taking place virtually on Zoom. Title: Quantum Matter Adventure to Fundamental Physics and Mathematics (Continued)
Abstract: In 1956, Parity violation in Weak Interactions is confirmed in particle physics. The maximal parity violation now is a Standard Model physics textbook statement, but it goes without any down-to-earth explanation for long. Why? We will see how the recent physics development in Quantum Matter may guide us to give an adventurous story and possibly a new elementary
explanation. We will see how the topology and cobordism in mathematics may come into play of anomalies and non-perturbative interactions in
fundamental physics. Perhaps some of you (geometers, string theorists, etc.) can team up with me to understand the “boundary conditions” of the Standard Model and Beyond4/15/2020 Lars Andersson (Max-Planck Institute for Gravitational Physics)This meeting will be taking place virtually on Zoom. Title: Stability of spacetimes with supersymmetric compactifications
Abstract: Spacetimes with compact directions, which have special holonomy such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss the global, non-linear stability for the vacuum Einstein equations on a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. I will start by giving a brief overview of related stability problems which have received a lot of attention recently, including the black hole stability problem. This is based on joint work with Pieter Blue, Zoe Wyatt and Shing-Tung Yau.
4/22/2020 William Minicozzi (MIT) This meeting will be taking place virtually on Zoom. Title: Mean curvature flow in high codimension
Abstract: I will talk about joint work with Toby Colding on higher codimension mean curvature flow. Some of the ideas come from function theory on manifolds with Ricci curvature bounds.
4/29/2020 Gerhard Huisken (Tübingen University / MFO) This meeting will be taking place virtually on Zoom. Title: Mean curvature flow of mean-convex embedded 2-surfaces in 3-manifolds
Abstract: The lecture describes joint work with Simon Brendle on the deformation of embedded surfaces with positive mean curvature in Riemannian 3-manifolds in direction of their mean curvature vector. It is described how to find long-time solutions of this flow, possibly including singularities that are overcome by surgery, leading to a comprehensive description of embedded mean-convex surfaces and the regions they bound in a 3-manifold. The flow can be used to sweep out the region between space-like infinity and the outermost horizon in asymptotically flat 3-manifolds arising in General Relativity. (Joint with Simon Brendle.)
5/6/2020 Lydia Bieri (UMich) This meeting will be taking place virtually on Zoom. Title: Energy, Mass and Radiation in General Spacetimes
Abstract: In Mathematical General Relativity (GR) the Einstein equations describe the laws of the universe. Isolated gravitating systems such as binary stars, black holes or galaxies can be described in GR by asymptotically flat (AF) solutions of these equations. These are solutions that look like flat Minkowski space outside of spatially compact regions. There are well-defined notions for energy and mass for such systems. The energy-matter content as well as the dynamics of such a system dictate the decay rates at which the solution tends to the flat one at infinity. Interesting questions occur for very general AF systems of slow decay. We are also interested in spacetimes with pure radiation. In this talk, I will review what is known for these systems. Then we will concentrate on spacetimes with pure radiation. In particular, we will compare the situations of incoming radiation and outgoing radiation under various circumstances and what we can read off from future null infinity.
5/13/2020 Mikhail Lukin (Harvard) This meeting will be taking place virtually on Zoom. Title: Exploring New Frontiers of Quantum Science with Programmable Atom Arrays
Abstract: We will discuss recent work at a new scientific interface between many-body physics and quantum information science. Specifically, we will describe the advances involving programmable, coherent manipulation of quantum many-body systems using atom arrays excited into Rydberg states. Within this system we performed quantum simulations of one dimensional spin models, discovered a new type of non-equilibrium quantum dynamics associated with the so-called many body scars and created large-scale entangled states. We will also describe the most recent developments that now allow the control over 200 atoms in two-dimensional arrays. Ongoing efforts to study exotic many-body phenomena and to realize and test quantum optimization algorithms within such systems will be discussed.
5/20/2020 This meeting will be taking place virtually on Zoom. Fall 2019
Date Speaker Title/Abstract 9/18/2019 Bill Helton (UC San Diego) Title: A taste of noncommutative convex algebraic geometry Abstract: The last decade has seen the development of a substantial noncommutative (in a free algebra) real and complex algebraic geometry. The aim of the subject is to develop a systematic theory of equations and inequalities for (noncommutative) polynomials or rational functions of matrix variables. Such issues occur in linear systems engineering problems, in free probability (random matrices), and in quantum information theory. In many ways the noncommutative (NC) theory is much cleaner than classical (real) algebraic geometry. For example,
◦ A NC polynomial, whose value is positive semidefinite whenever you plug matrices into it, is a sum of squares of NC polynomials.
◦ A convex NC semialgebraic set has a linear matrix inequality representation.
◦ The natural Nullstellensatz are falling into place.
The goal of the talk is to give a taste of a few basic results and some idea of how these noncommutative problems occur in engineering. The subject is just beginning and so is accessible without much background. Much of the work is joint with Igor Klep who is also visiting CMSA for the Fall of 2019.
9/25/2019 Pavel Etingof (MIT) Title: Double affine Hecke algebras Abstract: Double affine Hecke algebras (DAHAs) were introduced by I. Cherednik in the early 1990s to prove Macdonald’s conjectures. A DAHA is the quotient of the group algebra of the elliptic braid group attached to a root system by Hecke relations. DAHAs and their degenerations are now central objects of representation theory. They also have numerous connections to many other fields — integrable systems, quantum groups, knot theory, algebraic geometry, combinatorics, and others. In my talk, I will discuss the basic properties of double affine Hecke algebras and touch upon some applications.
10/2/2019 Spiro Karigiannis (University of Waterloo) Title: Cohomologies on almost complex manifolds and their applications Abstract: We define three cohomologies on an almost complex manifold (M, J), defined using the Nijenhuis-Lie derivations induced from the almost complex structure J and its Nijenhuis tensor N, regarded as vector-valued forms on M. One of these can be applied to distinguish non-isomorphic non-integrable almost complex structures on M. Another one, the J-cohomology, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex structures. The J-cohomology encodes whether a complex manifold satisfies the “del-delbar-lemma”, and more generally in the non-integrable case the J-cohomology encodes whether (M, J) satisfies a generalization of this lemma. We also mention some other potential cohomologies on almost complex manifolds, related to an interesting question involving the Nijenhuis tensor. This is joint work with Ki Fung Chan and Chi Cheuk Tsang.
10/9/2019 Hans Lindblad (Johns Hopkins University) Title: Global Existence and Scattering for Einstein’s equations and related equations satisfying the weak null condition Abstract: Einstein’s equations in harmonic or wave coordinates are a system of nonlinear wave equations for a Lorentzian metric, that in addition satisfy the preserved wave coordinate condition.
Christodoulou-Klainerman proved global existence for Einstein vacuum equations for small asymptotically flat initial data. Their proof avoids using coordinates since it was believed the metric in harmonic coordinates would blow up for large times.
John had noticed that solutions to some nonlinear wave equations blow up for small data, whereas lainerman came up with the ‘null condition’, that guaranteed global existence for small data. However Einstein’s equations do not satisfy the null condition.
Hormander introduced a simplified asymptotic system by neglecting angular derivatives which we expect decay faster due to the rotational invariance, and used it to study blowup. I showed that the asymptotic system corresponding to the quasilinear part of Einstein’s equations does not blow up and gave an example of a nonlinear equation of this form that has global solutions even though it does not satisfy the null condition.
Together with Rodnianski we introduced the ‘weak null condition’ requiring that the corresponding asymptotic system have global solutions and we showed that Einstein’s equations in wave coordinates satisfy the weak null condition and we proved global existence for this system. Our method reduced the proof to afraction and has now been used to prove global existence also with matter fields.
Recently I derived precise asymptotics for the metric which involves logarithmic corrections to the radiation field of solutions of linear wave equations. We are further imposing these asymptotics at infinity and solve the equationsbackwards to obtain global solutions with given data at infinity.
10/16/2019 Aram Harrow (MIT) Title: Monogamy of entanglement and convex geometry Abstract: The SoS (sum of squares) hierarchy is a flexible algorithm that can be used to optimize polynomials and to test whether a quantum state is entangled or separable. (Remarkably, these two problems are nearly isomorphic.) These questions lie at the boundary of P, NP and the unique games conjecture, but it is in general open how well the SoS algorithm performs. I will discuss how ideas from quantum information (the “monogamy” property of entanglement) can be used to understand this algorithm. Then I will describe an alternate algorithm that relies on apparently different tools from convex geometry that achieves similar performance. This is an example of a series of remarkable parallels between SoS algorithms and simpler algorithms that exhaustively search over carefully chosen sets. Finally, I will describe known limitations on SoS algorithms for these problems.
10/23/2019 No talk 10/30/2019 Nima Arkani-Hamed (IAS) Title: Spacetime, Quantum Mechanics and Positive Geometry at Infinity 11/6/2019 Kevin Costello (Perimeter Institute) Title: A unified perspective on integrability Abstract: Two dimensional integrable field theories, and the integrable PDEs which are their classical limits, play an important role in mathematics and physics. I will describe a geometric construction of integrable field theories which yields (essentially) all known integrable theories as well as many new ones. Billiard dynamical systems will play a surprising role. Based on work (partly in progress) with Gaiotto, Lee, Yamazaki, Witten, and Wu.
11/13/2019 Heather Harrington (University of Oxford) Title: Algebra, Geometry and Topology of ERK Enzyme Kinetics Abstract: In this talk I will analyse ERK time course data by developing mathematical models of enzyme kinetics. I will present how we can use differential algebra and geometry for model identifiability and topological data analysis to study these the wild type dynamics of ERK and ERK mutants. This work is joint with Lewis Marsh, Emilie Dufresne, Helen Byrne and Stanislav Shvartsman.
11/20/2019 Xi Yin (Harvard) Title: An Introduction to the Non-Perturbative Bootstrap Abstract: I will discuss non-perturbative definitions of quantum field theories, some properties of correlation functions of local operators, and give a brief overview of some results and open questions concerning the conformal bootstrap
11/25/2019 Monday
Madhu Sudan (Harvard) Abstract: The task of manipulating randomness has been a subject of intense investigation in the theory of computer science. The classical definition of this task consider a single processor massaging random samples from an unknown source and trying to convert it into a sequence of uniform independent bits.In this talk I will talk about a less studied setting where randomness is distributed among different players who would like to convert this randomness to others forms with relatively little communication. For instance players may be given access to a source of biased correlated bits, and their goal may be to get a common random bit out of this source. Even in the setting where the source is known this can lead to some interesting questions that have been explored since the 70s with striking constructions and some surprisingly hard questions. After giving some background, I will describe a recent work which explores the task of extracting common randomness from correlated sources with bounds on the number of rounds of interaction.
Based on joint works with Mitali Bafna (Harvard), Badih Ghazi (Google) and Noah Golowich (Harvard).
12/4/2019 Xiao-Gang Wen (MIT)
VideoTitle: Emergence of graviton-like excitations from a lattice model Abstract: I will review some construction of lattice rotor model which give rise to emergent photons and graviton-like excitations. The appearance of vector-like charge and symmetric tensor field may be related to gapless fracton phases.
2018-2019
Date Speaker Title/Abstract 9/26/2018 Xiao-Gang Wen (MIT) Title: A classification of low dimensional topological orders and fully extended TQFTs Abstract: In this talk, I will review the recent progress on classification of gapped phases of quantum matter (ie topological orders) in 1,2, and 3 spatial dimensions for boson systems. In 1-dimension, there is no non-trivial topological orders. In 2-dimensions, the topological orders are classified by modular tensor category theory. In 3-dimensions, the topological orders are classified by a simple class of braided fusion 2-categories. The classification of topological orders may correspond to a classification of fully extended unitary TQFTs.
10/03/2018 Richard Schoen (Stanford) Title: Perspectives on the scalar curvature Abstract: This will be a general talk concerning the role that the scalar curvature plays in Riemannian geometry and general relativity. We will describe recent work on extending the known results to all dimensions, and other issues which are being actively studied.
10/10/2018 Justin Solomon (MIT) Title: Correspondence and Optimal Transport for Geometric Data Processing Abstract: Correspondence problems involving matching of two or more geometric domains find application across disciplines, from machine learning to computer vision. A basic theoretical framework involving correspondence along geometric domains is optimal transport (OT). Dating back to early economic applications, the OT problem has received renewed interest thanks to its applicability to problems in machine learning, computer graphics, geometry, and other disciplines. The main barrier to wide adoption of OT as a modeling tool is the expense of optimization in OT problems. In this talk, I will summarize efforts in my group to make large-scale transport tractable over a variety of domains and in a variety of application scenarios, helping transition OT from theory to practice. In addition, I will show how OT can be used as a unit in algorithms for solving a variety of problems involving the processing of geometrically-structured data.
10/17/2018 Jeremy England (MIT) Title: Wisdom of the Jumble Abstract: There are certain, specific behaviors that are particularly distinctive of life. For example, living things self-replicate, harvest energy from challenging environmental sources, and translate experiences of past and present into actions that accurately anticipate the predictable parts of their future. What all of these activities have in common from a physics standpoint is that they generally take place under conditions where the pronounced flow of heat sharpens the arrow of time. We have therefore sought to use thermodynamics to understand the emergence and persistence of life-like phenomena in a wide range of messy systems made of many interacting components.
In this talk I will discuss some of the recent insights we have gleaned from studying emergent fine-tuning in disordered collections of matter exposed to complexly patterned environments. I will also point towards future possible applications in the design of new, more life-like ways of computing that have the potential to either be cheaper or more powerful than existing means.
10/31/2018 Moon Duchin (Tufts) Title: Exploring the (massive) space of graph partitions Abstract: The problem of electoral redistricting can be set up as a search of the space of partitions of a graph (representing the units of a state or other jurisdiction) subject to constraints (state and federal rules about the properties of districts). I’ll survey the problem and some approaches to studying it, with an emphasis on the deep mathematical questions it raises, from combinatorial enumeration to discrete differential geometry to dynamics.
11/14/2018 Dusa McDuff (Columbia) Title: The virtual fundamental class in symplectic geometry Abstract: Essential to many constructions and applications of symplectic geometry is the ability to count J-holomorphic curves. The moduli spaces of such curves have well understood compactifications, and if cut out transversally are oriented manifolds of dimension equal to the index of the problem, so that they a fundamental class that can be used to count curves. In the general case, when the defining equation is not transverse, there are various different approaches to constructing a representative for this class, We will discuss and compare different approaches to such a construction e.g. using polyfolds or various kinds of finite dimensional reduction. Most of this is joint work with Katrin Wehrheim.
11/19/2018 Xiaoqin Wang (Johns Hopkins) Title: Computational Principles of Auditory Cortex Abstract: Auditory cortex is located at the top of a hierarchical processing pathway in the brain that encodes acoustic information. This brain region is crucial for speech and music perception and vocal production. Auditory cortex has long been considered a difficult brain region to study and remained one of less understood sensory cortices. Studies have shown that neural computation in auditory cortex is highly nonlinear. In contrast to other sensory systems, the auditory system has a longer pathway between sensory receptors and the cerebral cortex. This unique organization reflects the needs of the auditory system to process time-varying and spectrally overlapping acoustic signals entering the ears from all spatial directions at any given time. Unlike visual or somatosensory cortices, auditory cortex must also process and differentiate sounds that are externally generated or self-produced (during speaking). Neural representations of acoustic information in auditory cortex are shaped by auditory feedback and vocal control signals during speaking. Our laboratory has developed a unique and highly vocal non-human primate model (the common marmoset) and quantitative tools to study neural mechanisms underlying audition and vocal communication.
11/28/2018 Robert Haslhofer (University of Toronto) Title: Recent progress on mean curvature flow Abstract: A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution in extrinsic geometry and shares many features with Hamilton’s Ricci flow from intrinsic geometry. In the first half of the talk, I will give an overview of the well developed theory in the mean convex case, i.e. when the mean curvature vector everywhere on the surface points inwards. Mean convex mean curvature flow can be continued through all singularities either via surgery or as level set solution, with a precise structure theory for the singular set. In the second half of the talk, I will report on recent progress in the general case without any curvature assumptions. Namely, I will describe our solution of the mean convex neighborhood conjecture and the nonfattening conjecture, as well as a general classification result for all possible blowup limits near spherical or cylindrical singularities. In particular, assuming Ilmanen’s multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed. This is joint work with Kyeongsu Choi and Or Hershkovits.
12/5/2018 Robert McCann (University of Toronto) Title: Displacement convexity of Boltzmann’s entropy characterizes positive energy in general relativity Abstract: Einstein’s theory of gravity is based on assuming that the fluxes of a energy and momentum in a physical system are proportional to a certain variant of the Ricci curvature tensor on a smooth 3+1 dimensional spacetime. The fact that gravity is attractive rather than repulsive is encoded in the positivity properties which this tensor is assumed to satisfy. Hawking and Penrose (1971) used this positivity of energy to give conditions under which smooth spacetimes must develop singularities. By lifting fractional powers of the Lorentz distance between points on a globally hyperbolic spacetime to probability measures on spacetime events, we show that the strong energy condition of Hawking and Penrose is equivalent to convexity of the Boltzmann-Shannon entropy along the resulting geodesics of probability measures. This new characterization of the strong energy condition on globally hyperbolic manifolds also makes sense in (non-smooth) metric measure settings, where it has the potential to provide a framework for developing a theory of gravity which admits certain singularities and can be continued beyond them. It provides a Lorentzian analog of Lott, Villani and Sturm’s metric-measure theory of lower Ricci bounds, and hints at new connections linking gravity to the second law of thermodynamics.
Preprint available at http://www.math.toronto.edu/mccann/papers/GRO.pdf
12/12/2018 Zhiwei Yun (MIT) Title: Shtukas: what and why Abstract: This talk is of expository nature. Drinfeld introduced the notion of Shtukas and the moduli space of them. I will review how Shtukas compare to more familiar objects in geometry, how they are used in the Langlands program, and what remains to be done about them.
1/30/2019 Richard Freeman (Harvard) Title: Innovation in Cell Phones in the US and China: Who Improves Technology Faster? Abstract: Cell phones are the archetypical modern consumer innovation, spreading around the world at an incredible pace, extensively used for connecting people with the Internet and diverse apps. Consumers report spending from 2-5 hours a day at their cell phones, with 44% of Americans saying “couldn’t go a day without their mobile devices.” Cell phone manufacturers introduce new models regularly, embodying additional features while other firms produce new applications that increase demand for the phones. Using newly developed data on the prices, attributes, and sales of different models in the US and China, this paper estimates the magnitude of technological change in the phones in the 2000s. It explores the problems of analyzing a product with many interactive attributes in the standard hedonic price regression model and uses Principal Components Regression to reduce dimensionality. The main finding is that technology improved the value of cell phones at comparable rates in the US and China, despite different market structures and different evaluations of some attributes and brands. The study concludes with a discussion of ways to evaluate the economic surplus created by the cell phones and their contribution to economic well-being.
2/7/2019 *Thursday*
Ulrich Mueller (Princeton) Title: Inference for the Mean Abstract: Consider inference about the mean of a population with finite variance, based on an i.i.d. sample. The usual t-statistic yields correct inference in large samples, but heavy tails induce poor small sample behavior. This paper combines extreme value theory for the smallest and largest observations with a normal approximation for the t-statistic of a truncated sample to obtain more accurate inference. This alternative approximation is shown to provide a refinement over the standard normal approximation to the full sample t-statistic under more than two but less than three moments, while the bootstrap does not. Small sample simulations suggest substantial size improvements over the bootstrap.
2/13/2019 Christian Santangelo (UMass Amherst) Title: 4D printing with folding forms Abstract: 4D printing is the name given to a set of advanced manufacturing techniques for designing flat materials that, upon application of a stimulus, fold and deform into a target three-dimensional shapes. The successful design of such structures requires an understanding of geometry as it applies to the mechanics of thin, elastic sheets. Thus, 4D printing provides a playground for both the development of new theoretical tools as well as old tools applied to new problems and experimental challenges in soft materials. I will describe our group’s efforts to understand and design structures that can fold from an initially flat sheet to target three-dimensional shapes. After reviewing the state-of-the-art in the theory of 4D printing, I will describe recent results on the folding and misfolding of flat structures and highlight the challenges remaining to be overcome.
2/20/2019 Michael Woodford (Columbia) Title: Optimally Imprecise Memory and Biased Forecasts Abstract: We propose a model of optimal decision making subject to a memory constraint. The constraint is a limit on the complexity of memory measured using Shannon’s mutual information, as in models of rational inattention; the structure of the imprecise memory is optimized (for a given decision problem and noisy environment) subject to this constraint. We characterize the form of the optimally imprecise memory, and show that the model implies that both forecasts and actions will exhibit idiosyncratic random variation; that beliefs will fluctuate forever around the rational-expectations (perfect-memory) beliefs with a variance that does not fall to zero; and that more recent news will be given disproportionate weight. The model provides a simple explanation for a number of features of observed forecast bias in laboratory and field settings.
[authors: Rava Azeredo da Silveira (ENS) and Michael Woodford (Columbia)]
2/27/2019 2:30pm
Ian Martin (LSE) Title: Sentiment and Speculation in a Market with Heterogeneous Beliefs Abstract: We present a dynamic model featuring risk-averse investors with heterogeneous beliefs. Individual investors have stable beliefs and risk aversion, but agents who were correct in hindsight become relatively wealthy; their beliefs are overrepresented in market sentiment, so “the market” is bullish following good news and bearish following bad news. Extreme states are far more important than in a homogeneous economy. Investors understand that sentiment drives volatility up, and demand high risk premia in compensation. Moderate investors supply liquidity: they trade against market sentiment in the hope of capturing a variance risk premium created by the presence of extremists. [with Dimitris Papadimitriou]
3/6/2019 2:30pm
Philippe Sosoe (Cornell) Title: A sharp transition for Gibbs measures associated to the nonlinear Schrödinger equation Abstract: In 1987, Lebowitz, Rose and Speer (LRS) showed how to construct formally invariant measures for the nonlinear Schrödinger equation on the torus. This seminal contribution spurred a large amount of activity in the area of partial differential equations with random initial data. In this talk, I will explain LRS’s result, and discuss a sharp transition in the construction of the Gibbs-type invariant measures considered by these authors. (Joint work with Tadahiro Oh and Leonardo Tolomeo)
3/13/2019 5:15pm
Greg Galloway (University of Miami) Title: On the geometry and topology of initial data sets in General Relativity Abstract: A theme of long standing interest (to the speaker!) concerns the relationship between the topology of spacetime and the occurrence of singularities (causal geodesic incompleteness). Many results concerning this center around the notion of topological censorship, which has to do with the idea that the region outside all black holes (and white holes) should be simple. The aim of the results to be presented is to provide support for topological censorship at the pure initial data level, thereby circumventing difficult issues of global evolution. The proofs rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry. The talk will begin with a brief overview of general relativity and topological censorship. The talk is based primarily on joint work with various collaborators: Lars Andersson, Mattias Dahl, Michael Eichmair and Dan Pollack.
3/20/2019 Sonia Jaffe (Microsoft) Title: Quality Externalities on Platforms: The Case of Airbnb Abstract: We explore quality externalities on platforms: when buyers have limited information, a seller’s quality affects whether her buyers return to the platform, thereby impacting other sellers’ future business. We propose an intuitive measure of this externality, applicable across a range of platforms. Guest Return Propensity (GRP) is the aggregate propensity of a seller’s customers to return to the platform. We validate this metric using Airbnb data: matching customers to listings with a one standard deviation higher GRP causes them to take 17% more subsequent trips. By directing buyers to higher-GRP sellers, platforms may be able to increase overall seller surplus. (Joint work with Peter Coles, Steven Levitt, and Igor Popov.)
3/27/2019 5:15pm
Tatyana Sharpee (Salk Institute for Biological Studies) Title: Hyperbolic geometry of the olfactory space. Abstract: The sense of smell can be used to avoid poisons or estimate a food’s nutrition content because biochemical reactions create many by-products. Thus, the presence of certain bacteria in the food becomes associated with the emission of certain volatile compounds. This perspective suggests that it would be convenient for the nervous system encode odors based on statistics of their co-occurrence within natural mixtures rather than based on the chemical structure per se. I will discuss how this statistical perspective makes it possible to map odors to points in a hyperbolic space. Hyperbolic coordinates have a long but often underappreciated history of relevance to biology. For example, these coordinates approximate distance between species computed along dendograms, and more generally between points within hierarchical tree-like networks. We find that these coordinates, which were generated purely based on the statistics of odors in the natural environment, provide a contiguous map of human odor pleasantness. Further, a separate analysis of human perceptual descriptions of smells indicates that these also generate a three dimensional hyperbolic representation of odors. This match in geometries between natural odor statistics and human perception can help to minimize distortions that would otherwise arise when mapping odors to perception. We identify three axes in the perceptual space that are aligned with odor pleasantness, its molecular boiling point and acidity. Because the perceptual space is curved, one can predict odor pleasantness by knowing the coordinates along the molecular boiling point and acidity axes.
4/3/2019 2:30pm
Sarah Moshary (Chicago Booth) Title: Deregulation through Direct Democracy: Lessons from Liquor Abstract: This paper examines the merits of state control versus private provision of spirits retail, using the 2012 deregulation of liquor sales in Washington state as an event study. We document effects along a number of dimensions: prices, product variety, convenience, substitution to other goods, state revenue, and consumption externalities. We estimate a demand system to evaluate the net effect of privatization on consumer welfare. Our findings suggest that deregulation harmed the median Washingtonian, even though residents voted in favor of deregulation by a 16% margin. Further, we find that vote shares for the deregulation initiative do not reflect welfare gains at the ZIP code level. We discuss implications of our findings for the efficacy of direct democracy as a policy tool.
4/10/2019 2:30pm
Pietro Veronesi (Chicago Booth) Title: Inequality Aversion, Populism, and the Backlash Against Globalization Abstract: Motivated by the recent rise of populism in western democracies, we develop a model in which a populist backlash emerges endogenously in a growing economy. In the model, voters dislike inequality, especially the high consumption of “elites.” Economic growth exacerbates inequality due to heterogeneity in risk aversion. In response to rising inequality, rich-country voters optimally elect a populist promising to end globalization. Countries with more inequality, higher financial development, and current account deficits are more vulnerable to populism, both in the model and in the data. Evidence on who voted for Brexit and Trump in 2016 also supports the model.
4/17/2019 Yi-Zhuang You (UCSD) Title: Machine Learning Physics: From Quantum Mechanics to Holographic Geometry Abstract: Inspired by the “third wave” of artificial intelligence (AI), machine learning has found rapid applications in various topics of physics research. Perhaps one of the most ambitious goals of machine learning physics is to develop novel approaches that ultimately allows AI to discover new concepts and governing equations of physics from experimental observations. In this talk, I will present our progress in applying machine learning technique to reveal the quantum wave function of Bose-Einstein condensate (BEC) and the holographic geometry of conformal field theories. In the first part, we apply machine translation to learn the mapping between potential and density profiles of BEC and show how the concept of quantum wave function can emerge in the latent space of the translator and how the Schrodinger equation is formulated as a recurrent neural network. In the second part, we design a generative model to learn the field theory configuration of the XY model and show how the machine can identify the holographic bulk degrees of freedom and use them to probe the emergent holographic geometry.
.
[1] C. Wang, H. Zhai, Y.-Z. You. Uncover the Black Box of Machine Learning Applied to Quantum Problem by an Introspective Learning Architecture https://arxiv.org/abs/1901.11103
[2] H.-Y. Hu, S.-H. Li, L. Wang, Y.-Z. You. Machine Learning Holographic Mapping by Neural Network Renormalization Group https://arxiv.org/abs/1903.00804
[3] Y.-Z. You, Z. Yang, X.-L. Qi. Machine Learning Spatial Geometry from Entanglement Features https://arxiv.org/abs/1709.01223
4/24/2019 Shengwu Li (Harvard) Title: Credible MechanismsAbstract: Consider an extensive-form mechanism, run by an auctioneer who communicates sequentially and privately with agents. Suppose the auctioneer can deviate from the rules provided that no single agent detects the deviation. A mechanism is credible if it is incentive-compatible for the auctioneer to follow the rules. We study the optimal auctions in which only winners pay, under symmetric independent private values. The first-price auction is the unique credible static mechanism. The ascending auction is the unique credible strategy-proof mechanism.Date………… Speaker Title 02-09-2018 *Friday Fan Chung (UCSD)
Sequences: random, structured or something in between There are many fundamental problems concerning sequences that arise in many areas of mathematics and computation. Typical problems include finding or avoiding patterns;
testing or validating various `random-like’ behavior; analyzing or comparing different statistics, etc. In this talk, we will examine various notions of regularity or irregularity for sequences and mention numerous open problems.
02-14-2018 Zhengwei Liu (Harvard Physics)
A new program on quantum subgroups Abstract: Quantum subgroups have been studied since the 1980s. The A, D, E classification of subgroups of quantum SU(2) is a quantum analogue of the McKay correspondence. It turns out to be related to various areas in mathematics and physics. Inspired by the quantum McKay correspondence, we introduce a new program that our group at Harvard is developing.
02-21-2018 Don Rubin (Harvard)
Essential concepts of causal inference — a remarkable history Abstract: I believe that a deep understanding of cause and effect, and how to estimate causal effects from data, complete with the associated mathematical notation and expressions, only evolved in the twentieth century. The crucial idea of randomized experiments was apparently first proposed in 1925 in the context of agricultural field trails but quickly moved to be applied also in studies of animal breeding and then in industrial manufacturing. The conceptual understanding seemed to be tied to ideas that were developing in quantum mechanics. The key ideas of randomized experiments evidently were not applied to studies of human beings until the 1950s, when such experiments began to be used in controlled medical trials, and then in social science — in education and economics. Humans are more complex than plants and animals, however, and with such trials came the attendant complexities of non-compliance with assigned treatment and the occurrence of “Hawthorne” and placebo effects. The formal application of the insights from earlier simpler experimental settings to more complex ones dealing with people, started in the 1970s and continue to this day, and include the bridging of classical mathematical ideas of experimentation, including fractional replication and geometrical formulations from the early twentieth century, with modern ideas that rely on powerful computing to implement aspects of design and analysis.
02-26-2018 *Monday Tom Hou (Caltech)
Computer-assisted analysis of singularity formation of a regularized 3D Euler equation Abstract: Whether the 3D incompressible Euler equation can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. In a recent joint work with Dr. Guo Luo, we provided convincing numerical evidence that the 3D Euler equation develops finite time singularities. Inspired by this finding, we have recently developed an integrated analysis and computation strategy to analyze the finite time singularity of a regularized 3D Euler equation. We first transform the regularized 3D Euler equation into an equivalent dynamic rescaling formulation. We then study the stability of an approximate self-similar solution. By designing an appropriate functional space and decomposing the solution into a low frequency part and a high frequency part, we prove nonlinear stability of the dynamic rescaling equation around the approximate self-similar solution, which implies the existence of the finite time blow-up of the regularized 3D Euler equation. This is a joint work with Jiajie Chen, De Huang, and Dr. Pengfei Liu.
03-07-2018 Richard Kenyon (Brown)
Harmonic functions and the chromatic polynomial Abstract: When we solve the Dirichlet problem on a graph, we look for a harmonic function with fixed boundary values. Associated to such a harmonic function is the Dirichlet energy on each edge. One can reverse the problem, and ask if, for some choice of conductances on the edges, one can find a harmonic function attaining any given tuple of edge energies. We show how the number of solutions to this problem is related to the chromatic polynomial, and also discuss some geometric applications. This talk is based on joint work with Aaron Abrams and Wayne Lam.
03-14-2018 03-21-2018 03-28-2018 Andrea Montanari (Stanford) A Mean Field View of the Landscape of Two-Layers Neural Networks Abstract: Multi-layer neural networks are among the most powerful models in machine learning and yet, the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a highly non-convex and high-dimensional objective (risk function), a problem which is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the first case, does this happen because local minima are absent, or because SGD somehow avoids them? In the second, why do local minima reached by SGD have good generalization properties?
We consider a simple case, namely two-layers neural networks, and prove that –in a suitable scaling limit– the SGD dynamics is captured by a certain non-linear partial differential equation. We then consider several specific examples, and show how the asymptotic description can be used to prove convergence of SGD to network with nearly-ideal generalization error. This description allows to `average-out’ some of the complexities of the landscape of neural networks, and can be used to capture some important variants of SGD as well.
[Based on joint work with Song Mei and Phan-Minh Nguyen]03-30-2018 04-04-2018 Ramesh Narayan (Harvard)
Black Holes and Naked Singularities Abstract: Black Hole solutions in General Relativity contain Event Horizons and
Singularities. Astrophysicists have discovered two populations of
black hole candidates in the Universe: stellar-mass objects with
masses in the range 5 to 30 solar masses, and supermassive objects
with masses in the range million to several billion solar
masses. There is considerable evidence that these objects have Event
Horizons. It thus appears that astronomical black hole candidates are
true Black Holes. Direct evidence for Singularities is much harder to
obtain since, at least in the case of Black Holes, the Singularities
are hidden inside the Event Horizon. However, General Relativity also
permits Naked Singularities which are visible to external
observers. Toy Naked Singularity models have been constructed, and
some observational features of accretion flows in these spacetimes
have been worked out.04-11-2018 Pablo Parrilo (MIT)
Graph Structure in Polynomial Systems: Chordal Networks Abstract: The sparsity structure of a system of polynomial equations or an optimization problem can be naturally described by a graph summarizing the interactions among the decision variables. It is natural to wonder whether the structure of this graph might help in computational algebraic geometry tasks (e.g., in solving the system). In this lecture we will provide a gentle introduction to this area, focused on the key notions of chordality and treewidth, which are of great importance in related areas such as numerical linear algebra, database theory, constraint satisfaction, and graphical models. In particular, we will discuss “chordal networks”, a novel representation of structured polynomial systems that provides a computationally convenient decomposition of a polynomial ideal into simpler (triangular) polynomial sets, while maintaining its underlying graphical structure. As we will illustrate through examples from different application domains, algorithms based on chordal networks can significantly outperform existing techniques. Based on joint work with Diego Cifuentes (MIT).
04-18-2018 Washington Taylor (MIT)
On the fibration structure of known Calabi-Yau threefolds Abstract: In recent years, there is increasing evidence from a variety of directions, including the physics of F-theory and new generalized CICY constructions, that a large fraction of known Calabi-Yau manifolds have a genus one or elliptic fibration. In this talk I will describe recent work with Yu-Chien Huang on a systematic analysis of the fibration structure of known toric hypersurface Calabi-Yau threefolds. Among other results, this analysis shows that every known Calabi-Yau threefold with either Hodge number exceeding 150 is genus one or elliptically fibered, and suggests that the fraction of Calabi-Yau threefolds that are not genus one or elliptically fibered decreases roughly exponentially with h_{11}. I will also make some comments on the connection with the structure of triple intersection numbers in Calabi-Yau threefolds.
04-25-2018 Xi Yin (Harvard)
How we can learn what we need to know about M-theory Abstract: M-theory is a quantum theory of gravity that admits an eleven dimensional Minkowskian vacuum with super-Poincare symmetry and no dimensionless coupling constant. I will review what was known about M-theory based on its relation to superstring theories, then comment on a number of open questions, and discuss how they can be addressed from holographic dualities. I will outline a strategy for extracting the S-matrix of M-theory from correlation functions of dual superconformal field theories, and in particular use it to recover the 11D R^4 coupling of M-theory from ABJM theory.
05-02-2018 05-09-2018 2016-2017
Date Name Title/Abstract 01-25-17 Sam Gershman, Harvard Center for Brain Science, Department of Psychology Title: Spectral graph theory of cognitive maps
Abstract: The concept of a “cognitive map” has played an important role in neuroscience and psychology. A cognitive map is a representation of the environment that supports navigation and decision making. A longstanding question concerns the precise computational nature of this map. I offer a new mathematical foundation for the cognitive map, based on ideas at the intersection of spectral graph theory and reinforcement learning. Empirical data from neural recordings and behavioral experiments supports this theory.
02-01-17 Sean Eddy, Harvard Department of Molecular and Cellular Biology Title: Biological sequence homology searches: the future of deciphering the past Abstract: Computational recognition of distant common ancestry of biological sequences is a key to studying ancient events in molecular evolution.The better our sequence analysis methods are, the deeper in evolutionary time we can see. A major aim in the field is to improve the resolution of homology recognition methods by building increasingly realistic, complex, parameter-rich models. I will describe current and future research in homology search algorithms based on probabilistic inference methods, using hidden Markov models(HMMs) and stochastic context-free grammars (SCFGs). We make these methods available in the HMMER and Infernal software from my laboratory, in collaboration with database teams at the EuropeanBioinformatics Institute in the UK.
02-08-17 Matthew Headrick, Brandeis University Title: Quantum entanglement, classical gravity, and convex programming: New connections Abstract: In recent years, developments from the study of black holes and quantum gravity have revealed a surprising connection between quantum entanglement and classical general relativity. The theory of convex programming, applied in the differential-geometry setting, turns out to be useful for understanding what’s behind this correspondence. We will describe these developments, giving the necessary background in quantum information theory and convex programming along the way.
02-15-17 Masahito Yamazaki, IMPU Title: Geometry of 3-manifolds and Complex Chern-Simons Theory Abstract: The geometry of 3-manifolds has been a fascinating subject in mathematics. In this talk I discuss a “quantization” of 3-manifold geometry, in the language of complex Chern-Simons theory. This Chern-Simons theory in turn is related to the physics of 30dimensional supersymmetric field theories through the so-called 3d/3d correspondence, whose origin can be traced back to a mysterious theory on the M5-branes. Along the way I will also comment on the connection with a number of related topics, such as knot theory, hyperbolic geometry, quantum dilogarithm and cluster algebras.
02-22-17 Steven Rayan, University of Saskatchewan Title: Higgs bundles and the Hitchin system
Abstract: I will give an informal introduction to the Hitchin system, an object lying at the crossroads of geometry and physics. As a moduli space, the Hitchin system parametrizes semistable Higgs bundles on a Riemann surface up to equivalence. From this point of view, the Hitchin map and spectral curves emerge. We’ll use these to form an impression of what the moduli space “looks like”. I will also outline the appearances of the Hitchin system in dynamics, hyperkaehler geometry, and mirror symmetry.
03-01-17 Jun Liu, Harvard University Title: Expansion of biological pathways by integrative Genomics Abstract: The number of publicly available gene expression datasets has been growing dramatically. Various methods had been proposed to predict gene co-expression by integrating the publicly available datasets. These methods assume that the genes in the query gene set are homogeneously correlated and consider no gene-specific correlation tendencies, no background intra-experimental correlations, and no quality variations of different experiments. We propose a two-step algorithm called CLIC (CLustering by Inferred Co-expression) based on a coherent Bayesian model to overcome these limitations. CLIC first employs a Bayesian partition model with feature selection to partition the gene set into disjoint co-expression modules (CEMs), simultaneously assigning posterior probability of selection to each dataset. In the second step, CLIC expands each CEM by scanning the whole reference genome for candidate genes that were not in the input gene set but co-expressed with the genes in this CEM. CLIC is capable of integrating over thousands of gene expression datasets to achieve much higher coexpression prediction accuracy compared to traditional co-expression methods. Application of CLIC to ~1000 annotated human pathways and ~6000 poorly characterized human genes reveals new components of some well-studied pathways and provides strong functional predictions for some poorly characterized genes. We validated the predicted association between protein C7orf55 and ATP synthase assembly using CRISPR knock-out assays.
Based on the joint work with Yang Li and the Vamsi Mootha lab.
03-08-17 Gabor Lippner, Northeastern University Title: Evolution of cooperation in structured populations Abstract: Understanding how the underlying structure affects the evolution of a population is a basic, but difficult, problem in the evolutionary dynamics. Evolutionary game theory, in particular, models the interactions between individuals as games, where different traits correspond to different strategies. It is one of the basic approaches to explain the emergence of cooperative behavior in Darwinian evolution.
In this talk I will present new results about the model where the population is represented by an interaction network. We study the likelihood of a random mutation spreading through the entire population. The main question is to understand how the network influences this likelihood. After introducing the model, I will explain how the problem is connected to the study of meeting times of random walks on graphs, and based on this connection, outline a general method to analyze the model on general networks.03-15-17 Spring Break: No session 03-22-17 Gunther Uhlmann, University of Washington Abstract: We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times ofwaves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also applications in optics and medical imaging among others.The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.We will also describe some recent results, joint with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.03-29-17 Leslie Greengard, Courant Institute Title: Inverse problems in acoustic scattering and cryo-electron microscopy Abstract: A variety of problems in image reconstruction give rise to large-scale, nonlinear and non-convex optimization problems. We will show how recursive linearization combined with suitable fast solvers are bringing such problems within practical reach, with an emphasis on acoustic scattering and protein structure determination via cryo-electron microscopy.
NOTE: This talk will begin at 4:00pm
04-05-17 Gongjie Li, Harvard University Title: Unveiling the Origin of Planetary Systems by Dynamical and Statistical Approaches Abstract: The unexpected diversity of observed extrasolar planetary systems has posed new challenges to our classical understanding of planetary formation. A lot of these challenges can be addressed by a deeper understanding of the dynamics in planetary systems, which will also allow us to construct more accurate planetary formation theories consistent with observations. In this talk, I will first explain the origin of counter orbiting planets using a new dynamical mechanism I discovered, which also has wide implications in other astrophysical systems, such as the enhancement of tidal disruption rates near supermassive black hole binaries. In addition, I will discuss the architectural properties of circumbinary planetary systems from selection biases using statistical methods, and infer the origin of such systems.
04-12-17 Shlomo Razamat, Israel Institute of Technology Title: Complicated four-dimensional physics and simple mathematics Abstract: We will discuss SCFTs in four dimensions obtained from compactifications of six dimensional models. We will discuss the relation of the partition functions, specifically the supersymmetric index, of the SCFTs to certain special functions, and argue that the partition functions are expected to be naturally expressed in terms of eigenfunctions of generalizations of Ruijsenaars-Schneider models. We will discuss how the physics of the compactifications implies various precise mathematical identities involving the special functions, most of which are yet to be proven.
04-19-17 Cumrun Vafa, Harvard University Title: String Swampland Abstract: In this talk I review the idea behind identification of the string swampland. In particular I discuss the weak gravity conjecture as one such criterion and explain a no-go theorem for non-supersymmetric AdS/CFT holography.
04-27-17 Mehran Kardar, MIT Title: Levitation by Casimir forces in and out of equilibrium Abstract: Equilibrium fluctuation-induced forces are abundant in nature, ranging from quantum electrodynamic (QED) Casimir and van der Waals forces, to their thermal analogs in fluctuating soft matter. Repulsive Casimir forces have been proposed for a variety of shapes and materials. A generalization of Earnshaw’s theorem constrains the possibility of levitation by Casimir forces in equilibrium. The scattering formalism, which forms the basis of this proof, can be used to study fluctuation-induced forces for different materials, diverse geometries, both in and out of equilibrium. Conformal field theory methods suggest that critical (thermal) Casimir forces are not subject to a corresponding constraint.
Note: This talk will begin at 3:00pm
05-02-17 Simona Cocco, Laboratoire de Physique Statistique de l’ENS Title: Reverse modeling of protein sequence data: from graphical models to structural and functional predictions Body: A fundamental yet largely open problem in biology and medicine is to understand the relationship between the amino-acid sequence of a protein and its structure and function. Protein databases such as Pfam, which collect, align, and classify protein sequences into families containing
similar (homologous) sequences are growing at a fast pace thanks to recent advances in sequencing technologies. What kind of information about the structure and function of proteins can be obtained from the statistical distribution of sequences in a protein family? To answer this question I will describe recent attempts to infer graphical models able to reproduce the low-order statistics of protein sequence data, in particular amino acid conservation and covariation. I will also review how those models
have led to substantial progress in protein structural and functional
predictions.Note: This talk will begin at 4:00pm
05-03-17 Xue-Mei Li, University of Warwick Title: Perturbation to conservation law and stochastic averaging Abstract: A deterministic or random system with a conservation law is often used to
approximate dynamics that are also subjected to smaller deterministic or random influences. Consider for example dynamical descriptions for Brownian motions and singular perturbed operators arising from rescaled Riemmannian metrics. In both cases the conservation laws, which are maps with values in a manifold, are used to separate the slow and fast variables. We discuss stochastic averaging and diffusion creation arising from these contexts. Our overarching question is to describe stochastic dynamics associated with the convergence of Riemannian manifolds and metric spaces.Note: This talk will be held in the Science Center, Room 507
05-10-17 05-17-17 Kwok Wai Chan, Chinese University of Hong Kong Title: Scattering diagrams from asymptotic analysis on Maurer-Cartan equations Abstract: In 2005, a program was set forth by Fukaya aiming at investigating SYZ mirror symmetry by asymptotic analysis on Maurer-Cartan equations. In this talk, I will explain some results which implement part of Fukaya’s program. More precisely, I will show how semi-classical limits of Maurer-Cartan solutions give rise naturally to consistent scattering diagrams, which are known to encode Gromov-Witten data on the mirror side and have played an important role in the works of Kontsevich-Soibelman and Gross-Siebert on the reconstruction problem in mirror symmetry. This talk is based on joint work with Conan Leung and Ziming Ma, which was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK14302015).
05-24-17 NO COLLOQUIUM 05-31-17 Peter Michor, University of Vienna Title: Geometry of shape spaces and diffeomorphism groups and some of their uses Abstract: This talk is devoted to shape spaces, Riemannian metrics on them, their geodesics and distance functions, and some of their uses, mainly in computational anatomy. The simplest Riemannian metrics have vanishing geodesic distance, so one has to use, for example, higher order Sobolev metrics on shape spaces. These have curvature, which complicates statistics on these spaces.
Date Name Title 09-09-16 Bong Lian, Brandeis
Title: Riemann-Hilbert Problem and Period Integrals
Abstract: Period integrals of an algebraic manifolds are certain special functions that describe, among other things, deformations of the variety. They were originally studied by Euler, Gauss and Riemann, who were interested in analytic continuation of these objects. In this lecture, we will discuss a number of long-standing problems on period integrals in connection with mirror symmetry and Calabi-Yau geometry. We will see how the theory of D-modules have led us to solutions and insights into some of these problems.
09-14-16 Sze-Man Ngai, Georgia Southern University Title: The multifractal formalism and spectral asymptotics of self-similar measures with overlaps Abstract: Self-similar measures form a fundamental class of fractal measures, and is much less understood if they have overlaps. The multifractal formalism, if valid, allows us to compute the Hausdorff dimension of the multifractal components of the measure through its Lq-spectrum. The asymptotic behavior of the eigenvalue counting function for the associated Laplacians is closely related to the multifractal structure of the measure. Throughout this talk, the infinite Bernoulli convolution associated with the golden ratio will be used as a basic example to describe some of the results.
09-21-16 Prof. L. Mahadevan, Harvard SEAS Title: “Morphogenesis: Biology, Physics and Mathematics” Abstract: A century since the publication of Darcy Thompson’s classic “On growth and form,” his vision has finally begun to permeate into the fabric of modern biology. Within this backdrop, I will discuss some simple questions inspired by the onset of form in biology wherein mathematical models and computations, in close connection with experiments allow us to begin unraveling the physical basis for morphogenesis in the context of examples such as tendrils, leaves, guts, and brains. I will also try and indicate how these problems enrich their roots, creating new questions in mathematics, physics, and biology.
09-28-16 Hong Liu, MIT Title: A new theory of fluctuating hydrodynamics Despite its long and glorious history, hydrodynamics has so far been formulated mostly at the level of equations of motion, which is inadequate for capturing fluctuations. In a fluid, however, fluctuations occur spontaneously and continuously, at both the quantum and statistical levels, the understanding of which is important for a wide variety of physical problems. Another unsatisfactory aspect of the current formulation of hydrodynamics is that the equations of motion are constrained by various phenomenological conditions on the solutions, which need to be imposed by hand. One of such constraints is the local second law of thermodynamics, which plays a crucial role, yet whose physical origin has been obscure.
We present a new theory of fluctuating hydrodynamics which incorporates fluctuations systematically and reproduces all the phenomenological constraints from an underlying Z_2 symmetry. In particular, the local second law of thermodynamics is derived. The theory also predicts new constraints which can be considered as nonlinear generalizations of Onsager relations. When truncated to Gaussian noises, the theory recovers various nonlinear stochastic equations.
Curiously, to describe thermal fluctuations of a classical fluid consistently one needs to introduce anti-commuting variables and the theory exhibits an emergent supersymmetry.
10-05-16 Alexander Logunov, Tel-Aviv University
Title: Zeroes of harmonic functions and Laplace eigenfunctions Abs: Nadirashvili conjectured that for any non-constant harmonic function in R^3 its zero set has infinite area. This question was motivated by the Yau conjecture on zero sets of Laplace eigenfunctions. Both conjectures can be treated as an attempt to control the zero set of a solution of elliptic PDE in terms of growth of the solution. For holomorhpic functions such kind of control is possible only from one side: there is a plenty of holomorphic functions that have no zeros. While for a real-valued harmonic function on a plane the length of the zero set can be estimated (locally) from above and below by the frequency, which is a characteristic of growth of the harmonic function. We will discuss the notion of frequency, its properties and applications to zero sets in the higher dimensional case, where the understanding is far from being complete.
10-12-16 Conan Nai Chung Leung, CUHK Title: Coisotropic A-branes and their SYZ transform
Abstract: “Kapustin introduced coisotropic A-branes as the natural boundary condition for strings in A-model, generalizing Lagrangian branes and argued that they are indeed needed to for homological mirror symmetry. I will explain in the semiflat case that the Nahm transformation along SYZ fibration will transform fiberwise Yang-Mills holomorphic bundles to coisotropic A-branes. This explains SYZ mirror symmetry away from the large complex structure limit.”
10-19-16 Vaughan Jones, UC Berkeley Title: Are the Thompson groups any good as a model for Diff(S^1)? Abstract. The Thompson groups are by definition groups of piecewise linear
diffeomorphisms of the circle. A result of Ghys-Sergiescu says that a Thompson group can
be conjugated to a group of smooth diffeomorphisms. That’s the good news.
The bad news is that there is an important central extension of Diff(S^1) which requires a certain amount of smoothness for its definition. And Ghys-Sergiescu show that, no matter how the Thompson groups are embedded in Diff(S^1), the restriction of the central extension splits. Is it possible to obtain central extensions of the Thompson groups by any
procedure analogous to the constructions of the central extension of Diff(S^1)?
I will define all the players in this game, explain this question in detail,and present some failed attempts to answer it.10-26-16 Henry Cohn, Microsoft
Sums of squares, correlation functions, and exceptional geometric structures
Some exceptional structures such as the icosahedron or E_8 root system have remarkable optimality properties in settings such as packing, energy minimization, or coding. How can we understand and prove their optimality? In this talk, I’ll interweave this story with two other developments in recent mathematics (without assuming familiarity with either): how semidefinite optimization and sums of squares have expanded the scope of optimization, and how representation theory has shed light on higher correlation functions for particle systems.
11-02-16 Christian Borgs, Microsoft
Title: Graphon processes and limits of sparse graph sequences
Abstract: The theory of graph limits for dense graphs is by now well established, with graphons describing both the limit of a sequence of deterministic graphs, and a model for so-called exchangeable random graphs. Here a graphon is a function defined over a “feature space’’ equipped with some probability measure, the measure describing the distribution of features for the nodes, and the graphon describing the probability that two nodes with given features form a connection. While there are rich models of sparse random graphs based on graphons, they require an additional parameter, the edge density, whose dependence on the size of the graph has either to be postulated as an additional function, or considered as an empirical observed quantity not described by the model.
In this talk I describe a new model, where the underlying probability space is replaced by a sigma-finite measure space, leading to both a new random model for exchangeable graphs, and a new notion of graph limits. The new model naturally produces a graph valued stochastic process indexed by a continuous time parameter, a “graphon process”, and describes graphs which typically have degree distributions with long tails, as observed in large networks in real life.
11-09-16 TIME CHANGE: 4PM
Norden E. Huang, National Central University, (Taiwan)
Title: On Holo-Hilbert Spectral Analysis Traditionally, spectral analysis is defined as transform the time domain data to frequency domain. It is achieved through integral transforms based on additive expansions of a priori determined basis, under linear and stationary assumptions. For nonlinear processes, the data can have both amplitude and frequency modulations generated by intra-wave and inter-wave interactions involving both additive and nonlinear multiplicative processes. Under such conditions, the additive expansion could not fully represent the physical processes resulting from multiplicative interactions. Unfortunately, all existing spectral analysis methods are based on additive expansions, based either on a priori or adaptive bases. While the adaptive Hilbert spectral analysis could accommodate the intra-wave nonlinearity, the inter-wave nonlinear multiplicative mechanisms that include cross-scale coupling and phase lock modulations are left untreated. To resolve the multiplicative processes, we propose a full informational spectral representation: The Holo-Hilbert Spectral Analysis (HHSA), which would accommodate all the processes: additive and multiplicative, intra-mode and inter-mode, stationary and non-stationary, linear and nonlinear interactions, through additional dimensions in the spectrum to account for both the variations in frequency and amplitude modulations (FM and AM) simultaneously. Applications to wave-turbulence interactions and other data will be presented to demonstrate the usefulness of this new spectral representation.
11-16-16 Tristan Collins, Harvard University TIME CHANGE: 3:30PM
Title: Restricted volumes and finite time singularities of the Kahler-Ricci flow
Abstract: I will discuss the relationship between restricted volumes, as defined algebraically or analytically, and the finite time singularities of the Kahler-Ricci flow. This is joint work with Valentino Tosatti.
11-22-16 TUESDAY TIME CHANGE: 4-5PM
Xiangfeng Gu, Stonybrook
Title: Differential Geometric Methods for Engineering Applications
Abstract: With the development of virtual reality and augmented reality, many challenging problems raised in engineering fields. Most of them are with geometric nature, and can be explored by modern geometric means. In this talk, we introduce our approaches to solve several such kind of problems: including geometric compression, shape classification, surface registration, cancer detection, facial expression tracking and so on, based on surface Ricci flow and optimal mass transportation.
11-30-16 TIME CHANGE: 4:20PM
Sharad Ramanathan, Harvard MCB & SEAS
Title: Finding co-ordinate systems to monitor the development of mammalian embryos 12-07-16 Valentino Tosatti, Northwestern
Title: Metric limits of hyperkahler manifolds
Abstract: I will discuss a proof of a conjecture of Kontsevich-Soibelman and Gross-Wilson about the behavior of unit-diameter Ricci-flat Kahler metrics on hyperkahler manifolds (fibered by holomorphic Lagrangian tori) near a large complex structure limit. The collapsed Gromov-Hausdorff limit is a special Kahler metric on a half-dimensional complex projective space, away from a singular set of Hausdorff codimension at least 2. The resulting picture is also compatible with the Strominger-Yau-Zaslow mirror symmetry. This is joint work with Yuguang Zhang.
12-14-16 2015-2016
Workshop on Dynamics, Randomness, and Control in Molecular and Cellular Networks
On November 12-14, 2019 the CMSA will be hosting a workshop on Dynamics, Randomness, and Control in Molecular and Cellular Networks. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Biological cells are the fundamental units of life, and predictive modeling of cellular dynamics is essential for understanding a myriad of biological processes and functions. Rapid advances in technologies have made it possible for biologists to measure many variables and outputs from complex molecular and cellular networks with various inputs and environmental conditions. However, such advances are far ahead of the development of mathematical theory, models and methods needed to secure a deep understanding of how high-level robust behaviors emerge from the interactions in complex structures, especially in dynamic and stochastic environments. This workshop will bring together mathematicians and biological scientists involved in developing mathematical theories and methods for understanding, predicting and controlling dynamic behavior of molecular and cellular networks. Particular emphasis will be placed on efforts directed towards discovering underlying biological principles that govern function, adaptation and evolution, and on the development of associated mathematical theories.
Organizers: Jeremy Gunawardena (Harvard) and Ruth Williams (University of California, San Diego)
A limited amount of funding is available to help in defraying the travel costs of early career researchers, women, and underrepresented minorities, participating the workshop. Early career researchers are researchers who received their Ph.D. in 2014 or later, or who are Ph.D. students expecting to complete their Ph.D. by the end of 2020.
To apply, please send a CV, a statement of why you wish to attend, and, if you are a grad student, a letter of support from your advisor to Sarah LaBauve at slabauve@math.harvard.edu
All applications received by 5pm, EDT, October 28, 2019 will receive full consideration.
Speakers:
- David Anderson, University of Wisconsin | Slides
- James Collins, MIT
- Domitilla Del Vecchio, MIT | Slides
- Olga K. Dudko, UC San Diego
- Massimiliano Esposito, University of Luxembourg | Slides
- John Fricks, Arizona State University | Slides
- Heather Harrington, University of Oxford
- Joe Howard, Yale University
- Krešimir Josić, University of Houston
- Samuel Kou, Harvard University
- Tom Kurtz, University of Wisconsin | Slides
- Andrew Murray, Harvard University
- Antonis Papachristodoulou, University of Oxford
- Johan Paulsson, Harvard University
- Lea Popovic, Concordia University
- Sharad Ramanathan, Harvard University
- Eduardo Sontag, Northeastern University
- Jörg Stelling, ETH Zurich | Slides
- Pieter Rein ten Wolde, AMOLF | Slides
Videos from the workshop can be found in the Youtube playlist.
Learning from health data in the million genome era
On November 1, 2019 the CMSA will be hosting a conference organized by Seven Bridges Genomics. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Projects currently underway around the world are collecting detailed health and genomic data from millions of volunteers. In parallel, numerous healthcare systems have announced commitments to integrate genomic data into the standard of care for select patients. These data have the potential to reveal transformative insights into health and disease. However, to realize this promise, novel approaches are required across the full life cycle of data analysis. This symposium will include discussion of advanced statistical and algorithmic approaches to draw insights from petabyte scale genomic and health data; success stories to date; and a view towards the future of clinical integration of genomics in the learning health system.
Speakers:
- Heidi Rehm, Ph.D.
Chief Genomics Officer, MGH; Professor of Pathology, MGH, BWH & Harvard Medical School; Medical Director, Broad Institute Clinical Research Sequencing Platform. - Saiju Pyarajan, Ph.D.
Director, Centre for Data and Computational Sciences,VABHS, and Department of Medicine, BWH and HMS - Tianxi Cai, Sci.D
John Rock Professor of Population and Translational Data Sciences, Department of Biostatistics, Harvard School of Public Health - Susan Redline, M.D., M.P.H
Farrell Professor of Sleep MedicineHarvard Medical School, Brigham and Women’s Hospital and Beth Israel Deaconess Medical Center - Avinash Sahu, Ph.D.
Postdoctoral Research Fellow, Dana Farber Cancer Institute, Harvard School of Public Health - Peter J. Park, Ph.D.
Professor of Biomedical Informatics, Department of Biomedical Informatics, Harvard Medical School - David Roberson
Community Engagement Manager, Seven Bridges
Registration & Schedule
Rank-Based Independence Testing in Near Linear Time
Speaker: Chaim Even-Zohar (Alan Turing Institute, London)
Title: Rank-Based Independence Testing in Near Linear Time
Abstract: In 1948 Hoeffding proposed a nonparametric test that detects dependence between two continuous random variables (X,Y), based on the ranking of n paired samples (Xi,Yi). The computation of this commonly-used test statistic requires O(n log n) time. Hoeffding’s test is consistent against any dependent probability density f(x,y), but can be fooled by other bivariate distributions with continuous margins. Variants of this test with stronger consistency have been considered in works by Blum, Kiefer, and Rosenblatt, Yanagimoto, and Bergsma and Dassios, and others. The so far best known algorithms to compute them have required quadratic time.
We present an algorithm that computes these improved tests in time O(n log n). It is based on a new combinatorial approach for counting pattern occurrences in a given permutation, which we call corner tree formulas, and will be explained in the talk.Joint work with Calvin Leng.
Symmetry types in QFT and the CRT theorem
Title: Symmetry types in QFT and the CRT theorem
Abstract: I will discuss ideas around symmetry and Wick rotation contained in joint work with Mike Hopkins (https://arxiv.org/abs/1604.06527). This includes general symmetry types for relativistic field theories and their Wick rotation. I will then indicate how the basic CRT theorem works for general symmetry types, focusing on the case of the pin groups. In particular, I expand on a subtlety first flagged by Greaves-Thomas.
Spacetime and Quantum Mechanics Master Class Workshop
As part of the program on Spacetime and Quantum Mechanics, Total Positivity and Motives, the CMSA will host a “Master Class Workshop” on October 28-30, 2019. Each day of the workshop will feature an intensive full day of pedagogical lectures, with the aim of bringing actively interested but non-expert physicists and mathematicians up to speed on the featured topics.
Everyone is welcome to attend the lectures.
The master class workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Photos of the event
Organizers:
- Nima Arkani-Hamed (IAS)
- Lauren Williams (Harvard) | Slides 1 | Slides 2 | Slides 3
- Alex Postnikov (MIT)
- Thomas Lam (Michigan)
02-03-2017 CMSA Members’ Seminar
Hansol Hong, Harvard
Title: Homological Mirror Functors
Abstract: I will first give a brief introduction to mirror symmetry, which intertwines symplectic geometry and complex geometry of a pair of Kahler manifolds, and explain mirror construction using formal deformation of a Lagrangian submanifold. We will see that counting of holomorphic discs bounding Lagrangian naturally gives rise to a mirror space (Landau-Ginzburg model) and a functor from Fukaya category to its mirror matrix factorization category. I will mainly focus on one specific example to give a concrete description of the construction.
Applications of instantons, sphalerons and instanton-dyons in QCD
Title: Applications of instantons, sphalerons and instanton-dyons in QCD
Abstract: I start with a general map of gauge topology, including monopoles, instantons and instanton-dyons. Then comes reminder of the “topological landscape”, the minimal energy gauge field configurations, as a function of Chern-Simons number Ncs and r.m.s. size. It includes “valleys” at integer Ncs separated by mountain ridges. The meaning of instantons, instanton-antiinstanton “streamlines” or thimbles, and sphalerons are reminded, together with some proposal to produce sphalerons at LHC and RHIC.
Applications of instanton ensembles, as a model of QCD vacuum, are mostly related to their fermionic zero modes and t’Hooft effective Lagrangian, which explains explicit and spontaneous breaking of chiral symmetries. Recent applications are related with hadronic wave functions, at rest and in the light front (LFWFs). Two application would be spin-dependent forces and the so called “flavor asymmetry of antiquark sea” of the nucleons. At temperatures comparable to deconfinement transition, instantons get split into constituents called instanton-dyons. Studies of their ensemble explains both deconfinement and chiral transitions, in ordinary and deformed QCD.
Oscillations in the thermal conductivity of a spin liquid*
Title: Oscillations in the thermal conductivity of a spin liquid*
Abstract: The layered honeycomb magnet alpha-RuCl3 orders below 7 K in a zigzag phase in zero field. An in-plane magnetic field H||a suppresses the zigzag order at 7 Tesla, leaving a spin-disordered phase widely believed to be a quantum spin liquid (QSL) that extends to ~12 T. We have observed oscillations in the longitudinal thermal conductivity Kxx vs. H from 0.4 to 4 K. The oscillations are periodic in 1/H (with a break-in-slope at 7 T). The amplitude function is maximal in the QSL phase (7 –11.5 T). I will describe a benchmark for crystalline disorder, the reproducibility and intrinsic nature of the oscillations, and discuss implications for the QSL state. I will also show detailed data on the thermal Hall conductivity Kxy measured from 0.4 K to 10 K and comment on recent half-quantization results.
*Czajka et al., Nature Physics 17, 915 (2021).
Collaborators: Czajka, Gao, Hirschberger, Lampen Kelley, Banerjee, Yan, Mandrus and Nagler.
Line defects in CFTs: Renormalization group flows and semiclassical limits
Title: Line defects in CFTs: Renormalization group flows and semiclassical limits
Abstract: I will discuss line defects in d-dimensional Conformal Field Theories (CFTs). In the first part of the talk, I will argue that the ambient CFT places nontrivial constraints on Renormalization Group (RG) flows on such line defects. I will show that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. In the second part of the talk, I will present some applications. In particular, I will discuss impurities with large isospin S for some O(3) symmetric theories in the epsilon expansion. For sufficiently large S diagrammatic perturbation theory breaks down, and these are studied in a semiclassical expansion at fixed epsilon S.
Quantum Information Workshop
Please note, this workshop has been postponed to a later date. Details will be posted to this page when they are available.
The CMSA will host a workshop on Quantum Information. This workshop will be held virtually using Zoom.
The workshop on Quantum information is organized by Mikhail Lukin, Horng-Tzer Yau, and Norman Yao.
More information to follow.
A tour of categorical symmetry
Title: A tour of categorical symmetry
Abstract: I will discuss some perspectives on symmetry coming from the study of topological defects in quantum field theory. I will argue that we should take topological defects themselves to define the symmetries of QFT. This gives us a view of the “category of QFTs”. I will describe some examples of these “categorical symmetries”, their applications, and some open problems.
Integrability and chaos of 1+1d chiral edge states
Speaker: Biao Lian (Princeton)
Title: Integrability and chaos of 1+1d chiral edge states
Abstract: I will talk about the integrability and chaos of 1+1d interacting chiral edge states, which may arise on the edge of 2+1d topological phases. We show that integrable chiral Luttinger liquid is not always a good low energy description of the edge states, and marginal interactions can significantly affect their spectrum and integrability. We first study N identical chiral Majorana fermion modes with random 4-fermion interactions, where we show that the system undergoes a transition from integrable to quantum chaotic as N increases. The large N limit defines a chiral SYK model where the Lyapunov exponent in the out-of-time-ordered correlation can be solved analytically. I will also present a chiral SY model consisting of N interacting SU(M)_1 WZW models, which host anyons and exhibits similar quantum chaos for Abelian anyons. Lastly, I will talk about the analytical and numerical study of the 4/3 FQH edge theory, which shows unusual behavior in its integrability.
Anomaly resolution via decomposition
Speaker: Eric Sharpe (Virginia Tech)
Title: Anomaly resolution via decomposition
Abstract: In this talk we will discuss a method of anomaly resolution due to Wang-Wen-Witten in the special case of (1+1) dimensional theories. Briefly, for our purposes, Wang-Wen-Witten argued that an ill-defined anomalous orbifold [X/G] could be resolved by extending G to a larger group and adding suitable phases. We analyze this process from the perspective of decomposition, a property of (1+1)-dimensional theories with “one-form symmetries” first described in 2006. Examples of such theories include orbifolds with trivially-acting subgroups, of which the extensions of [X/G] are examples. After a review of decomposition, we will see that decomposition implies that in (1+1) dimensions, the Wang-Wen-Witten procedure results in orbifolds that are equivalent to disjoint unions of orbifolds of X by explicitly nonanomalous subgroups of G.
General Relativity Seminar, Wednesdays
The Seminar on General Relativity will take place every Wednesday from 12pm – 1pm in CMSA Building, 20 Garden Street, G10.
The list of speakers is below and will be updated as details are confirmed.
Date Name Title 04-06-2016 Mihalis Dafermos (Princeton) The black hole stability problem: the inside story 04-13-2016 Felix Finster, University of Regensburg Linear stability of Kerr black holes 04-20-2016 Paul Chesler, Harvard Physics Numerical relativity in asymptotically anti-de Sitter spacetime 04-27-2016 Andy Strominger (Harvard Physics) & Mihalis Dafermos (Princeton University) The Scattering Problem in General Relativity 05-04-2016 Robert Penna, MIT BMS invariance and the membrane paradigm 05-11-2016 Piotr T. Chruściel, University of Vienna Gluing things in general relativity 05-18-2016 Achilleas Porfyriadis, Harvard Physics Gravitational waves from the Kerr/CFT correspondence 05-25-2016 Scott Hughes, MIT The gravitational-wave event GW150914: What we learned, and how we learned it CMSA Math-Science Literature Lecture: Immersions of manifolds and homotopy theory
Ralph Cohen (Stanford University)
Title: Immersions of manifolds and homotopy theory
Abstract: The interface between the study of the topology of differentiable manifolds and algebraic topology has been one of the richest areas of work in topology since the 1950’s. In this talk I will focus on one aspect of that interface: the problem of studying embeddings and immersions of manifolds using homotopy theoretic techniques. I will discuss the history of this problem, going back to the pioneering work of Whitney, Thom, Pontrjagin, Wu, Smale, Hirsch, and others. I will discuss the historical applications of this homotopy theoretic perspective, going back to Smale’s eversion of the 2-sphere in 3-space. I will then focus on the problems of finding the smallest dimension Euclidean space into which every n-manifold embeds or immerses. The embedding question is still very much unsolved, and the immersion question was solved in the 1980’s. I will discuss the homotopy theoretic techniques involved in the solution of this problem, and contributions in the 60’s, 70’s and 80’s of Massey, Brown, Peterson, and myself. I will also discuss questions regarding the best embedding and immersion dimensions of specific manifolds, such has projective spaces. Finally, I will end by discussing more modern approaches to studying spaces of embeddings due to Goodwillie, Weiss, and others. This talk will be geared toward a general mathematical audience.
Talk chair: Michael Hopkins
The Inside View: Raymarching and the Thurston Geometries
On Wednesday, December 16 at 12:00 p.m. EST, WAM and CMSA will host a holiday seminar featuring Sabetta Matsumoto, Georgia Institute of Technology who will present The Inside View: Raymarching and the Thurston Geometries.
The properties of euclidean space seem natural and obvious to us, to the point that it took mathematicians over two thousand years to see an alternative to Euclid’s parallel postulate. The eventual discovery of hyperbolic geometry in the 19th century shook our assumptions, revealing just how strongly our native experience of the world blinded us from consistent alternatives, even in a field that many see as purely theoretical. Non-euclidean spaces are still seen as unintuitive and exotic, but with direct immersive experiences we can get a better intuitive feel for them. The latest wave of virtual reality hardware, in particular the HTC Vive, tracks both the orientation and the position of the headset within a room-sized volume, allowing for such an experience. We create realtime rendering to explore the three-dimensional geometries of the Thurston/Perelman geometrization theorem. In this talk, we use the “inside view” of each manifold to try to understand its geometry and what life might be like on the inside. Joint work with Rémi Coulon, Henry Segerman and Steve Trettel.
Register to access this event here
Cosmic Road to New Physics
The CMSA will host a 3-day workshop on cosmological signatures of fundamental physics. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
The next decade will see a wealth of new cosmological data, which can lead to new insights for fundamental physics. This upcoming data will span the entire history of the cosmos, from the era prior to big-bang nucleosynthesis to the inner Galactic structure today, including the intervening eras of recombination and cosmic dawn. Often, beyond-standard-model (BSM) physics will leave imprints in more than one of these eras. Thus, it is timely to gather experts in BSM physics across the entire cosmic history to exchange ideas and develop joint and powerful probes of new physics. For this program, it will be crucial to have an overlap of particle physicists, astrophysicists and cosmologists. There are a number of tools and techniques being actively developed across these disciplines. The workshop aims to provide a platform for efficient exchange of these new ideas.
The first day we will discuss sub-Galactic probes, including Gaia data and gravitational waves. The second day we will cover cosmological probes, such as the cosmic microwave background and the 21-cm line. The third day we will discuss early Universe probes, such as inflation and phase transitions. Every day the meeting will begin with a pedagogical blackboard talk plus an overview talk, followed by about 4 talks on more specific topics.
Organizers:
Scientific Advisory:
- Lisa Randall
- Matt Reece
- Shing-Tung Yau
Speakers:
- Yacine Ali-Haimoud, NYU
- Mustafa Amin, Rice University
- Xingang Chen, Harvard University
- Francis-Yan Cyr-Racine, University of New Mexico
- Francesca Chadha-Day, University of Cambridge
- Jiji Fan, Brown University
- Daniel Grin, Institute for Advanced Study
- Anson Hook, University of Maryland
- Junwu Huang, Perimeter Institute
- Hongwan Liu, NYU
- Gustavo Marques-Tavares, University of Maryland
- Guilherme Pimentel, University of Amsterdam
- Tracy Slatyer, MIT
- Lian-Tao Wang, University of Chicago
Compactification for cluster varieties without frozen variables of finite type
CMSA, 20 Garden Street, Cambridge, MA 02138 USASpeaker: Man-Wai Cheung
Title: Compactification for cluster varieties without frozen variables of finite type
Abstract: Cluster varieties are blow up of toric varieties. They come in pairs $(A,X)$, with $A$ and $X$ built from dual tori. Compactifications of $A$, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties while the compactifications of X, studied by Fock and Goncharov, generalize the fan construction. The conjecture is that the $A$ and the $X$ cluster varieties are mirrors to each other. Together with Tim Magee, we have shown that there exists a positive polytope for the type $A$ cluster varieties which give us a hint to the Batyrev–Borisov construction.
Special Lecture Series on Donaldson-Thomas and Gromov-Witten Theories
From March 8 to April 19, the Center of Mathematical Sciences and Applications will be hosting a special lecture series on Donaldson-Thomas and Gromov-Witten Theories. Artan Sheshmani (QGM Aarhus and CMSA Harvard) will give eight talks on the topic on Wednesdays and Fridays from 9:00-10:30 am, which will be recorded and promptly available on CMSA’s Youtube Channel.
2019 Ding Shum Lecture
On October 22, 2019, the CMSA will be hosting our third annual Ding Shum lecture. This year’s lecture will be a talk on “Election Security” by Ronald L. Rivest (MIT). The lecture will take place from 4:30-5:30pm in Science Center, Hall A.
Ronald L. Rivest is an Institute Professor at the Massachusetts Institute of Technology. He is a member of the Electrical Engineering and Computer Science Department and the Computer Science and Artificial Intelligence Laboratory (CSAIL) and a founder of the Cryptography and Information Security research group within CSAIL. His research has been in the areas of algorithms, machine learning, cryptography, and election security, for which he has received multiple awards, including: the ACM Turing Award (with Adleman and Shamir), the BBVA Frontiers of Knowledge Award, National Inventor’s Hall of Fame membership, and the Marconi Prize.
Prof. Rivest is also well-known as a co-author of the textbook “Introduction to Algorithms” (with Cormen, Leiserson, and Stein), and as a co-inventor of the RSA public-key cryptosystem (with Adleman and Shamir). He is a co-founder of RSA and of Verisign.He has served on the Technical Guidelines Development Committee (advisory to the Election Assistance Commission), in charge of the Security subcommittee. He is a member of the CalTech/MIT Voting Technology Project, on the Board of Verified Voting, and an advisor to the Electronic Privacy Information Center. Additionally, he has served on the Technical Guidelines Development Committee (advisory to the Election Assistance Commission), as a member of the CalTech/MIT Voting Technology Project, and as an advisor to the Electronic Privacy Information Center.
Last year featured Eric Maskin, who spoke on “How to Improve Presidential Elections: the Mathematics of Voting.” The first Ding Shum lecture took place on October 10, 2017, featuring Leslie Valiant on “Learning as a Theory of Everything.”
This event is made possible by the generous funding of Ding Lei and Harry Shum.
Noncommutative Analysis, Computational Complexity, and Quantum Information
On October 16-18, 2019 the CMSA will be hosting a workshop on Noncommutative Analysis, Computational Complexity, and Quantum Information.
This workshop will focus on linking three different rapidly developing areas: noncommutative real algebraic geometry (RAG), theory of computation and quantum information theory. This mix of overlapping but independently developing topics should lead to a stimulating flow of tools and important problems into several disciplines. Given the different communities there will be an emphasis on tutorials and making the lectures broadly understandable.
The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by Boaz Barak, Bill Helton, Pablo Parrilo, Tselil Schramm.
Please register here
Speakers:
- Jason Altschuler, MIT | Video
- Boaz Barak, Harvard | Video
- Ankit Garg, Microsoft Research | Slides | Video
- David Gosset, University of Waterloo | Video
- Aram Harrow, MIT | Video
- Igor Klep, University of Ljubljana
- Salma Kuhlmann, Universität Konstanz | Video
- Scott McCullough, University of Florida | Slides
- Ion Nechita, Laboratoire de Physique Théorique | Slides | Video
- Rafael Oliveira, University of Toronto | Video
- Vern Paulsen, University of Waterloo | Video
- Suvrit Sra, MIT | Video
- Victor Vinnikov, Ben Gurion University | Video
- Jurij Volčič, Texas A&M University | Slides | Video
- Adam Bene Watts, MIT
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Conference on Differential Geometry, Calabi-Yau theory and General Relativity: A conference in honor of the 70th Birthday of Shing-Tung Yau
1 Oxford Street, Cambridge MA 02138On May 2-5, 2019 the Harvard Mathematics Department hosted a Conference on Differential Geometry, Calabi-Yau Theory and General Relativity: A conference in honor of the 70th Birthday of Shing-Tung Yau. The conference was held in the Science Center, Lecture Hall C.
Organizers:
- Horng-Tzer Yau (Harvard)
- Wilfried Schmid (Harvard)
- Clifford Taubes (Harvard)
- Cumrun Vafa (Harvard)
Speakers:
- Lydia Bieri, University of Michigan
- Tristan Collins, MIT
- Simon Donaldson, Imperial College
- Fan Chung Graham, UC San Diego
- Nigel Hitchin, Oxford University
- Jun Li, Stanford University
- Kefeng Liu, UCLA
- Chiu-Chu Melissa Liu, Columbia University
- Alina Marian, Northeastern University
- Xenia de la Ossa, Oxford University
- Duong H. Phong, Columbia University
- Richard Schoen, UC Irvine
- Andrew Strominger, Harvard University
- Nike Sun, MIT
- Clifford Taubes, Harvard University
- Chuu-Lian Terng, UC Irvine
- Valentino Tosatti, Northwestern University
- Karen Uhlenbeck, University of Texas
- Cumrun Vafa, Harvard University
- Mu Tao Wang, Columbia University
- Edward Witten, IAS
- Stephen Yau, Tsinghua University, P.R. China
CMSA Math-Science Literature Lecture: A personal story of the 4D Poincare conjecture
Michael Freedman (Microsoft – Station Q)
Title: A personal story of the 4D Poincare conjecture
Abstract: The proof of PC4 involved the convergence of several historical streams. To get started: high dimensional manifold topology (Smale), a new idea on how to study 4-manifolds (Casson), wild “Texas” topology (Bing). Once inside the proof: there are three submodules: Casson towers come to life (in the sense of reproduction), a very intricate explicit shrinking argument (provided by Edwards), and the “blind fold” shrinking argument (which in retrospect is in the linage of Brown’s proof of the Schoenflies theorem). Beyond those mentioned: Kirby, Cannon, Ancel, Quinn, and Starbird helped me understand my proof. I will discuss the main points and how they fit together.
Talk Chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: From string theory and Moonshine to vertex algebras
Bong Lian (Brandeis)
Title: From string theory and Moonshine to vertex algebras
Abstract: This is a brief survey of the early historical development of vertex algebras, beginning in the seventies from Physics and Representation Theory. We shall also discuss some of the ideas that led to various early formulations of the theory’s foundation, and their relationships, as well as some of the subsequent and recent developments. The lecture is aimed at a general audience.
CMSA Math-Science Literature Lecture: Four-dimensional topology
Ciprian Manolescu (Stanford)
Title: Four-dimensional topology
Abstract: I will outline the history of four-dimensional topology. Some major events were the work of Donaldson and Freedman from 1982, and the introduction of the Seiberg-Witten equations in 1994. I will discuss these, and then move on to what has been done in the last 20 years, when the focus shifted to four-manifolds with boundary and cobordisms. Floer homology has led to numerous applications, and recently there have also been a few novel results (and proofs of old results) using Khovanov homology. The talk will be accessible to a general mathematical audience.
Conference on Algebraic Geometry, Representation theory and Mathematical Physics
From April 29 to May 1, 2019 the CMSA will be hosting a Conference on Algebraic Geometry, Representation theory and Mathematical Physics. This workshop is organized by Bong Lian (Brandeis) and Artan Sheshmani (CMSA) . The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Videos
Speakers:
- Dan Abramovich, Brown
- Roman Bezrukavnikov, MIT
- Fedor Bogomolov, NYU
- Qile Chen, Boston College
- Dawei Chen, Boston College
- Alexander Efimov, Moscow
- Pavel Etingof, MIT
- Maksym Fedorchuk, Boston College
- Dennis Gaitsgory, Harvard
- Amin Gholampour, Maryland
- Brendan Hassett, Brown
- Ludmil Katzarkov, Miami & Moscow
- Si Li, Tsinghua
- Andrei Negut, MIT
- Yuri Tschinkel, NYU
- Wei Zhang, MIT
Monday, April 29
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 10:00am Wei Zhang, MIT Title: The arithmetic fundamental lemma for diagonal cycles Abstract: I’ll recall the Gross–Zagier theorem and a high dimensional generalization, the arithmetic Gan-Gross-Prasad conjecture, which relates the height pairing of arithmetic diagonal cycles on certain shimura varieties to the first order derivative of certain L-functions. The arithmetic fundamental lemma conjecture arises from the relative trace formula approach to this conjecture. I will recall the statement of the arithmetic fundamental lemma and outline a proof.
10:00 – 10:30am Break 10:30 – 11:30am Yuri Tschinkel, NYU Title: Equivariant birational geometry and modular symbols Abstract: We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups (joint with M. Kontsevich and V. Pestun).
11:30 – 1:30pm Lunch 1:30 – 2:30pm Alexander Efimov, Moscow Title: Torsionness for regulators of canonical extensions Abstract: I will sketch a generalization of the results of Iyer and Simpson arXiv:0707.0372 to the general case of a normal-crossings divisor at infinity.
2:30 – 3:00pm Break 3:00 – 4:00pm Amin Gholampour, Maryland Title: Euler Characteristics of punctual quot schemes on threefolds Abstract: Let F be a homological dimension 1 torsion free sheaf on a nonsingular quasi-projective threefold. The first cohomology of the derived dual of F is a 1-dimension sheaf G supported on the singular locus of F. We prove a wall-crossing formula relating the generating series of the Euler characteristics of Quot(F, n) and Quot(G,n), where Quot(-,n) denotes the quot scheme of length n quotients. We will use this relation in studying the Euler characteristics of the moduli spaces of stable torsion free sheaves on nonsingular projective threefolds. This is a joint work with Martijn Kool.
4:00 – 4:30pm Break 4:30 – 5:30pm Maksym Fedorchuck, BC Title: Stability of one-parameter families of weighted hypersurfaces Abstract: We define a notion of stability for fibrations over a curve with generic fibers being weighted hypersurfaces (in some weighted projective space) generalizing Kollár’s stability for families of hypersurfaces in a projective space. The stability depends on a choice of an effective line bundle on the parameter space of weighted hypersurfaces and different choices pick out different birational model of the total space of the fibration. I will describe enumerative geometry that goes into understanding these stability conditions, and, if time permits, examples where this machinery can be used to produce birational models with good properties. Joint work with Hamid Ahmadinezhad and Igor Krylov.
Tuesday, April 30
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 10:00am Brendan Hassett, Brown Title: Rationality for geometrically rational threefolds Abstract: We consider rationality questions for varieties over non-closed fields that become rational over an algebraic closure, like smooth complete intersections of two quadrics. (joint with Tschinkel)
10:00 – 10:30am Break 10:30 – 11:30am Dennis Gaitsgory, Harvard Title: The Fundamental Local Equivalence in quantum geometric Langlands Abstract: The Fundamental Local Equivalence is statement that relates the q-twisted Whittaker category of the affine Grassmannian for the group G and the category of modules over the Langlands dual “big” quantum group. The non-triviaiity of the statement lies is the fact that the relationship between the group and its dual is combinatorial, so to prove the FLE one needs to express both sides in combinatorial terms. In the talk we will indicate the proof of a related statement for the “small” quantum group. The combinatorial link is provided by the category of factorization modules over a certain factorization algebra, which in itself is a geometric device that concisely encodes the root data.
11:30 – 1:00pm Lunch 1:00- 2:00pm Andrei Negut, MIT Title: AGT relations in geometric representation theory Abstract: I will survey a program that seeks to translate the Alday-Gaiotto-Tachikawa correspondence (between gauge theory on R^4 and conformal field theory) into the language of algebraic geometry. The objects of study become moduli spaces of sheaves on surfaces, and the goal is to connect them with the W-algebra of type gl_n.
2:00 – 2:15pm Break 2:15 – 3:15pm Dan Abramovich, Brown Title: Resolution in characteristic 0 using weighted blowing up Abstract: Given a variety $X$, one wants to blow up the worst singular locus, show that it gets better, and iterate until the singularities are resolved.
Examples such as the whitney umbrella show that this iterative process cannot be done by blowing up smooth loci – it goes into a loop.
We show that there is a functorial way to resolve varieties using \emph{weighted} blowings up, in the stack-theoretic sense. To an embedded variety $X \subset Y$ one functorially assigns an invariant $(a_1,\ldots,a_k)$, and a center locally of the form $(x_1^{a_1} , \ldots , x_k^{a_k})$, whose stack-theoretic weighted blowing up has strictly smaller invariant under the lexicographic order.
This is joint work with Michael Tëmkin (Jerusalem) and Jaroslaw Wlodarczyk (Purdue), a side product of our work on functorial semistable reduction. A similar result was discovered by G. Marzo and M. McQuillan.
3:15 – 3:30pm Break 3:30 – 4:30pm Fedor Bogomolov, NYU Title: On the base of a Lagrangian fibration for a compact hyperkahler manifold. Abstract: In my talk I will discuss our proof with N. Kurnosov that the base of such fibration for complex projective manifold hyperkahler manifold of dimension $4$ is always a projective plane $P^2$. In fact we show that the base of such fibration can not have a singular point of type $E_8$. It was by the theorem of Matsushita and others that only quotient singularities can occur and if the base is smooth then the it is isomorphic to $P^2$. The absence of other singularities apart from $E_8$ has been already known and we show that $E-8$ can not occur either. Our method can be applied to other types of singularities for the study of Lagrangian fibrations in higher dimensions More recently similar result was obtained by Huybrechts and Xu.
4:30 – 4:45pm Break 4:45 – 5:45pm Dawei Chen, BC Title: Volumes and intersection theory on moduli spaces of Abelian differentials Abstract: Computing volumes of moduli spaces has significance in many fields. For instance, Witten’s conjecture regarding intersection numbers on moduli spaces of Riemann surfaces has a fascinating connection to the Weil-Petersson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. In this talk I will introduce an analogue of Witten’s intersection numbers on moduli spaces of Abelian differentials to compute the Masur-Veech volumes induced by the flat metric associated with Abelian differentials. This is joint work with Moeller, Sauvaget, and Zagier (arXiv:1901.01785).
Wednesday, May 1
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 10:00am Pavel Etingof, MIT Title: Short star-products for filtered quantizations Abstract: PDF
This is joint work with Eric Rains and Douglas Stryker.
10:00 – 10:30am Break 10:30 – 11:30am Roman Bezrukavnikov, MIT Title: Stability conditions and representation theory Abstract: I will recall the concept of real variation of stabilities (introduced in my work with Anno and Mirkovic)
and its relation to modular Lie algebra representations. I will also address a potential generalization of that picture
to modular representations of affine Lie algebras related to the classical limit of geometric Langlands duality and its local counterpart.11:30 – 11:45am Break 11:45 – 12:45pm Qile Chen, BC Title: Counting curves in critical locus via logarithmic compactification Abstract: An R-map consists of a pre-stable map to possibly non-GIT quotient together with sections of certain spin bundles. The moduli of R-maps are in general non-compact. When the target of R-maps is equipped with a super-potential W with compact critical locus, using Kiem-Li cosection localization it has been proved by many authors in various settings that the virtual cycle of R-maps can be represented by the cosection localized virtual cycle which is supported on the proper locus consisting of R-maps in the critical locus of W. Though the moduli of R-maps is equipped with a natural torus action by scaling of the spin bundles, the non-compactness of the R-maps moduli makes such powerful torus action useless.
In this talk, I will introduce a logarithmic compactification of the moduli of R-maps using certain modifications of stable logarithmic maps. The logarithmic moduli space carries a canonical virtual cycle from the logarithmic deformation theory. In the presence of a super-potential with compact critical locus, it further carries a reduced virtual cycle. We prove that (1) the reduced virtual cycle of the compactification can be represented by the cosection localized virtual cycle; and (2) the difference of the canonical and reduced virtual cycles is another reduced virtual cycle supported along the logarithmic boundary. As an application, one recovers the Gromov-Witten invariants of the critical locus as the invariants of logarithmic R-maps of its ambient space in an explicit form. The latter can be calculated using the spin torus action.
This is a joint work with Felix Janda and Yongbin Ruan.
12:45 – 2:30pm Lunch 2:30 – 3:30pm Si Li, Tsinghua Title: Semi-infinite Hodge structure: from BCOV theory to Seiberg-Witten geometry Abstract: I will explain how the semi-infinite Hodge theory extends Kodaira-Spencer gravity (Bershadsky-Cecotti-Ooguri-Vafa theory of B-twisted closed topological string field theory) into a full solution of Batalin-Vilkovisky master equation. This allows us to formulate quantum B-model via a rigorous BV quantization method and construct integrable hierarchies arising naturally from the background symmetry. In the second part of the talk, I will explain the recent discovery of the connection between K.Saito’s primitive form and 4d N=2 Seiberg-Witten geometry arising from singularity theory.
3:30 – 4:00pm Break 4:00 – 5:00pm Ludmil Katzarkov, Moscow Title: PDE’s non commutative motives and HMS. Abstract: In this talk we will discuss the theory of central manifolds and the new structures in geometry it produces. Application to Bir. Geometry will be discussed.
Workshop on Mirror Symmetry and Stability
This three-day workshop will take place at Harvard University on March 18-20, 2019 in Science Center room 507. The main topic will be stability conditions in homological mirror symmetry. This workshop is funded by the Simons Collaboration in Homological Mirror Symmetry.
Organizers: Denis Auroux, Yu-Wei Fan, Hansol Hong, Siu-Cheong Lau, Bong Lian, Shing-Tung Yau, Jingyu Zhao
Speakers:
Dylan Allegretti (Sheffield)
Tristan Collins (MIT)
Naoki Koseki (Tokyo)
Chunyi Li (Warwick)
Jason Lo (CSU Northridge)
Emanuele Macrì (NEU & IHES)
Genki Ouchi (Riken iTHEMS)
Pranav Pandit (ICTS)
Laura Pertusi (Edinburgh)
Jacopo Stoppa (SISSA)
Alex Takeda (UC Berkeley)
Xiaolei Zhao (UC Santa Barbara)More details will be added later.
Visit the event page for more information.
Stochastic PDE as scaling limits of interacting particle systems
Abstract: Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models. In this talk, I will illustrate how this challenge can be overcome by elucidating the probabilistic connections between models of different levels of detail. These connections explain how stochastic partial differential equations (SPDE) arise naturally from particle models.
I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics.
The Festina Lente Bound
Abstract: I will explain what the Festina Lente bound means and where it comes from. Then I discuss its possible implications for phenomenology, both top-down and bottom-up.
CMSA Math-Science Literature Lecture: Deep Networks from First Principles
Yi Ma (University of California, Berkeley)
Title: Deep Networks from First Principles
Abstract: In this talk, we offer an entirely “white box’’ interpretation of deep (convolution) networks from the perspective of data compression (and group invariance). In particular, we show how modern deep layered architectures, linear (convolution) operators and nonlinear activations, and even all parameters can be derived from the principle of maximizing rate reduction (with group invariance). All layers, operators, and parameters of the network are explicitly constructed via forward propagation, instead of learned via back propagation. All components of so-obtained network, called ReduNet, have precise optimization, geometric, and statistical interpretation. There are also several nice surprises from this principled approach: it reveals a fundamental tradeoff between invariance and sparsity for class separability; it reveals a fundamental connection between deep networks and Fourier transform for group invariance – the computational advantage in the spectral domain (why spiking neurons?); this approach also clarifies the mathematical role of forward propagation (optimization) and backward propagation (variation). In particular, the so-obtained ReduNet is amenable to fine-tuning via both forward and backward (stochastic) propagation, both for optimizing the same objective. This is joint work with students Yaodong Yu, Ryan Chan, Haozhi Qi of Berkeley, Dr. Chong You now at Google Research, and Professor John Wright of Columbia University.
Talk chair: Harry Shum
On singular Hilbert schemes of points
Abstract: It is well known that the Hilbert schemes of points on smooth surfaces are smooth. In higher dimensions the Hilbert schemes of points are in general singular. In this talk we will present some examples and conjectures on the local structures of the Hilbert scheme of points on $\mathbb{P}^3$. As an application we study a conjecture of Wang-Zhou on the Euler characteristics of the tautological sheaves on Hilbert schemes of points.
A mirror theorem for GLSMs
Abstract: A gauged linear sigma model (GLSM) consists roughly of a complex vector space V, a group G acting on V, a character \theta of G, and a G-invariant function w on V. This data defines a GIT quotient Y = [V //_\theta G] and a function on that quotient. GLSMs arise naturally in a number of contexts, for instance as the mirrors to Fano manifolds and as examples of noncommutative crepant resolutions. GLSMs provide a broad setting in which it is possible to define an enumerative curve counting theory, simultaneously generalizing FJRW theory and the Gromov-Witten theory of hypersurfaces. Despite a significant effort to rigorously define the enumerative invariants of a GLSM, very few computations of these invariants have been carried out. In this talk I will describe a new method for computing generating functions of GLSM invariants. I will explain how these generating functions arise as derivatives of generating functions of Gromov-Witten invariants of Y.
Cytoskeletal Energetics and Energy Metabolism
Abstract: Life is a nonequilibrium phenomenon. Metabolism provides a continuous flux of energy that dictates the form and function of many subcellular structures. These subcellular structures are active materials, composed of molecules which use chemical energy to perform mechanical work and locally violate detailed balance. One of the most dramatic examples of such a self-organizing structure is the spindle, the cytoskeletal based assembly which segregates chromosomes during cell division. Despite its central role, very little is known about the nonequilibrium thermodynamics of active subcellular matter, such as the spindle. In this talk, I will describe ongoing work from my lab aimed at understanding the flows of energy which drive the nonequilibrium behaviors of the cytoskeleton in vitro and in vivo.
Simons Collaboration Workshop, April 5-7, 2018
The CMSA will be hosting a three-day Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory on April 5-7, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Please click here to register for this event. We have space for up to 30 registrants on a first come, first serve basis.
We may be able to provide some financial support for grad students and postdocs interested in this event. If you are interested in funding, please send a letter of support from your mentor to Hansol Hong at hansol84@gmail.com.
Confirmed Speakers:
- Jacob Bourjaily (Niels Bohr Institute)
- Mandy Cheung (Havard University)
- Tristan Collins (Harvard University)
- Yoosik Kim (Boston University)
- Yu-Shen Lin (Harvard University)
- Cheuk-Yu Mak (Cambridge University)
- Yu Pan (MIT)
- Mauricio Romo (Tsinghua University)
- Shu-Heng Shao (IAS)
- Zack Sylvan (Columbia University)
- Dmitry Vaintrob (IAS)
Exploring the Holographic Swampland
Abstract: I describe our work looking at `traditional’ scenarios of moduli stabilisation from a holographic perspective. This reveals some interesting structure that is not apparent from the top-down perspective. For vacua in the extreme regions of moduli space, such as LVS in type IIB or the DGKT flux vacua in type IIA, the dual moduli conformal dimensions reduce to fixed values – in a certain sense, the low-conformal dimension part of the CFT is unique and independent of the large number of flux choices. For the DGKT flux vacua these conformal dimensions are also integer, for reasons we do not understand.
What do bounding chains look like, and why are they related to linking numbers?
Abstract: Gromov-Witten invariants count pseudo-holomorphic curves on a symplectic manifold passing through some fixed points and submanifolds. Similarly, open Gromov-Witten invariants are supposed to count disks with boundary on a Lagrangian, but in most cases such counts are not independent of some choices as we would wish. Motivated by Fukaya’11, J. Solomon and S. Tukachinsky constructed open Gromov-Witten invariants in their 2016 papers from an algebraic perspective of $A_{\infty}$-algebras of differential forms, utilizing the idea of bounding chains in Fukaya-Oh-Ohta-Ono’06. On the other hand, Welschinger defined open invariants on sixfolds in 2012 that count multi-disks weighted by the linking numbers between their boundaries. We present a geometric translation of Solomon-Tukachinsky’s construction. From this geometric perspective, their invariants readily reduce to Welschinger’s.
Convexity of Charged Operators in CFTs and the Weak Gravity Conjecture
Abstract: In this talk I will introduce a particular formulation of the Weak Gravity Conjecture in AdS space in terms of the self-binding energy of a particle. The holographic CFT dual of this formulation corresponds to a certain convex-like structure for operators charged under continuous global symmetries. Motivated by this, we propose a conjecture that this convexity is a general property of all CFTs, not just those with weakly-curved gravitational duals. It is possible to test this in simple CFTs, the conjecture passes all the tests performed so far.
CMSA Math-Science Literature Lecture: My life and times with the sporadic simple groups
Robert Griess (University of Michigan)
Title: My life and times with the sporadic simple groups
Abstract: Five sporadic simple groups were proposed in 19th century and 21 additional ones arose during the period 1965-1975. There were many discussions about the nature of finite simple groups and how sporadic groups are placed in mathematics. While in mathematics grad school at University of Chicago, I became fascinated with the unfolding story of sporadic simple groups. It involved theory, detective work and experiments. During this lecture, I will describe some of the people, important ideas and evolution of thinking about sporadic simple groups. Most should be accessible to a general mathematical audience.
The many phases of a cell
Abstract: I will begin by introducing an emerging paradigm of cellular organization – the dynamic compartmentalization of biochemical pathways and molecules by phase separation into distinct and multi-phase condensates. Motivated by this, I will discuss two largely orthogonal problems, united by the theme of phase separation in multi-component and chemically active fluid mixtures.
1. I will propose a theoretical model based on Random-Matrix Theory, validated by phase-field simulations, to characterizes the rich emergent dynamics, compositions, and steady-state properties that underlie multi-phase coexistence in fluid mixtures with many randomly interacting components.
2. Motivated by puzzles in gene-regulation and nuclear organization, I will propose a role for how liquid-like nuclear condensates can be organized and regulated by the active process of RNA synthesis (transcription) and RNA-protein coacervation. Here, I will describe theory and simulations based on a Landau formalism and recent experimental results from collaborators.
Knot homology and sheaves on the Hilbert scheme of points on the plane.
Abstract: The knot homology (defined by Khovavov, Rozansky) provide us with a refinement of the knot polynomial knot invariant defined by Jones. However, the knot homology are much harder to compute compared to the polynomial invariant of Jones. In my talk I present recent developments that allow us to use tools of algebraic geometry to compute the homology of torus knots and prove long-standing conjecture on the Poincare duality the knot homology. In more details, using physics ideas of Kapustin-Rozansky-Saulina, in the joint work with Rozansky, we provide a mathematical construction that associates to a braid on n strands a complex of sheaves on the Hilbert scheme of n points on the plane. The knot homology of the closure of the braid is a space of sections of this sheaf. The sheaf is also invariant with respect to the natural symmetry of the plane, the symmetry is the geometric counter-part of the mentioned Poincare duality.
CMSA Math-Science Literature Lecture: Rationality questions in algebraic geometry
Joe Harris (Harvard)
Title: Rationality questions in algebraic geometry
Abstract: Over the course of the history of algebraic geometry, rationality questions — motivated by both geometric and arithmetic problems — have often driven the subject forward. The rationality or irrationality of cubic hypersurfaces in particular have led to the development of abelian integrals (dimension one), birational geometry (dimension two) and Hodge theory (dimension 3). But there remained much we didn’t understand about the condition of rationality, such as how it behaves in families. However, there has been recent progress: work of Hassett, Tschinkel, Pirutka and others, working with examples in dimension 4, showed that it is in general neither an open condition nor a closed one, but does behave well with respect to specialization. In this talk I’ll try to give an overview of the history of rationality and the current state of our knowledge.
Combinatorics & Complexity Seminar, Fridays
The seminar on Combinatorics and Complexity will be held every Friday from 1:00-4:00pm in CMSA Building, 20 Garden Street, Room G10.
The list of speakers for the upcoming academic year will be posted below and updated as details are confirmed. Titles and abstracts for the talks will be added as they are received.
Additional information on CMSA’s Combinatorics and Complexity program can be found here.
Date Name Title/Abstract 09-08-17 TBA 09-15-2017 TBA 09-22-17 TBA 09-29-17 TBA 10-06-17 TBA 10-13-2017 TBA 10-20-2017 TBA 10-27-2017 TBA 11-03-2017 TBA 11-10-2017 TBA 11-17-2017 TBA 11-24-2017 TBA 12-01-2017 TBA 12-08-2017 TBA Derived projectivizations of two-term complexes
Abstract: For a given two-term complex of vector bundles on a derived scheme (or stack), there are three natural ways to define its “derived projectivizations”: (i) as the derived base-change of the classical projectivization of Grothendieck; (ii) as the derived moduli parametrizing one-dimensional locally free quotients; (iii) as the GIT quotient of the total space by $\mathbb{G}_m$-action. In this talk, we first show that these three definitions are equivalent. Second, we prove a structural theorem about the derived categories of derived projectivizations and study the corresponding mutation theory. Third, we apply these results to various moduli situations, including the moduli of certain stable pairs on curves and the Hecke correspondences of one-point modification of moduli of stable sheaves on surfaces. If time allowed, we could also discuss the generalizations of these results to the derived Quot schemes of locally free quotients.
D-critical structure(s) on Quot schemes of points of Calabi-Yau 3-folds
Abstract: D-critical schemes and Artin stacks were introduced by Joyce in 2015, and play a central role in Donaldson-Thomas theory. They typically occur as truncations of (-1)-shifted symplectic derived schemes, but the problem of constructing the d-critical structure on a “DT moduli space” without passing through derived geometry is wide open. We discuss this problem, and new results in this direction, when the moduli space is the Hilbert (or Quot) scheme of points on a Calabi-Yau 3-fold. Joint work with Michail Savvas.
Drivers of Morphological Complexity
Abstract: During development, organisms interact with their natural habitats while undergoing morphological changes, yet we know little about how the interplay between developing systems and their environments impacts animal morphogenesis. Cnidaria, a basal animal lineage that includes sea anemones, corals, hydras, and jellyfish, offers unique insight into the development and evolution of morphological complexity. In my talk, I will introduce our research on “ethology of morphogenesis,” a novel concept that links the behavior of organisms to the development of their size and shape at both cellular and biophysical levels, opening new perspectives about the design principle of soft-bodied animals. In addition, I will discuss a fascinating feature of cnidarian biology. For humans, our genetic code determines that we will grow two arms and two legs. The same fate is true for all mammals. Similarly, the number of fins of a fish or legs and wings of an insect is embedded in their genetic code. I will describe how sea anemones defy this rule.
References
Anniek Stokkermans, Aditi Chakrabarti, Ling Wang, Prachiti Moghe, Kaushikaram Subramanian, Petrus Steenbergen, Gregor Mönke, Takashi Hiiragi, Robert Prevedel, L. Mahadevan, and Aissam Ikmi. Ethology of morphogenesis reveals the design principles of cnidarian size and shape development. bioRxiv 2021.08.19.456976Ikmi A, Steenbergen P, Anzo M, McMullen M, Stokkermans M, Ellington L, and Gibson M (2020). Feeding-dependent tentacle development in the sea anemone Nematostella vectensis. Nature communications, Sept 02; 11:4399
He S, Del Viso F, Chen C, Ikmi A, Kroesen A, Gibson MC (2018). An axial Hox code controls tissue segmentation and body patterning in Nematostella vectensis. Science, Vol. 361, Issue 6409, pp. 1377-1380.
Ikmi A, McKinney SA, Delventhal KM, Gibson MC (2014). TALEN and CRISPR/Cas9 mediated genome editing in the early-branching metazoan Nematostella vectensis. Nature communications. Nov 24; 5:5486.The Mirror Clemens-Schmid Sequence
Abstract: I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a “mirror P=W” conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting. This is joint work with Charles F. Doran.
Eppur si muovono: rotations in active matter
Abstract: Living matter relies on the self organization of its components into higher order structures, on the molecular as well as on the cellular, organ or even organism scale. Collective motion due to active transport processes has been shown to be a promising route for attributing fascinating order formation processes on these different length scales. Here I will present recent results on structure formation on actively transported actin filaments on lipid membranes and vesicles, as well as the cell migration induced structure formation in the developmental phase of mammary gland organoids. For both systems spherical structures with persistent collective rotations are observed.
9/28/2021 Combinatorics, Physics and Probability Seminar
Title: The hypersimplex and the m=2 amplituhedron
Abstract: I’ll discuss a curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). Along the way, we prove the sign-flip description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers (inspired by an analogous hypersimplex decomposition).
Second Annual STAR Lab Conference
The second annual STAR Lab conference is running 10/29/-10/30/2015 at the Harvard Business School. This event is co-sponsored by the Center of Mathematical Sciences and Applications.
For more information, please consult the event’s website.
Categorification and applications
Abstract: I will give a survey of the program of categorification for quantum groups, some of its recent development and applications to representation theory.
2016 Big Data Conference & Workshop
1 Oxford Street, Cambridge MA 02138! LOCATION CHANGE: The conference will be in Science Center Hall C on Tuesday, Aug.23, 2016.
The Center of Mathematical Sciences and Applications will be hosting a workshop on Big Data from August 12 – 21, 2016 followed by a two-day conference on Big Data from August 22 – 23, 2016.
Big Data Conference features many speakers from the Harvard Community as well as many scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics. This is the second conference on Big Data the Center will host as part of our annual events. The 2015 conference was a huge success.
The conference will be hosted at Harvard Science Center Hall A (Monday, Aug.22) & Hall C (Tuesday, Aug.23): 1 Oxford Street, Cambridge, MA 02138.
The 2016 Big Data conference is sponsored by the Center of Mathematical Sciences and Applications at Harvard University and the Alfred P. Sloan Foundation.
Conference Speakers:
- Jörn Boehnke, Harvard CMSA
- Joan Bruna, UC Berkeley [Video]
- Tamara Broderick, MIT [Video]
- Justin Chen, MIT [Video]
- Yiling Chen, Harvard University [Video]
- Amir Farbin, UT Arlington [Video]
- Doug Finkbeiner, Harvard University [Video]
- Andrew Gelman, Columbia University [Video]
- Nina Holden, MIT [Video]
- Elchanan Mossel, MIT
- Alex Peysakhovich, Facebook
- Alexander Rakhlin, University of Pennsylvania [Video]
- Neal Wadhwa, MIT [Video]
- Jun Yin, University of Wisconsin
- Harry Zhou, Yale University [Video]
Please click Conference Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Please click here for registration.
Conference Schedule:
August 22 – Day 1 8:30am Breakfast 8:55am Opening remarks 9:00am – 9:50am Yiling Chen, “Machine Learning with Strategic Data Sources” [Video] 9:50am – 10:40am Andrew Gelman, “Taking Bayesian Inference Seriously” [Video] 10:40am – 11:10am Break 11:10am – 12:00pm Harrison Zhou, “A General Framework for Bayes Structured Linear Models” [Video] 12:00pm – 1:30pm Lunch 1:30pm – 2:20pm Douglas Finkbeiner, “Mapping the Milky Way in 3D with star colors” [Video] 2:20pm – 3:10pm Nina Holden, “Sparse exchangeable graphs and their limits” [Video] 3:10pm – 3:40pm Break 3:40pm – 4:30pm Alex Peysakhovich, “How social science methods inform personalization on Facebook News Feed” [Video] 4:30pm – 5:20pm Amir Farbin, “Deep Learning in High Energy Physics” [Video] August 23 – Day 2 8:45am Breakfast 9:00am – 9:50am Joan Bruna Estrach, “Addressing Computational and Statistical Gaps with Deep Networks” [Video] 9:50am – 10:40am Justin Chen & Neal Wadhwa, “Smaller Than the Eye Can See: Big Engineering from Tiny Motions in Video” [Video] 10:40am – 11:10am Break 11:10am – 12:00pm Alexander Rakhlin, “How to Predict When Estimation is Hard: Algorithms for Learning on Graphs” [Video] 12:00pm – 1:30pm Lunch 1:30pm – 2:20pm Tamara Broderick, “Fast Quantification of Uncertainty and Robustness with Variational Bayes” [Video] 2:20pm – 3:10pm Elchanan Mossel, “Phylogenetic Reconstruction – a Rigorous Model of Deep Learning” 3:10pm – 3:40pm Break 3:40pm – 4:30pm Jörn Boehnke, “Amazon’s Price and Sales-rank Data: What can one billion prices on 150 thousand products tell us about the economy?” Workshop Participants:
Richard Freeman’s Group:
- Sen Chai, ESSEC
- Brock Mendel, Harvard University
- Raviv Muriciano-Goroff, Stanford University
- Sifan Zhou, CMSA
Scott Kominer’s Group:
- Bradly Stadie, UC Berkeley
- Neal Wadhwa, MIT [Video]
- Justin Chen
Christopher Rogan’s Group:
- Amir Farbin, UT Arlington [Video]
- Paul Jackson, University of Adelaide
For more information about the workshops, please reach out directly to the individual group leaders.
* This event is sponsored by CMSA Harvard University and the Alfred P. Sloan Foundation.
2/16/2021 Computer Science for Mathematicians
Speaker: Michael P. Kim (UC Berkeley)
Title: Outcome Indistinguishability
Abstract: Prediction algorithms assign numbers to individuals that are popularly understood as individual “probabilities” — e.g., what is the probability of 5-year survival after cancer diagnosis? — and which increasingly form the basis for life-altering decisions. The understanding of individual probabilities in the context of such unrepeatable events has been the focus of intense study for decades within probability theory, statistics, and philosophy. Building off of notions developed in complexity theory and cryptography, we introduce and study Outcome Indistinguishability (OI). OI predictors yield a model of probabilities that cannot be efficiently refuted on the basis of the real-life observations produced by Nature.
We investigate a hierarchy of OI definitions, whose stringency increases with the degree to which distinguishers may access the predictor in question. Our findings reveal that OI behaves qualitatively differently than previously studied notions of indistinguishability. First, we provide constructions at all levels of the hierarchy. Then, leveraging recently-developed machinery for proving average-case fine-grained hardness, we obtain lower bounds on the complexity of the more stringent forms of OI. The hardness result provides scientific grounds for the political argument that, when inspecting algorithmic risk prediction instruments, auditors should be granted oracle access to the algorithm, not simply historical predictions.
Joint work with Cynthia Dwork, Omer Reingold, Guy N. Rothblum, Gal Yona; to appear at STOC 2021.
Current Developments in Mathematics 2019
Friday, Nov. 22, 2019 1:30 pm – 5:20 pm
Saturday, Nov. 23, 2019 9:00 am – 5:00 pm
Harvard University Science Center, Hall C
Speakers:
· Svetlana Jitomirskaya (UC Irvine)
· Subash Khot (NYU)
· Jun Li (Stanford)
· André Neves (Chicago)
· Geordie Williamson (U Sidney)
Free and open to the public – registration is required.
Please register in advance online at www.math.harvard.edu/cdmConcluding Conference of the Special Program on Nonlinear Equations, April 8 – 10, 2016
The Center of Mathematical Sciences and Applications will be hosting a concluding conference on April 8-10, 2016 to accompany the year-long program on nonlinear equations. The conference will have 15 speakers and will be hosted at Harvard CMSA Building: Room G10 20 Garden Street, Cambridge, MA 02138
Speakers:
- Lydia Bieri (University of Michigan)
- Luis Caffarelli (University of Texas at Austin)
- Mihalis Dafermos (Princeton University)
- Camillo De Lellis (Universität Zürich)
- Pengfei Guan (McGill University)
- Slawomir Kolodziej (Jagiellonian University)
- Melissa Liu (Columbia University)
- Duong H. Phong (Columbia University)
- Richard Schoen (UC Irvine)
- Cliff Taubes (Harvard University)
- Blake Temple (UC Davis)
- Valentino Tosatti (Northwestern University)
- Tai-Peng Tsai (University of British Columbia)
- Mu-Tao Wang (Columbia University)
- Xu-jia Wang (Australian National University)
Please click NLE Conference Schedule with Abstracts for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Schedule:
April 8 – Day 1 8:30am Breakfast 8:45am Opening remarks 9:00am – 10:00am Camillo De Lellis, “A Nash Kuiper theorem for $C^{1,1:5}$ isometric immersions of disks“ 10:00am – 10:15am Break 10:15am – 11:15am Xu-Jia Wang, “Monge’s mass transport problem“ 11:15am – 11:30am Break 11:30am – 12:30pm Peng-Fei Guan, “The Weyl isometric embedding problem in general $3$ d Riemannian manifolds“ 12:30pm – 2:00pm Lunch 2:00pm – 3:00pm Blake Temple, “An instability in the Standard Model of Cosmology“ 3:00pm – 3:15pm Break 3:15pm – 4:15pm Lydia Bieri, “The Einstein Equations and Gravitational Radiation“ 4:15pm – 4:30pm Break 4:30pm – 5:30pm Valentino Tosatti, “Adiabatic limits of Ricci flat Kahler metrics“ April 9 – Day 2 8:45am Breakfast 9:00am – 10:00am D.H. Phong, “On Strominger systems and Fu-Yau equations” 10:00am – 10:15am Break 10:15am – 11:15am Slawomir Kolodziej, “Stability of weak solutions of the complex Monge-Ampère equation on compact Hermitian manifolds” 11:15am – 11:30am Break 11:30am – 12:30pm Luis Caffarelli, “Non local minimal surfaces and their interactions” 12:30pm – 2:00pm Lunch 2:00pm – 3:00pm Mihalis Dafermos, “The interior of dynamical vacuum black holes and the strong cosmic censorship conjecture in general relativity” 3:00pm – 3:15pm Break 3:15pm – 4:15pm Mu-Tao Wang, “The stability of Lagrangian curvature flows” 4:15pm – 4:30pm Break 4:30pm – 5:30pm Melissa Liu, “Counting curves in a quintic threefold” April 10 – Day 3 8:45am Breakfast 9:00am – 10:00am Rick Schoen, “Metrics of fixed area on high genus surfaces with largest first eigenvalue” 10:00am – 10:15am Break 10:15am – 11:15am Cliff Taubes, “The zero loci of Z/2 harmonic spinors in dimensions 2, 3 and 4” 11:15am – 11:30am Break 11:30am – 12:30pm Tai-Peng Tsai, “Forward Self-Similar and Discretely Self-Similar Solutions of the 3D incompressible Navier-Stokes Equations” * This event is sponsored by National Science Foundation (NSF) and CMSA Harvard University.
Random Matrix & Probability Theory Seminar
Beginning immediately, until at least December 31, all seminars will take place virtually, through Zoom.
In the 2020-2021 AY, the Random Matrix and Probability Theory Seminar will take place on select Wednesdays from 2:00 – 3:00pm virtually. This seminar is organized by Christian Brennecke (brennecke@math.harvard.edu ).
To learn how to attend this seminar, please fill out this form.
The schedule below will be updated as the details are confirmed.
Spring 2021:
Date Speaker Title/Abstract 3/31/2021 Philippe Sosoe, Cornell University Title: Fluctuation bounds for O’Connell-Yor type systems Abstract: The O’Connell-Yor polymer is a fundamental model of a polymer in a random environment. It corresponds to the positive temperature version of Brownian Last Passage percolation. Although much is known about this model thanks to remarkable algebraic structure uncovered by O’Connell, Yor and others, basic estimates for the behavior of the tails of the centered partition function for finite N that are available for zero temperature models are missing. I will present an iterative estimate to obtain strong concentration and localization bounds for the O’Connell-Yor polymer on an almost optimal scale N^{1/3+\epsilon}.
In the second part of the talk, I will introduce a system of interacting diffusions describing the successive increments of partition functions of different sizes. For this system, the N^{2/3} variance upper bound known for the OY polymer can be proved for a general class of interactions which are not expected to correspond to integrable models.
Joint work with Christian Noack and Benjamin Landon.
4/7/2021 Yue M. Lu, Harvard Title: Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Random Matrices
Abstract: In many problems in statistical learning, random matrix theory, and statistical physics, one needs to simulate dynamics on random matrix ensembles. A classical example is to use iterative methods to compute the extremal eigenvalues/eigenvectors of a (spiked) random matrix. Other examples include approximate message passing on dense random graphs, and gradient descent algorithms for solving learning and estimation problems with random initialization. We will show that all such dynamics can be simulated by an efficient matrix-free scheme, if the random matrix is drawn from an ensemble with translation-invariant properties. Examples of such ensembles include the i.i.d. Gaussian (i.e. the rectangular Ginibre) ensemble, the Haar-distributed random orthogonal ensemble, the Gaussian orthogonal ensemble, and their complex-valued counterparts.A “direct” approach to the simulation, where one first generates a dense n × n matrix from the ensemble, requires at least O(n^2) resource in space and time. The new algorithm, named Householder Dice (HD), overcomes this O(n^2) bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are O(nT) and O(n T^2), respectively, with T being the number of iterations. When T ≪ n, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint.Finally, the HD algorithm is not just a computational trick. I will show how its construction can serve as a simple proof technique for several problems in high-dimensional estimation4/14/2021 Canceled 4/16/2021
FridayPatrick Lopatto (IAS) Title: Fluctuations in local quantum unique ergodicity for generalized Wigner matrices Abstract: In a disordered quantum system, delocalization can be understood in many ways. One of these is quantum unique ergodicity, which was proven in the random matrix context by Bourgade and Yau. It states that for a given eigenvector and set of coordinates J, the mass placed on J by the eigenvector tends to N^{-1}|J|, the mass placed on those coordinates by the uniform distribution. Notably, this convergence holds for any size of J, showing that the eigenvectors distribute evenly on all scales.
I will present a result which establishes that the fluctuations of these averages are Gaussian on scales where |J| is asymptotically less than N, for generalized Wigner matrices with smooth entries. The proof uses new eigenvector observables, which are analyzed dynamically using the eigenvector moment flow and the maximum principle.
This is joint work with Lucas Benigni.
4/21/2021 Jean-Christophe Mourrat, Courant Institute, NYU Title: Mean-field spin glasses: beyond Parisi’s formula?
Abstract: Spin glasses are models of statistical mechanics encoding disordered interactions between many simple units. One of the fundamental quantities of interest is the free energy of the model, in the limit when the number of units tends to infinity. For a restricted class of models, this limit was predicted by Parisi, and later rigorously proved by Guerra and Talagrand. I will first show how to rephrase this result using an infinite-dimensional Hamilton-Jacobi equation. I will then present partial results suggesting that this new point of view may allow to understand limit free energies for a larger class of models, focusing in particular on the case in which the units are organized over two layers, and only interact across layers.Fall 2020:
Date Speaker Title/Abstract 9/9/2020 Yukun He (Zurich) Title: Single eigenvalue fluctuations of sparse Erdős–Rényi graphs Abstract: I discuss the fluctuations of individual eigenvalues of the adjacency matrix of the Erdös-Rényi graph $G(N,p)$. I show that if $N^{-1}\ll p \ll N^{-2/3}, then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. The main technical tool of the proof is a rigidity bound of accuracy $N^{-1-\varepsilon}p^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from previous works. Joint work with Antti Knowles.
10/14/2020 David Belius (University of Basel) Title: The TAP approach to mean field spin glasses Abstract: The Thouless-Anderson-Palmer (TAP) approach to the Sherrington-Kirkpatrick mean field spin glass model was proposed in one of the earliest papers on this model. Since then it has complemented subsequently elaborated methods in theoretical physics and mathematics, such as the replica method, which are largely orthogonal to the TAP approach. The TAP approach has the advantage of being interpretable as a variational principle optimizing an energy/entropy trade-off, as commonly encountered in statistical physics and large deviations theory, and potentially allowing for a more direct characterization of the Gibbs measure and its “pure states”. In this talk I will recall the TAP approach, and present preliminary steps towards a solution of mean field spin glass models entirely within a TAP framework.
10/28/2020 Giuseppe Genovese (University of Basel) Title: Non-convex variational principles for the RS free energy of restricted Boltzmann machines Abstract: From the viewpoint of spin glass theory, restricted Boltzmann machines represent a veritable challenge, as to the lack of convexity prevents us to use Guerra’s bounds. Therefore even the replica symmetric approximation for the free energy presents some challenges. I will present old and new results around the topic along with some open problems.
11/4/2020 Benjamin Landon (MIT) Title: Fluctuations of the spherical Sherrington-Kirkpatrick model Abstract: The SSK model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual SK model with Ising spins. Fluctuations of its observables may be related to quantities from random matrix theory using integral representations. In this informal talk we discuss some results on fluctuations of this model at critical temperature and with a magnetic field
11/11/2020
3:00 – 4:00pmLucas Benigni (University of Chicago) Title: Optimal delocalization for generalized Wigner matrices Abstract: We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.
11/18/2020 Simone Warzel (Technical University of Munich) Title: Hierarchical quantum spin glasses
Abstract: Hierarchical spin glasses such as the generalised random energy model are known to faithfully model typical energy landscapes in the classical theory of mean-field spin glasses. Their built-in hierarchical structure is known to emerge spontaneously in the spin-glass phase of, e.g., the Sherrington-Kirkpatrick model. In this talk, I will review recent results on the effects of a transversal magnetic field on such hierarchical quantum spin glasses.
In particular, I will present a formula of Parisi-type for their free energy which allows to make predictions about the phase diagram.12/2/2020 Sabine Jansen (LMU Munich) Title: Thermodynamics of a hierarchical mixture of cubes
Abstract: The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes of side-lengths 2^j, j\in \N_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).For information on previous seminars, click here
The schedule will be updated as details are confirmed.
Topological Insulators and Mathematical Science – Conference and Program
The CMSA will be hosting a conference on the subject of topological insulators and mathematical science on September 15-17. Seminars will take place each day from 2:00-7:00pm in Science Center Hall D, 1 Oxford Street, Cambridge, MA.
Homological Mirror Symmetry Seminar
The seminar series, Homological Mirror Symmetry, will be held on selected Thursdays from 2PM – 4pm in CMSA Building, 20 Garden Street, Room G10.
The list of speakers is below and will be updated as details are confirmed.
Date Name Title 09-15-16 09-22-16 Netanel Blaier, Brandeis “Intro to HMS.” Abstract: This is the first talk of the seminar series. We survey the statement of Homological Mirror Symmetry (introduced by Kontsevich in 1994) and some known results, as well as briefly discussing its importance, and the connection to other formulations of Mirror Symmetry and the SYZ conjecture. Following that, we will begin to review the definition of the A-side (namely, the Fukaya category) in some depth. No background is assumed! Also, in the last half hour, we will divide papers and topics among participants.
09-29-16 Netanel Blaier, Brandeis “Intro to HMS 2.” Abstract: In the second talk, we review (some) of the nitty-gritty details needed to construct a Fukaya categories. This include basic Floer theory, the analytic properties of J-holomorphic curves and cylinders, Gromov compactness and its relation to metric topology on the compactified moduli space, and Banach setup and perturbation schemes commonly used in geometric regularization. We then proceed to recall the notion of an operad, Fukaya’s differentiable correspondences, and how to perform the previous constructions coherently in order to obtain $A_\infty$-structures. We will try to demonstrate all concepts in the Morse theory ‘toy model’.
10-06-16 Hansol Hong, CMSA
Title: Homological mirror symmetry for elliptic curves Abstract:
We survey the proof of homological mirror symmetry by Polishchuk and Zaslow. Some of more recent methods to prove HMS for elliptic curves will be discussed also,
which use homological algebra techniques and formal deformation theory of Lagrangians etc.Notes
Notes (Baris)
10-13-16 Yu-Wei Fan, Harvard
Title: Semi-flat mirror symmetry and Fourier-Mukai transform
Abstract: We will review the semi-flat mirror symmetry setting in Strominger-Yau-Zaslow, and discuss the correspondence between special Lagrangian sections on the A-side and deformed Hermitian-Yang-Mills connections on the B-side using real Fourier-Mukai transform, following Leung-Yau-Zaslow.
10-20-16 Tim Large, MIT
Title: “Symplectic cohomology and wrapped Fukaya categories” Abstract: While mirror symmetry was originally conjectured for compact manifolds, the phenomenon applies to non-compact manifolds as well. In the setting of Liouville domains, a class of open symplectic manifolds including affine varieties, cotangent bundles and Stein manifolds, there is an A-infinity category called the wrapped Fukaya category, which is easier to define and often more amenable to computation than the original Fukaya category. In this talk I will construct it, along with symplectic cohomology (its closed-string counterpart), and compute some examples. We will then discuss how compactifying a symplectic manifold corresponds, on the B-side of mirror symmetry, to turning on a Landau-Ginzburg potential.
10-27-16 Philip Engel, Columbia
Title: Mirror symmetry in the complement of an anticanonical divisor”
According to the SYZ conjecture, the mirror of a Calabi-Yau variety can be constructed by dualizing the fibers of a special Lagrangian fibration. Following Auroux, we consider this rubric for an open Calabi-Yau variety X-D given as the complement of a normal crossings anticanonical divisor D in X. In this talk, we first define the moduli space of special Lagrangian submanfiolds L with a flat U(1) connection in X-D, and note that it locally has the structure of a Calabi-Yau variety. The Fukaya category of such Lagrangians is obstructed, and the degree 0 part of the obstruction on L defines a holomorphic function on the mirror. This “superpotential” depends on counts of holomorphic discs of Maslov index 2 bounded by L. We then restrict to the surface case, where there are codimension 1 “walls” consisting of Lagrangians which bound a disc of Maslov index 0. We examine how the superpotential changes when crossing a wall and discuss how one ought to “quantum correct” the complex structure on the moduli space to undo the discontinuity introduced by these discs.
11-03-16 Yusuf Baris Kartal, MIT
I will present Auroux-Katzarkov-Orlov’s proof of one side of the homological mirror symmetry for Del Pezzo surfaces. Namely I will prove their derived categories are equivalent to the categories of vanishing cycles for certain LG-models together with B-fields. I plan to show how the general B-field corresponds to non-commutative Del Pezzo surfaces and time allowing may mention HMS for simple degenerations of Del Pezzo surfaces. The tools include exceptional collections( and mutations for degenerate case), explicit description of NC deformations, etc.
11-10-16 No seminar this week 12-08-16 Lino Amorim, Boston University
Title: The Fukaya category of a compact toric manifold
Abstract: In this talk I will discuss the Fukaya category of a toric manifold following the work of Fukaya-Oh-Ohta-Ono. I will start with an overview of the general structure of the Fukaya category of a compact symplectic manifold. Then I will consider toric manifolds in particular the Fano case and construct its mirror.
Math Science Lectures in Honor of Raoul Bott: Freddy Cachazo
1 Oxford Street, Cambridge MA 02138On April 2-3, the CMSA will be hosting two lectures by Freddy Cachazo (Perimeter Institute) on “Geometry and Combinatorics in Particle Interactions.” This will be the first of the new annual Bott Math Science Lecture Series hosted by the CMSA.
The lectures will take place from 4:30-5:30pm in Science Center, Hall D.
09-22-2016 Homological Mirror Symmetry Seminar
References:
- D. Auroux, A beginner’s introduction to Fukaya categories. arXiv:1301.7056
- I. Smith, A symplectic prolegomenon. arXiv:1401.0269
- D. Auroux, “Topics in geometry: mirror symmetry”, Fall 2009 (MIT Math 18.969)
- Nick Sheridan’s IAS and Jussieu lectures.
- Sheel Gantara “Topics in symplectic topology”, Spring 2016 (Stanford Math 257B)
Quantum Cohomology, Nakajima Varieties and Quantum groups
During the Spring 2018 Semester Artan Sheshmani (QGM/CMSA) will be teaching a CMSA special lecture series on Quantum Cohomology, Nakajima Vareties and Quantum groups. The lectures will be held Tuesdays and Thursdays beginning January 25th, from 1:00 to 3:00pm in room G10, CMSA Building.
You can watch Prof. Sheshmani describe the series here.
The Syllabus is as follows:
Date……….. Topic Video/Audio 1-25-2018 Gromov-Witten invariants Definition, examples via algebraic geometry I
Video / Audio / Combined
*due to technical difficulties the audio and video are split for this lecture.2-01-2018 Gromov-Witten invariants Virtual Fundamental Class I (definition)
Video / Audio / Combined
*due to technical difficulties the audio and video are split for this lecture2-13-2018 Gromov-Witten invariants Virtual Fundamental Class II (computation in some cases)
2-15-2018 Computing GW invariants Three level GW classes
Genus zero invariants of the projective plane
2-20-2018 Quantum Cohomology Small Quantum Cohomology (Definition and Properties) I
2-22-2018 Quantum Cohomology Small Quantum Cohomology (Definition and Properties) II
2-27-2018 Quantum Cohomology Big Quantum Cohomology I
3-1-2018 Quantum Cohomology Big Quantum Cohomology II
GW potential
WDVV equation
3-6-2018 GW invariants via Quantum Cohomology The Quintic threefold case
The P^2 case
GW invariants via Quantum Cohomology Dubrovin (quantum) connection
Nakajima varieties -Algebraic and symplectic reduction
Nakajima varieties Quasi maps to Nakajima varieties
Quantum cohomology of Nakajima varieties Small Quantum Cohomology of Hilb^n (C2) I
Quantum cohomology of Nakajima varieties Small Quantum Cohomology of Hilb^n (C2) II
Quantum cohomology of Nakajima varieties Small Quantum Cohomology of Hilb^n (C2) III
Quantum cohomology of Nakajima varieties Big Quantum Cohomology of Hilb^n (C2) I
Quantum cohomology of Nakajima varieties Big Quantum Cohomology of Hilb^n (C2) II
Quantum cohomology of Nakajima varieties Big Quantum Cohomology of Hilb^n (C2) III
Quantum cohomology of Nakajima varieties Big Quantum Cohomology of Hilb^n (C2) IV
Existence of Canonical Metrics on Non-Kähler Geometry
On Wednesday September 9, CMSA director Prof. Shing-Tung Yau gave a lecture for the Simons foundation on “Existence of Canonical Metrics on Non-Kähler Geometry.“
In this lecture, Prof. Yau surveys the existence of canonical balanced metrics on non-Kähler complex manifolds through the Hull-Strominger system, which was motivated by string theory on compactifications. He discusses works by Jun Li of Fudan University in Shanghai, Ji-Xiang Fu of Fudan University, Ivan Smith of the University of Cambridge, Richard P. Thomas of Imperial College London, Tristan C. Collins of the Massachusetts Institute of Technology, French mathematician Émile Picard, Teng Fei of Rutgers University in Newark, New Jersey, Adam Jacob of the University of California, Davis, and Duong H. Phong of Columbia University.
More information about this talk can be found on the Simons Foundation website.
1/13/2022 Interdisciplinary Science Seminar
Title: A universal triangulation for flat tori
Abstract: A celebrated theorem of Nash completed by Kuiper implies that every smooth Riemannian surface has a C¹ isometric embedding in the Euclidean 3-space E³. An analogous result, due to Burago and Zalgaller, states that every polyhedral surface, obtained by gluing Euclidean triangles, has an isometric PL embedding in E³. In particular, this provides PL isometric embeddings for every flat torus (a quotient of E² by a rank 2 lattice). However, the proof of Burago and Zalgaller is partially constructive, relying on the Nash-Kuiper theorem. In practice, it produces PL embeddings with a huge number of vertices, moreover distinct for every flat torus. Based on a construction of Zalgaller and on recent works by Arnoux et al. we exhibit a universal triangulation with less than 10.000 vertices, admitting for any flat torus an isometric embedding that is linear on each triangle. Based on joint work with Florent Tallerie.
Hyperbolic Geometry and Quantum Invariants
Abstract: There are two very different approaches to 3-dimensional topology, the hyperbolic geometry following the work of Thurston and the quantum invariants following the work of Jones and Witten. These two approaches are related by a sequence of problems called the Volume Conjectures. In this talk, I will explain these conjectures and present some recent joint works with Ka Ho Wong related to or benefited from this relationship.
A Mathematical Language
Speaker: Thomas Hales, Univ. of Pittsburgh Dept. of Mathematics
Title: A Mathematical Language
Abstract: A controlled natural language for mathematics is an artificial language that is designed in an explicit way with precise computer-readable syntax and semantics. It is based on a single natural language (which for us is English) and can be broadly understood by mathematically literate English speakers. This talk will describe the design of a controlled natural language for mathematics that has been influenced by the Lean theorem prover, by TeX, and by earlier controlled natural languages. The semantics are provided by dependent type theory.
Special Lecture Series on Derived Algebraic/Differential Geometry
In the Spring 2019 Semester, the CMSA will be hosting a special lecture series on Derived algebraic/differential geometry run by Artan Sheshmani, with lectures given by Prof. Sheshmani and Dr. Dennis Borisov. The seminar will be held on Tuesdays and Thursdays from 3:00-4:30pm in CMSA, room G10.
Click here for reference material Schedule:
Section 1: Basic setting of derived geometry
The goal: To collect the minimum set of tools needed to do algebraic geometry in the derived context.
2/05/2019 Lecture 1: Model and с-categories Video 2/07/2019 Lecture 2: Grothendieck topologies and homotopy descent Video 2/12/2019 Lecture 3: Derived Artin stacks Video 2/14/2019 Lecture 4: Cotangent complexes Section 2: Loop spaces and differential forms
The goal: This is the algebraic heart of the course – here we learn the homological techniques that are needed for shifted symplectic forms.
2/19/2019 Lecture 5: De Rham complexes and S1-equivariant schemes (loop spaces) Video 2/21/2019 Lecture 6: Chern character Video 2/26/2019 Room G02
Lecture 7: Local structure of closed differential forms in the derived sense Part I Video 2/28/2019 Lecture 8: Local structure of closed differential forms in the derived sense Part II Video 3/05/2019 Lecture 9: Cyclic homology Video Section 3: Shifted symplectic structures
Goal: To see applications of the algebraic techniques from above in the geometric context of the actual moduli spaces.3/07/2019 Lecture 10: Definition and existence results Video 3/12/2019 Lecture 11: Lagrangians and Lagrangian fibrations Video 3/14/2019 Room G02
Lecture 12: Lagrangians and Lagrangian fibrations Video 3/26/2019 Lecture 13: Intersections of Lagrangians Video 3/28/2019 Room G02
Lecture 14: Examples and applications 2 (Part I) Video 4/02/2019 Lecture 15: Examples and applications 2 (Part II) Video Section 4: Uhlenbeck–Yau construction and correspondence
4/04/2019 Lecture 16: Examples and applications 2 (Part III) Video 4/09/2019 Room G02
Lecture 17: Uhlenbeck–Yau construction and correspondence Examples (Part I) Video CMSA Math-Science Literature Lecture: Quantum Groups
Pavel Etingof (MIT)
Title: Quantum Groups
Abstract: The theory of quantum groups developed in mid 1980s from attempts to construct and understand solutions of the quantum Yang-Baxter equation, an important equation arising in quantum field theory and statistical mechanics. Since then, it has grown into a vast subject with profound connections to many areas of mathematics, such as representation theory, the Langlands program, low-dimensional topology, category theory, enumerative geometry, quantum computation, algebraic combinatorics, conformal field theory, integrable systems, integrable probability, and others. I will review some of the main ideas and examples of quantum groups and try to briefly describe some of the applications.
CMSA Math-Science Literature Lecture: The ADHM construction of Yang-Mills instantons
Simon Donaldson (Stony Brook)
Title: The ADHM construction of Yang-Mills instantons
Abstract: In 1978 (Physics Letters 65A) Atiyah, Hitchin, Drinfeld and Manin (ADHM) described a construction of the general solution of the Yang-Mills instanton equations over the 4-sphere using linear algebra. This was a major landmark in the modern interaction between geometry and physics, and the construction has been the scene for much research activity up to the present day. In this lecture we will review the background and the original ADHM proof, using Penrose’s twistor theory and results on algebraic vector bundles over projective 3-space. As time permits, we will also discuss some further developments, for example, the work of Nahm on monopoles and connections to Mukai duality for bundles over complex tori.
AI and Theorem Proving
Speaker: Josef Urban, Czech Technical University
Title: AI and Theorem Proving
Abstract: The talk will discuss the main approaches that combine machine learning with automated theorem proving and automated formalization. This includes learning to choose relevant facts for “hammer” systems, guiding the proof search of tableaux and superposition automated provers by interleaving learning and proving (reinforcement learning) over large ITP libraries, guiding the application of tactics in interactive tactical systems, and various forms of lemmatization and conjecturing. I will also show some demos of the systems, and discuss autoformalization approaches such as learning probabilistic grammars from aligned informal/formal corpora, combining them with semantic pruning, and using neural methods to learn direct translation from Latex to formal mathematics.
Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models
Speaker: Jason Rute, CIBO Technologies
Title: Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models
Abstract: Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT ({\bf P}roof {\bf A}rtifact {\bf C}o-{\bf T}raining), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32% to 48%.
2018 Ding Shum Lecture
On October 24, 2018, the CMSA will be hosting our second annual Ding Shum lecture. This event was made possible by the generous funding of Ding Lei and Harry Shum. Last year featured Leslie Valiant, who spoke on “learning as a Theory of Everything.”
This year will feature Eric Maskin, who will speak on “How to Improve Presidential Elections: the Mathematics of Voting.” This lecture will take place from 5:00-6:00pm in Science Center, Hall D.
Pictures of the event can be found here.
Algebraic Geometry Seminar, Thursdays
This seminar will not be held in the Spring 2018 Semester.
The Algebraic Geometry Seminar will be every Thursday from 3pm-4pm in CMSA Building, 20 Garden Street, Room G10.
The schedule will be updated as details are confirmed.
Date Name Title/Abstract 09-14-17 Yu-Wei Fan (Harvard Math) Entropy of an autoequivalence on Calami-Yau manifolds
Abstract: We will recall the notion of entropy of an autoequivalence on triangulated categories, and provide counterexamples of a conjecture by Kikuta-Takahashi.
11-1-17 *5:00pm, G10*
Shamil Shakirov, Harvard Math Undulation invariants of plane curves
Abstract: “One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has singularities or other distinctive features of interest). A classical example of such a problem, described by Cayley and Salmon in 1852, is to determine whether or not a given plane curve of degree r > 3 has undulation points — the points where the tangent line meets the curve with multiplicity four. Cayley proved that there exists an invariant of degree (r – 3)(3 r – 2) that vanishes if and only if the curve has undulation points. We construct this invariant explicitly for quartics (r=4) as the determinant of a 21 times 21 matrix with polynomial entries, and we conjecture a generalization for r = 5
11-2-17 Alexander Moll, IHES Hilbert Schemes from Geometric Quantization of Dispersive Periodic Benjamin-Ono Waves
ABSTRACT: By Grojnowski and Nakajima, Fock spaces are cohomology rings of Hilbert scheme of points in the plane. On the other hand, by Pressley-Segal, Fock spaces are spaces of J-holomorphic functions on the loop space of the real line that appear in geometric quantization with respect to the Kähler structure determined by the Sobolev regularity s= -1/2 and the Hilbert transform J. First, we show that the classical periodic Benjamin-Ono equation is a Liouville integrable Hamiltonian system with respect to this Kähler structure. Second, we construct an integrable geometric quantization of this system in Fock space following Nazarov-Sklyanin and describe the spectrum explicitly after a non-trivial rewriting of our coefficients of dispersion \ebar = e_1 + e_2 and quantization \hbar = – e_1 e_2 that is invariant under e_2 <-> e_1. As a corollary of Lehn’s theorem, our construction gives explicit creation and annihilation operator formulas for multiplication by new explicit universal polynomials in the Chern classes of the tautological bundle in the equivariant cohomology of our Hilbert schemes, in particular identifying \ebar with the deformation parameter of the Maulik-Okounkov Yangian and \hbar with the handle-gluing element. Our key ingredient is a simple formula for the Lax operators as elliptic generalized Toeplitz operators on the circle together with the spectral theory of Boutet de Monvel and Guillemin. As time permits, we discuss the relation of dispersionless \ebar -> 0 and semi-classical \hbar \rightarrow 0 limits to Nekrasov’s BPS/CFT Correspondence.
11-9-17 TBD TBD 11-16-17 TBD TBD 11-23-17 TBD TBD 11-30-17 TBD TBD 12-7-17 TBD TBD 12-15-17 TBD TBD Random Matrix & Probability Theory Seminar (2016-2017)
CMSA, 20 Garden Street, Cambridge, MA 02138 USAThe random matrix and probability theory will be every Wednesday from 3pm-4pm in CMSA Building, 20 Garden Street, Room G10.
Working Conference on Applications of Random Matrix Theory to Data Analysis, January 9-13, 2017
The Center of Mathematical Sciences and Applications will be hosting a working Conference on Applications of Random Matrix Theory to Data Analysis, January 9-13, 2017. The conference will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
Participants:
Gerard Ben Arous, Courant Institute of Mathematical Sciences
Alex Bloemendal, Broad Institute
Arup Chakraburty, MIT
Zhou Fan, Stanford University
Alpha Lee, Harvard University
Matthew R. McKay, Hong Kong University of Science and Technology (HKUST)
David R. Nelson, Harvard University
Nick Patterson, Broad Institute
Marc Potters, Capital Fund management
Yasser Roudi, IAS
Tom Trogdon, UC Irvine
Organizers:
Michael Brenner, Lucy Colwell, Govind Menon, Horng-Tzer Yau
Please click Program for a downloadable schedule with talk abstracts.
Please note that breakfast & lunch will be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants should you need recommendations for dinner.
Schedule:
January 9 – Day 1 9:30am – 10:00am Breakfast & Opening remarks 10:00am – 11:00am Marc Potters, “Eigenvector overlaps and the estimation of large noisy matrices” 11:00am – 12:00pm Yasser Roudi 12:00pm – 2:00pm Lunch 2:00pm Afternoon Discussion January 10 – Day 2 8:30am – 9:00am Breakfast 9:00am – 10:00am Arup Chakraburty, “The mathematical analyses and biophysical reasons underlying why the prevalence of HIV strains and their relative fitness are simply correlated, and pose the challenge of building a general theory that encompasses other viruses where this is not true.” 10:00am – 11:00am Tom Trogdon, “On the average behavior of numerical algorithms” 11:00am – 12:00pm David R. Nelson, “Non-Hermitian Localization in Neural Networks” 12:00pm – 2:00pm Lunch 2:00pm Afternoon Discussion January 11 – Day 3 8:30am – 9:00am Breakfast 9:00am – 10:00am Nick Patterson 10:00am – 11:00am Lucy Colwell 11:00am – 12:00pm Alpha Lee 12:00pm – 2:00pm Lunch 2:00pm-4:00pm Afternoon Discussion 4:00pm Gerard Ben Arous (Public Talk), “Complexity of random functions of many variables: from geometry to statistical physics and deep learning algorithms“ January 12 – Day 4 8:30am – 9:00am Breakfast 9:00am – 10:00am Govind Menon 10:00am – 11:00am Alex Bloemendal 11:00am – 12:00pm Zhou Fan, “Free probability, random matrices, and statistics” 12:00pm – 2:00pm Lunch 2:00pm Afternoon Discussion January 13 – Day 5 8:30am – 9:00am Breakfast 9:00am – 12:00pm Free for Working 12:00pm – 2:00pm Lunch 2:00pm Free for Working * This event is sponsored by CMSA Harvard University.
Members’ Seminar
The CMSA Members’ Seminar will occur every Friday at 9:30am ET on Zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. Please email the seminar organizers to obtain a link. This year’s seminar is organized by Tianqi Wu. The Schedule will be updated below.
Previous seminars can be found here.
Spring 2021:
Date Speaker Title/Abstract 1/29/2021 Cancelled 2/5/2021 Itamar Shamir Title: Boundary CFT and conformal anomalies Abstract: Boundary and defects in quantum field theory play an important role in many recent developments in theoretical physics. I will discuss such objects in the setting of conformal field theories, focusing mainly on conformal anomalies. Boundaries or defects can support various kinds of conformal anomalies on their world volume. Perhaps the one which is of greatest theoretical importance is associated with the Euler density in even dimensions. I will show how this anomaly is related to the one point function of exactly marginal deformations and how it arises explicitly from various correlation functions.
2/12/2021 Louis Fan Title: Joint distribution of Busemann functions in corner growth models Abstract: The 1+1 dimensional corner growth model with exponential weights is a centrally important exactly solvable model in the Kardar-Parisi-Zhang class of statistical mechanical models. While significant progress has been made on the fluctuations of the growing random shape, understanding of the optimal paths, or geodesics, is less developed. The Busemann function is a useful analytical tool for studying geodesics. We present the joint distribution of the Busemann functions, simultaneously in all directions of growth, in terms of mappings that represent FIFO (first-in-first-out) queues. As applications of this description we derive a marked point process representation for the Busemann function across a single lattice edge and point out its implication on structure of semi-infinite geodesics. This is joint work with Timo Seppäläinen.
2/19/2021 Daniel Junghans Title: Control issues of the KKLT scenario in string theory Abstract: The simplest explanation for the observed accelerated expansion of the universe is that we live in a 4-dimensional de Sitter space. We analyze to which extent the KKLT proposal for the construction of such de Sitter vacua in string theory is quantitatively controlled. As our main finding, we uncover and quantify an issue which one may want to call the “singular-bulk problem”. In particular, we show that, generically, a significant part of the manifold on which string theory is compactified in the KKLT scenario becomes singular. This implies a loss of control over the supergravity approximation on which the construction relies.
2/26/2021 Tsung-Ju Lee Title: SYZ fibrations and complex affine structures Abstract: Strominger–Yau–Zaslow conjecture has been a guiding principle in mirror symmetry. The conjecture predicts the existence of special Lagrangian torus fibrations of a Calabi–Yau manifold near a large complex structure limit point. Moreover, the mirror is given by the dual fibrations and the Ricci-flat metric is obtained from the semi-flat metric with corrections from holomorphic discs whose boundaries lie in a special Lagrangian fiber. By a result of Collins–Jacob–Lin, the complement of a smooth elliptic curve in the projective plane admits a SYZ fibration. In this talk, I will explain how to compute the complex affine structure induced from this SYZ fibration and show that it agrees with the affine structure used in Carl–Pumperla–Siebert. This is based on a joint work with Siu-Cheong Lau and Yu-Shen Lin.
3/5/2021 Cancelled 3/11/2021 9:00pm ET
Ryan Thorngren Title: Symmetry protected topological phases, anomalies, and their classification Abstract: I will give an overview of some mathematical aspects of the subject of symmetry protected topological phases (SPTs), especially as their theory relates to index theorems in geometry, cobordism of manifolds, and group cohomology.
3/18/2021 9:00pm ET
Ryan Thorngren Title: Symmetry protected topological phases, anomalies, and their classification
Abstract: I will give an overview of some mathematical aspects of the subject of symmetry protected topological phases (SPTs), especially as their theory relates to index theorems in geometry, cobordism of manifolds, and group cohomology.3/26/2021 8:30am ET
Aghil Alaee Title: Rich extra dimensions are hidden inside black holes Abstract: In this talk, I present an argument that shows why it is difficult to see rich extra dimensions in the Universe.
4/2/2021
8:30am ETEnno Keßler Title: Super Stable Maps of Genus Zero Abstract: I will report on a supergeometric generalization of J-holomorphic curves. Supergeometry is a mathematical theory of geometric spaces with anti-commuting coordinates and functions which is motivated by the concept of supersymmetry from theoretical physics. Super J-holomorphic curves and super stable maps couple the equations of classical J-holomorphic curves with a Dirac equation for spinors and might, in the future, lead to a supergeometric generalization of Gromov-Witten invariants.
4/9/2021 Juven Wang Title: Ultra Unification Abstract: Strong, electromagnetic, and weak forces were unified in the Standard Model (SM) with spontaneous gauge symmetry breaking. These forces were further conjectured to be unified in a simple Lie group gauge interaction in the Grand Unification (GUT). In this work, we propose a theory beyond the SM and GUT by adding new gapped Topological Phase Sectors consistent with the nonperturbative global anomaly matching and cobordism constraints (especially from the baryon minus lepton number B − L and the mixed gauge-gravitational anomaly). Gapped Topological Phase Sectors are constructed via symmetry extension, whose low energy contains unitary topological quantum field theories (TQFTs): either 3+1d non-invertible TQFT (long-range entangled gapped phase), or 4+1d invertible or non-invertible TQFT (short-range or long-range entangled gapped phase), or right-handed neutrinos, or their combinations. We propose that a new high-energy physics frontier beyond the conventional 0d particle physics relies on the new Topological Force and Topological Matter including gapped extended objects (gapped 1d line and 2d surface operators or defects, etc., whose open ends carry deconfined fractionalized particle or anyonic string excitations). I will also fill in the dictionary between math, QFT, and condensed matter terminology, and elaborate more on the nonperturbative global anomalies of Z2, Z4, Z16 classes useful for beyond SM. Work is based on arXiv:2012.15860, arXiv:2008.06499, arXiv:2006.16996, arXiv:1910.14668.
4/16/2021 Sergiy Verstyuk Title: Deep learning methods for economics Abstract: The talk discusses some recent developments in neural network models and their applicability to problems in international economics as well as macro-via-micro economics. Along the way, interpretability of neural networks features prominently.
4/23/2021 Yifan Wang Title: Virtues of Defects in Quantum Field Theories Abstract: Defects appear ubiquitously in many-body quantum systems as boundaries and impurities. They participate inextricably in the quantum dynamics and give rise to novel phase transitions and critical phenomena. Quantum field theory provides the natural framework to tackle these problems, where defects define extended operators over sub-manifolds of the spacetime and enrich the usual operator algebra. Much of the recent progress in quantum field theory has been driven by the exploration of general structures in this extended operator algebra, precision studies of defect observables, and the implications thereof for strongly coupled dynamics. In this talk, I will review selected developments along this line that enhance our understanding of concrete models in condensed matter and particle physics, and that open new windows to nonperturbative effects in quantum gravity.
4/30/2021 Yun Shi Title: D-critical locus structure for local toric Calabi-Yau 3-fold Abstract: Donaldson-Thomas (DT) theory is an enumerative theory which produces a count of ideal sheaves of 1-dimensional subschemes on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will give a brief introduction to motivic DT theory following the definition of Bussi-Joyce-Meinhardt, in particular the role of d-critical locus structure in the definition of motivic DT invariant. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint work in progress with Sheldon Katz.
5/7/2021 Thérèse Yingying Wu Title: Topological aspects of Z/2Z eigenfunctions for the Laplacian on S^2 Abstract: In this talk, I will present recent work with C. Taubes on an eigenvalue problem for the Laplacian on the round 2-sphere associated with a configuration of an even number of distinct points on that sphere, denoted as C_2n. I will report our preliminary findings on how eigenvalues and eigenfunctions change as a function of the configuration space. I will also discuss how the compactification of C_2n is connected to the moduli space of algebraic curves (joint work with S.-T. Yau). There is a supergeometry tie-in too.
5/14/2021 Du Pei Title: Three applications of TQFTs Abstract: Topological quantum field theories (TQFTs) often serve as a bridge between physics and mathematics. In this talk, I will illustrate how TQFTs that arise in physics can help to shed light on 1) the quantization of moduli spaces 2) quantum invariants of 3-manifolds, and 3) smooth structures on 4-manifolds.
5/21/2021 Farzan Vafa Title: Active nematic defects and epithelial morphogenesis Abstract: Inspired by recent experiments that highlight the role of topological defects in morphogenesis, we develop a minimal framework to study the dynamics of an active curved surface driven by its nematic texture (a rank 2 symmetric traceless tensor). Allowing the surface to evolve via relaxational dynamics (gradient flow) leads to a theory linking nematic defect dynamics, cellular division rates, and Gaussian curvature. Regions of large positive (negative) curvature and positive (negative) growth are colocalized with the presence of positive (negative) defects, and cells accumulate at positive defects and are depleted at negative defects. We also show that activity stabilizes a bound $+1$ defect state by creating an incipient tentacle, while a bound $+1$ defect state surrounded by two $-1/2$ defects can create a stationary ring configuration of tentacles, consistent with experimental observations. The talk is based on a recent paper with L Mahadevan [arXiv:2105.0106].
Fall 2020:
Date Speaker Title/Abstract 9/11/2020 Moran Koren Title: Observational Learning and Inefficiencies in Waitlists Abstract: Many scarce resources are allocated through waitlists without monetary transfers. We consider a model, in which objects with heterogeneous qualities are offered to strategic agents through a waitlist in a first-come-first-serve manner. Agents, upon receiving an offer, accept or reject it based on both a private signal about the quality of the object and the decisions of agents ahead of them on the list. This model combines observational learning and dynamic incentives, two features that have been studied separately. We characterize the equilibrium and quantify the inefficiency that arises due to herding and selectivity. We find that objects with intermediate expected quality are discarded while objects with a lower expected quality may be accepted. These findings help in understanding the reasons for the substantial discard rate of transplant organs of various qualities despite the large shortage of organ supply.
9/18/2020 Michael Douglas Title: A talk in two parts, on strings and on computers and math Abstract: I am dividing my time between two broad topics. The first is string theory, mostly topics in geometry and compactification. I will describe my current work on numerical Ricci flat metrics, and list many open research questions. The second is computation and artificial intelligence. I will introduce transformer models (Bert,GPT) which have led to breakthroughs on natural language processing, describe their potential for helping us do math, and sketch some related theoretical problems.
9/25/2020 Cancelled – Math Science Lecture 10/2/2020 Cancelled – Math Science Lecture 10/9/2020 Wai Tong (Louis) Fan Title: Stochastic PDE as scaling limits of interacting particle systems Abstract: Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.
In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics. Joint work with Rick Durrett.10/16/2020 Tianqi Wu Title: Koebe circle domain conjecture and the Weyl problem in hyperbolic 3-space Abstract: In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. The Weyl problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 into the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture, the Weyl problem and the work of Alexandrov and Thurston on convex surfaces. This is a joint work with Feng Luo.
10/23/2020 Changji Xu Title: Random Walk Among Bernoulli Obstacles Abstract: Place an obstacle with probability $1 – p$ independently at each vertex of $\mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. This is called random walk among Bernoulli obstacles. The most prominent feature of this model is a strong localization effect: the random walk will be localized in a very small region conditional on the event that it survives for a long time. In this talk, we will discuss some recent results about the behaviors of the conditional random walk, in quenched, annealed, and biased settings.
10/30/2020 Michael Simkin Title: The differential equation method in Banach spaces and the $n$-queens problem Abstract: The differential equation method is a powerful tool used to study the evolution of random combinatorial processes. By showing that the process is likely to follow the trajectory of an ODE, one can study the deterministic ODE rather than the random process directly. We extend this method to ODEs in infinite-dimensional Banach spaces.
We apply this tool to the classical $n$-queens problem: Let $Q(n)$ be the number of placements of $n$ non-attacking chess queens on an $n \times n$ board. Consider the following random process: Begin with an empty board. For as long as possible choose, uniformly at random, a space with no queens in its row, column, or either diagonal, and place on it a queen. We associate the process with an abstract ODE. By analyzing the ODE we conclude that the process almost succeeds in placing $n$ queens on the board. Furthermore, we can obtain a complete $n$-queens placement by making only a few changes to the board. By counting the number of choices available at each step we conclude that $Q(n) \geq (n/C)^n$, for a constant $C>0$ associated with the ODE. This is optimal up to the value of $C$.11/6/2020 Kenji Kawaguchi Title: Deep learning: theoretical results on optimization and mixup Abstract: Deep neural networks have achieved significant empirical success in many fields, including the fields of computer vision, machine learning, and artificial intelligence. Along with its empirical success, deep learning has been theoretically shown to be attractive in terms of its expressive power. However, the theory of the expressive power does not ensure that we can efficiently find an optimal solution in terms of optimization, robustness, and generalization, during the optimization process of a neural network. In this talk, I will discuss some theoretical results on optimization and the effect of mixup on robustness and generalization.
11/13/2020 Omri Ben-Eliezer Title: Sampling in an adversarial environment Abstract: How many samples does one need to take from a large population in order to truthfully “represent” the population? While this cornerstone question in statistics is very well understood when the population is fixed in advance, many situations in modern data analysis exhibit a very different behavior: the population interacts with and is affected by the sampling process. In such situations, the existing statistical literature does not apply.
We propose a new sequential adversarial model capturing these situations, where future data might depend on previously sampled elements; we then prove uniform laws of large numbers in this adversarial model. The results, techniques, and applications reveal close connections to various areas in mathematics and computer science, including VC theory, discrepancy theory, online learning, streaming algorithms, and computational geometry.
Based on joint works with Noga Alon, Yuval Dagan, Shay Moran, Moni Naor, and Eylon Yogev.
11/20/2020 Charles Doran Title: The Calabi-Yau Geometry of Feynman Integrals Abstract: Over the past 30 years Calabi-Yau manifolds have proven to be the key geometric structures behind string theory and its variants. In this talk, I will show how the geometry and moduli of Calabi-Yau manifolds provide a new framework for understanding and computing Feynman integrals. An important organizational principle is provided by mirror symmetry, and specifically the DHT mirror correspondence. This is joint work with Andrey Novoseltsev and Pierre Vanhove.
Strings, knots and quivers
Abstract: I will discuss a recently discovered relation between quivers and knots, as well as – more generally – toric Calabi-Yau manifolds. In the context of knots this relation is referred to as the knots-quivers correspondence, and it states that various invariants of a given knot are captured by characteristics of a certain quiver, which can be associated to this knot. Among others, this correspondence enables to prove integrality of LMOV invariants of a knot by relating them to motivic Donaldson-Thomas invariants of the corresponding quiver, it provides a new insight on knot categorification, etc. This correspondence arises from string theory interpretation and engineering of knots in brane systems in the conifold geometry; replacing the conifold by other toric Calabi-Yau manifolds leads to analogous relations between such manifolds and quivers.
Hodge and Noether-Lefschetz Loci Seminar
In the Fall 2018 Semester the CMSA will be hosting a seminar on Hodge and Noether-Lefschetz loci, with lectures given by Hossein Movasati (IMPA). The seminar will occur weekly on Wednesday at 1:30 in room G10 of the CMSA.
The schedule below will be updated as talks are confirmed.
Date Title/Abstract 11/7/2018 Title: Hodge and Noether-Lefschetz loci Abstract: Hodge cycles are topological cycles which are conjecturally (the millennium Hodge conjecture) supported in algebraic cycles of a given smooth projective complex manifold. Their study in families leads to the notion of Hodge locus, which is also known as Noether-Lefschetz locus in the case of surfaces. The main aim of this mini course is to introduce a computational approach to the study of Hodge loci for hypersurfaces and near the Fermat hypersurface. This will ultimately lead to the verification of the variational Hodge conjecture for explicit examples of algebraic cycles inside hypersurfaces and also the verification of integral Hodge conjecture for examples of Fermat hypersurfaces. Both applications highly depend on computer calculations of rank of huge matrices. We also aim to review some classical results on this topic, such as Cattani-Deligne-Kaplan theorem on the algebraicity of the components of the hodge loci, Deligne’s absolute Hodge cycle theorem for abelian varieties etc.
In the theoretical side another aim is to use the available tools in algebraic geometry and construct the moduli space of projective varieties enhanced with elements in their algebraic de Rham cohomology ring. These kind of moduli spaces have been useful in mathematical physics in order to describe the generating function of higher genus Gromov-Witten invariants, and it turns out that the Hodge loci in such moduli spaces are well-behaved, for instance, they are algebraic leaves of certain holomorphic foliations. Such foliations are constructed from the underlying Gauss-Manin connection. This lectures series involves many reading activities on related topics, and contributions by participants are most welcome.
11/14/2018 Title: Integral Hodge conjecture for Fermat varieties Abstract: We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same. This is a joint work with E. Aljovin and R. Villaflor.
11/21/2018 Title: Periods of algebraic cycles Abstract: The tangent space of the Hodge locus at a point can be described by the so called infinitesimal variation of Hodge structures and the cohomology class of Hodge cycles. For hypersurfaces of dimension $n$ and degree $d$ it turns out that one can describe it without any knowledge of cohomology theories and in a fashion which E. Picard in 1900’s wanted to study integrals/periods. The data of cohomology class is replaced with periods of Hodge cycles, and explicit computations of these periods, will give us a computer implementable description of the tangent space. As an application of this we show that for examples of $n$ and $d$, the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space.
11/28/2018 Title: Periods of Complete Intersection Algebraic Cycles Speaker: Roberto Villaflor
Abstract: In order to compute periods of algebraic cycles inside even dimensional smooth degree d hypersurfaces of the projective space, we restrict ourselves to cycles supported in a complete intersection subvariety. When the description of the complete intersection is explicit, we can compute its periods, and furthermore its cohomological class. As an application, we can use this data to describe the Zariski tangent space of the corresponding Hodge locus, as the degree d part of some Artinian Gorenstein ideal of the homogeneous coordinate ring of the projective space. Using this description, we can show that for d>5, the locus of hypersurfaces containing two linear cycles, is a reduced component of the Hodge locus in the underlying parameter space.
12/05/2018 Room G02
Title: Some explicit Hodge cycles Abstract: Explicit examples of Hodge cycles are due to D. Mumford and A. Weil in the case of CM abelian varieties. In this talk, I will describe few other examples for the Fermat variety. Effective verification of the Hodge conjecture for these cycles is not known.
12/12/2018 Title: A conjectural Hodge locus for cubic tenfold Abstract: In this talk we will consider the difference of two linear algebraic cycles of dimension 5 inside a smooth cubic tenfold and such that the dimension of their intersection is 3. We will show some computer assisted evidences to the fact that the corresponding Hodge locus is bigger than the expected locus of algebraic deformations of the cubic tenfold together with its linear cycles. A similar discussion will be also presented for cubic six and eightfold, for which we will prove that the corresponding second and third order infinitesimal Hodge loci are smooth. The main ingredient is a computer implementation of power series of periods of hypersurfaces.
1/16/2019 Title: Algebraic BCOV anomaly equation Abstract: We introduce the moduli space T of non-rigid compact Calabi-Yau threefolds enhanced with differential forms and a Lie algebra of vector fields in T. This will be used in order to give a purely algebraic interpretation of topological string partition functions and the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation (joint work with M. Alim, E. Scheidegger, S.-T. Yau). We will also define similar moduli spaces for even dimensional Calabi-Yau varieties, where we have the notion of Hodge locus.
1/23/2019 Title: A new model for modular curves Abstract: One of the non-trivial examples of a Hodge locus is the modular curve X_0(N), which is due to isogeny of elliptic curves (a Hodge/algebraic cycle in the product of two elliptic curves). After introducing the notion of enhanced moduli of elliptic curves, I will describe a new model for X_0(N) in the weighted projective space of dimension 4 and with weights (2,3,2,3,1). I will also introduce some elements in the defining ideal of such a model.
The talk is based on the article arXiv:1808.01689.
1/30/2019 Title: Constant Yukawa couplings Abstract: In this talk I will first introduce algebraic Yukawa couplings for any moduli of enhanced Calabi-Yau n-folds. Then I will list many examples in support of the following conjecture. A moduli of Calabi-Yau n-folds is a quotient of a Hermitian symmetric domain (constructed from periods) by an arithmetic group if and only if the corresponding Yukawa couplings are constants.
2/6/2019 Title: Integrality properties of CY modular forms Abstract: The integrality of the coefficients of the mirror map is a central problem in the arithmetic of Calabi-Yau varieties and it has been investigated by Lian-Yau (1996, 1998), Hosono-Lian-Yau (1996), Zudilin (2002), Kontsevich-Schwarz-Vologodsky (2006) Krattenthaler-Rivoal (2010). The central tool in most of these works has been the so called Dwork method. In this talk we use this method and classify all hypergeometric differential equations with a maximal unipotent monodromy whose mirror map has integral coefficients.
We also give a computable condition on the parameters of a hypergeometric function which conjecturally computes all the primes which appear in the denominators of the coefficients of the mirror map. This is a joint work with Kh. Shokri.
2/13/2019 Title: Foliations and Hodge loci Abstract: In this talk I will introduce a holomorphic foliation in a larger parameter space attached to families of enhanced projective varieties. Irreducible components of the Hodge locus with constant periods are algebraic leaves of such a foliation. Under the hypothesis that these are all the algebraic leaves, we get the fact that such algebraic leaves are defined over the algebraic closure of the base field and that Hodge classes are weak absolute in the sense of C. Voisin.
References:
- M. Alim, H. Movasati, E. Scheidegger, S.-T. Yau. Gauss-Manin connection in disguise: Calabi-Yau threefolds, Comm. Math. Phys. 344, (2016), no. 3, 889-914.
- E. H. Cattani, P. Deligne, and A. G. Kaplan. On the locus of Hodge classes. Amer. Math. Soc., 8(2):483–506, 1995.
- B. Haghighat H. Movasati, S.-T. Yau. Calabi-Yau modular forms in limit: Elliptic fibrations, Communications in Number Theory and Physics, Vol. 11, Number 4, 879-912, 2017.
- H. Movasati, Modular and automorphic forms & beyond, Book under preparation, 2019.
- H. Movasati. A Course in Hodge Theory: with Emphasis on Multiple Integrals.Book submitted,2018.
- H. Movasati, On elliptic modular foliation, II, 2018
- H. Movasati, R. Villaflor Loyola, Periods of linear algebraic cycles,, 2018.
- H. Movasati, Gauss-Manin connection in disguise: Calabi-Yau modular forms, Surveys in Modern Mathematics, Vol 13, International Press, Boston.
- H. Movasati, Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci, Asian Journal of Mathematics, Vol.21, No. 3, pp. 463-482, 2017.
- C. Voisin. Hodge loci and absolute Hodge classes. Compos. Math., 143(4):945–958, 2007.
- C. Voisin. Hodge loci. Handbook of moduli. Vol. III, volume 26 of Adv. Lect. Math. (ALM)}, pages 507–546. Int. Press, Somerville, MA, 2013.
Data Analysis Workshop, April 4 – 8, 2016
The Center of Mathematical Sciences and Applications will be hosting a 5-day workshop on Data Analysis and related areas on April 4 – 8, 2016.
Workshop Locations:
April 4 – 7 (Monday ~ Thursday)
Room G10,
20 Garden Street, Cambridge, MA 02138April 8 (Friday)
EPS Faculty Lounge, Room 409, 4th floor, Hoffman Lab
20 Oxford Street, Cambridge, MA 02138Participants:
- Peter Hubyers (Harvard University)
- Eli Tziperman (Harvard University)
- Andrew Rhines (University of Washington)
- Karen McKinnon (UCAR)
- Douglas MacMartin (Caltech)
- Thomas Laepple (Alfred Wegener Institute)
- Yossi Ashkenazy (Ben-Gurion University)
- Marlene Kretschamer (Potsdam Institute for Climate Impact Research)
- Natesh Pillai (Harvard University)
- Judah Cohen (Atmospheric and Environmental Research)
- Cristian Proistosescu (Harvard University)
Please click Workshop Agenda for a downloadable agenda.
* This event is sponsored by CMSA Harvard University.
Anisotropy, biased pairing theory and applications
Abstract: Not so long ago, the relations between algebraic geometry and combinatorics were strictly governed by the former party, with results like log-concavity of the coefficients of the characteristic polynomial of matroids shackled by intuitions and techniques from projective algebraic geometry, specifically Hodge Theory. And so, while we proved analogues for these results, combinatorics felt subjugated to inspirations from outside of it.
In recent years, a new powerful technique has emerged: Instead of following the geometric statements of Hodge theory about signature, we use intuitions from the Hall marriage theorem, translated to algebra: once there, they are statements about self-pairings, the non-degeneracy of pairings on subspaces to understand the global geometry of the pairing. This was used to establish Lefschetz type theorems far beyond the scope of algebraic geometry, which in turn established solutions to long-standing conjectures in combinatorics.I will survey this theory, called biased pairing theory, and new developments within it, as well as new applications to combinatorial problems. Reporting on joint work with Stavros Papadaki, Vasiliki Petrotou and Johanna Steinmeyer.
Mini-workshop on SYZ and Homological Mirror Symmetry
The Center of Mathematical Sciences and Applications will be hosting a 4-day workshop on SYZ and Homological Mirror Symmetry and related areas on November 28 – December 2, 2016 at Harvard CMSA Building: Room G10, 20 Garden Street, Cambridge, MA 02138.
Organizers:
Bong Lian (Brandeis University), Siu-Cheong Lau (Boston University), Shing-Tung Yau (Harvard University)
Speakers:
- Conan Leung, Chinese University of Hong Kong
- Junwu Tu, University of Missouri
- Jingyu Zhao, Columbia University
- David Treumann, Boston College
- Hiro Lee Tanaka, Harvard University
- Fabian Haiden, Harvard University
- Hansol Hong, Harvard CMSA/Brandeis University
- Netanel Blaier, Harvard CMSA/Brandeis University
- Garret Alston, The University of Oklahoma
Please click Workshop Program for a downloadable schedule with talk abstracts.
Conference Schedule:
Monday, November 28 – Day 1 10:30am –11:30am Hiro Lee Tanaka, “Floer theory through spectra” Lunch 1:00pm – 2:30pm Fabian Haiden, “Categorical Kahler Geometry” 2:30pm-2:45pm Break 2:45pm – 4:15pm Fabian Haiden, “Categorical Kahler Geometry” 4:30pm – 5:15pm Garret Alston, “Potential Functions of Non-exact fillings” Tuesday, November 29 – Day 2 10:30am –11:30am Conan Leung, “Remarks on SYZ” Lunch 1:00pm – 2:30pm Jingyu Zhao, “Homological mirror symmetry for open manifolds and Hodge theoretic invariants” 2:30pm-2:45pm Break 2:45pm – 4:15pm Hiro Lee Tanaka, “Floer theory through spectra” 4:30pm – 5:15pm Hansol Hong, “Mirror Symmetry for punctured Riemann surfaces and gluing construction” Wednesday, November 30 – Day 3 10:30am –11:30am Junwu Tu, “Homotopy L-infinity spaces and mirror symmetry” Lunch 1:00pm – 2:30pm Jingyu Zhao, “Homological mirror symmetry for open manifolds and Hodge theoretic invariants” 2:30-2:45pm Break 2:45pm – 4:15pm David Treumann, “Invariants of Lagrangians via microlocal sheaf theory” Thursday, December 1 – Day 4 10:30am –11:30am David Treumann, “Some examples in three dimensions” Lunch 1:00pm – 2:30pm Junwu Tu, “Homotopy L-infinity spaces and mirror symmetry” 2:30-2:45pm Break 2:45pm – 3:30pm Netanel Blaier, “The quantum Johnson homomorphism, and the symplectic mapping class group of 3-folds” * This event is sponsored by the Simons Foundation and CMSA Harvard University.
2020-2021 Colloquium, Wednesdays
During the Spring 2021 semester, and until further notice, all seminars will take place virtually.
The 2020-2021 Colloquium will take place every Wednesday from 9:00 to 10:00am ET virtually, using zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. Please email the seminar organizers to obtain a link. This year’s colloquium will be organized by Wei Gu and Sergiy Verstyuk. The schedule below will be updated as speakers are confirmed.
To learn how to attend, please fill out this form.
Information on previous colloquia can be found here.
Spring 2021:
Date Speaker Title/Abstract 1/27/2021 Evelyn Tang (Max Planck Institute for Dynamics and Self-Organization) Title: Topology protects chiral edge currents in stochastic systems Abstract: Living systems can exhibit time-scales much longer than those of the underlying components, as well as collective dynamical behavior. How such global behavior is subserved by stochastic constituents remains unclear. I will present two-dimensional stochastic networks that consist of out-of-equilibrium cycles at the molecular scale and support chiral edge currents in configuration space. I will discuss the topological properties of these networks and their uniquely non-Hermitian features such as exceptional points and vorticity. As these emergent edge currents are associated to macroscopic timescales and length scales, simply tuning a small number of parameters enables varied dynamical phenomena including a global clock, stochastic growth and shrinkage, and synchronization.
2/3/2021 André Luiz de Gouvêa (Northwestern) Title: The Brave Nu World Abstract: Neutrinos are the least understood of the fundamental particles that make up the so-called Standard Model of Particle Physics. Measuring neutrino properties and identifying how they inform our understanding of nature at the smallest distant scales is among the highest priorities of particle physics research today. I will discuss our current understanding of neutrinos, concentrating on the observation of neutrino oscillations and neutrino masses, along with all the open questions that came of these discoveries from the end of the 20th century.
2/10/2021 Mykhaylo Shkolnikov (Princeton) Title: Probabilistic approach to free boundary problems and applications Abstract: We will discuss a recently developed probabilistic approach to (singular) free boundary problems, such as the supercooled Stefan problem. The approach is based on a new notion of solution, referred to as probabilistic, which arises naturally in the context of large system limits of interacting particle systems. In the talk, I will give an example of how such interacting particle systems arise in applications (e.g., finance), then obtain a solution of a free boundary problem in the large system limit, and discuss how this solution can be analyzed mathematically (thereby answering natural questions about the systemic risk in financial systems and neural synchronization in the brain). The talk is based on recent and ongoing joint works with Sergey Nadtochiy, Francois Delarue, Jiacheng Zhang and Xiling Zhang
2/17/2021
9:00 – 10:00PM ETC. Seshadhri (UC Santa Cruz) Title: Studying the (in)effectiveness of low dimensional graph embeddings Abstract: Low dimensional graph embeddings are a fundamental and popular tool used for machine learning on graphs. Given a graph, the basic idea is to produce a low-dimensional vector for each vertex, such that “similarity” in geometric space corresponds to “proximity” in the graph. These vectors can then be used as features in a plethora of machine learning tasks, such as link prediction, community labeling, recommendations, etc. Despite many results emerging in this area over the past few years, there is less study on the core premise of these embeddings. Can such low-dimensional embeddings effectively capture the structure of real-world (such as social) networks? Contrary to common wisdom, we mathematically prove and empirically demonstrate that popular low-dimensional graph embeddings do not capture salient properties of real-world networks. We mathematically prove that common low-dimensional embeddings cannot generate graphs with both low average degree and large clustering coefficients, which have been widely established to be empirically true for real-world networks. Empirically, we observe that the embeddings generated by popular methods fail to recreate the triangle structure of real-world networks, and do not perform well on certain community labeling tasks. (Joint work with Ashish Goel, Caleb Levy, Aneesh Sharma, and Andrew Stolman.)
2/24/2021 David Ben-Zvi (U Texas) Title: Electric-Magnetic Duality for Periods and L-functions Abstract: I will describe joint work with Yiannis Sakellaridis and Akshay Venkatesh, in which ideas originating in quantum field theory are applied to a problem in number theory.
A fundamental aspect of the Langlands correspondence — the relative Langlands program — studies the representation of L-functions of Galois representations as integrals of automorphic forms. However, the data that naturally index the period integrals (spherical varieties for G) and the L-functions (representations of the dual group G^) don’t seem to line up.
We present an approach to this problem via the Kapustin-Witten interpretation of the [geometric] Langlands correspondence as electric-magnetic duality for 4-dimensional supersymmetric Yang-Mills theory. Namely, we rewrite the relative Langlands program as duality in the presence of supersymmetric boundary conditions. As a result the partial correspondence between periods and L-functions is embedded in a natural duality between Hamiltonian actions of the dual groups.3/3/2021 9:00pm ET
Omer Tamuz (Caltech) Title: Monotone Additive Statistics Abstract: How should a random quantity be summarized by a single number? We study mappings from random variables to real numbers, focussing on those with the following two properties: (1) monotonicity with respect to first-order stochastic dominance, and (2) additivity for sums of independent random variables. This problem turns out to be connected to the following question: Under what conditions on the random variables X and Y does there exist an independent Z so that X + Z first-order stochastically dominates Y + Z?
(Joint work with Tobias Fritz, Xiaosheng Mu, Luciano Pomatto and Philipp Strack.)
3/10/2021 9:00pm ET
Piotr Indyk (MIT) Title: Learning-Based Sampling and Streaming Abstract: Classical algorithms typically provide “one size fits all” performance, and do not leverage properties or patterns in their inputs. A recent line of work aims to address this issue by developing algorithms that use machine learning predictions to improve their performance. In this talk I will present two examples of this type, in the context of streaming and sampling algorithms. In particular, I will show how to use machine learning predictions to improve the performance of (a) low-memory streaming algorithms for frequency estimation (ICLR’19), and (b) sampling algorithms for estimating the support size of a distribution (ICLR’21). Both algorithms use an ML-based predictor that, given a data item, estimates the number of times the item occurs in the input data set. (The talk will cover material from papers co-authored with T Eden, CY Hsu, D Katabi, S Narayanan, R Rubinfeld, S Silwal, T Wagner and A Vakilian.
3/17/2021
9:00pm ETChiu-Chu Melissa Liu (Columbia) Title: Topological Recursion and Crepant Transformation Conjecture Abstract: The Crepant Transformation Conjecture (CTC), first proposed by Yongbin Ruan and later refined/generalized by others, relates Gromov-Witten (GW) invariants of K-equivalent smooth varieties or smooth Deligne-Mumford stacks. We will outline a proof of all-genus open and closed CTC for symplectic toric Calabi-Yau 3-orbifolds based on joint work with Bohan Fang, Song Yu, and Zhengyu Zong. Our proof relies on the Remodeling Conjecture (proposed by Bouchard-Klemm-Marino-Pasquetti and proved in full generality by Fang, Zong and the speaker) relating open and closed GW invariants of a symplectic toric Calabi-Yau 3-orbifold to invariants of its mirror curve defined by Chekhov-Eynard-Orantin Topological Recursion.
3/24/2021 Weinan E (Princeton) Title: Machine Learning and PDEs Abstract: I will discuss two topics:
(1) Machine learning-based algorithms and “regularity” theory for very high dimensional PDEs;
(2) Formulating machine learning as PDE (more precisely, integral-differental equation) problems.3/31/2021 Thore Graepel (DeepMind/UCL) Title: From AlphaGo to MuZero – Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model Abstract: Constructing agents with planning capabilities has long been one of the main challenges in the pursuit of artificial intelligence. Tree-based planning methods have enjoyed huge success in challenging domains, such as chess and Go, where a perfect simulator is available. However, in real-world problems the dynamics governing the environment are often complex and unknown. In this work we present the MuZero algorithm which, by combining a tree-based search with a learned model, achieves superhuman performance in a range of challenging and visually complex domains, without any knowledge of their underlying dynamics. MuZero learns a model that, when applied iteratively, predicts the quantities most directly relevant to planning: the reward, the action-selection policy, and the value function. When evaluated on 57 different Atari games – the canonical video game environment for testing AI techniques, in which model-based planning approaches have historically struggled – our new algorithm achieved a new state of the art. When evaluated on Go, chess and shogi, without any knowledge of the game rules, MuZero matched the superhuman performance of the AlphaZero algorithm that was supplied with the game rules.
4/7/2021 Kui Ren (Columbia) Title: Inversion via Optimization: Revisiting the Classical Least-Squares Formulation of Inverse Problems Abstract: The classical least-squares formulation of inverse problems has provided a successful framework for the computational solutions of those problems. In recent years, modifications and alternatives have been proposed to overcome some of the disadvantages of this classical formulation in dealing with new applications. This talk intends to provide an (likely biased) overview of the recent development in constructing new least-squares formulations for model and data-driven solutions of inverse problems.
4/14/2021 Siu-Cheong Lau (Boston U) Title: An algebro-geometric formulation of computing machines Abstract: Neural network in machine learning has obvious similarity with quiver representation theory. The main gap between the two subjects is that network functions produced from two isomorphic quiver representations are not equal, due to the presence of non-linear activation functions which are not equivariant under the automorphism group. This violates the important math/physics principle that isomorphic objects should produce the same results. In this talk, I will introduce a general formulation using moduli spaces of framed modules of (noncommutative) algebra and fix this gap. Metrics over the moduli space are crucial. I will also explain uniformization between spherical, Euclidean and hyperbolic moduli.
4/21/2021 Vasco Carvalho (Cambridge) Title: The Economy as a Complex Production Network
Abstract: A modern economy is an intricately linked web of specialized production units, each relying on the flow of inputs from their suppliers to produce their own output, which in turn is routed towards other downstream units. From this production network vantage point we: (i) present the theoretical foundations for the role of such input linkages as a shock propagation channel and as a mechanism for transforming micro-level shocks into macroeconomic, economy-wide fluctuations (ii) selectively survey both empirical and simulation-based studies that attempt to ascertain the relevance and quantitative bite of this argument and (time permitting) (iii) discuss a range of domains where this networked production view is currently being extended to.4/28/2021 9:00 – 10:00pm ET
Shamit Kachru (Stanford) Title: K3 Metrics from String Theory Abstract: Calabi-Yau manifolds have played a central role in important developments in string theory and mathematical physics. Famously, they admit Ricci flat metrics — but the proof of that fact is not constructive, and the metrics remain mysterious. K3 is perhaps the simplest non-trivial compact Calabi-Yau space. In this talk, I describe two different methods of constructing (smooth, Ricci flat) K3 metrics, and a string theory duality which relates them. The duality re-sums infinite towers of disc instanton corrections via a purely classical infinite-dimensional hyperkahler quotient construction, which can be practically implemented.
Fall 2020:
Date Speaker Title/Abstract 9/23/2020 David Kazhdan (Hebrew University) Title: On Applications of Algebraic Combinatorics to Algebraic Geometry Abstract: I present a derivation of a number of results on morphisms of a high Schmidt’s rank from a result in Algebraic Combinatorics. In particular will explain the flatness of such morphisms and show their fibers have rational singularities.
10/7/2020 10:00am
Mariangela Lisanti (Princeton University) Title: Mapping the Milky Way’s Dark Matter Halo with Gaia Abstract: The Gaia mission is in the process of mapping nearly 1% of the Milky Way’s stars—-nearly a billion in total. This data set is unprecedented and provides a unique view into the formation history of our Galaxy and its associated dark matter halo. I will review results based on the most recent Gaia data release, demonstrating how the evolution of the Galaxy can be deciphered from the stellar remnants of massive satellite galaxies that merged with the Milky Way early on. This analysis is an inherently “big data” problem, and I will discuss how we are leveraging machine learning techniques to advance our understanding of the Galaxy’s evolution. Our results indicate that the local dark matter is not in equilibrium, as typically assumed, and instead exhibits distinctive dynamics tied to the disruption of satellite galaxies. The updated dark matter map built from the Gaia data has ramifications for direct detection experiments, which search for the interactions of these particles in terrestrial targets.
10/14/2020 Gil Kalai (Hebrew University and IDC Herzliya) Title: Statistical, mathematical, and computational aspects of noisy intermediate-scale quantum computers Abstract: Noisy intermediate-scale quantum (NISQ) Computers hold the key for important theoretical and experimental questions regarding quantum computers. In the lecture I will describe some questions about mathematics, statistics and computational complexity which arose in my study of NISQ systems and are related to
a) My general argument “against” quantum computers,
b) My analysis (with Yosi Rinott and Tomer Shoham) of the Google 2019 “quantum supremacy” experiment.
Relevant papers:
Yosef Rinott, Tomer Shoham and Gil Kalai, Statistical aspects of the quantum supremacy demonstration, https://gilkalai.files.
wordpress.com/2019/11/stat-quantum2.pdf
Gil Kalai, The Argument against Quantum Computers, the Quantum Laws of Nature, and Google’s Supremacy Claims, https://gilkalai.files.
wordpress.com/2020/08/laws-blog2.pdf
Gil Kalai, Three puzzles on mathematics, computations, and games, https://gilkalai.files.
wordpress.com/2019/09/main-pr.pdf10/21/2020 Marta Lewicka (University of Pittsburgh) Title: Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models Abstract: We propose results that relate the following two contexts:
(i) Given a Riemann metric G on a thin plate, we study the question of what is its closest isometric immersion, with respect to the distance measured by energies E^h which are modifications of the classical nonlinear three-dimensional elasticity.
(ii) We perform the full scaling analysis of E^h, in the context of dimension reduction as the plate’s thickness h goes to 0, and derive the Gamma-limits of h^{-2n}E^h for all n. We show the energy quantization, in the sense that the even powers 2n of h are the only possible ones (all of them are also attained).
For each n, we identify conditions for the validity of the corresponding scaling, in terms of the vanishing of Riemann curvatures of G up to appropriate orders, and in terms of the matched isometry expansions. Problems that we discuss arise from the description of elastic materials displaying heterogeneous incompatibilities of strains that may be associated with growth, swelling, shrinkage, plasticity, etc. Our results display the interaction of calculus of variations,
geometry and mechanics of materials in the prediction of patterns and shape formation.10/28/2020 Jonathan Heckman (University of Pennsylvania) Title: Top Down Approach to Quantum Fields Abstract: Quantum Field theory (QFT) is the common language of particle physicists, cosmologists, and condensed matter physicists. Even so, many fundamental aspects of QFT remain poorly understood. I discuss some of the recent progress made in understanding QFT using the geometry of extra dimensions predicted by string theory, highlighting in particular the special role of seemingly “exotic” higher-dimensional supersymmetric QFTs with no length scales known as six-dimensional superconformal field theories (6D SCFTs). We have recently classified all examples of such 6D SCFTs, and are now using this to extra observables from strongly correlated systems in theories with more than four spacetime dimensions, as well as in spacetimes with four or fewer spacetime dimensions. Along the way, I will also highlight the remarkable interplay between physical and mathematical structures in the study of such systems
11/4/2020
9:00pm ETSurya Ganguli (Stanford) Title: Weaving together machine learning, theoretical physics, and neuroscience through mathematics Abstract: An exciting area of intellectual activity in this century may well revolve around a synthesis of machine learning, theoretical physics, and neuroscience. The unification of these fields will likely enable us to exploit the power of complex systems analysis, developed in theoretical physics and applied mathematics, to elucidate the design principles governing neural systems, both biological and artificial, and deploy these principles to develop better algorithms in machine learning. We will give several vignettes in this direction, including: (1) determining the best optimization problem to solve in order to perform regression in high dimensions; (2) finding exact solutions to the dynamics of generalization error in deep linear networks; (3) developing interpretable machine learning to derive and understand state of the art models of the retina; (4) analyzing and explaining the origins of hexagonal firing patterns in recurrent neural networks trained to path-integrate; (5) delineating fundamental theoretical limits on the energy, speed and accuracy with which non-equilibrium sensors can detect signals
Selected References:
M. Advani and S. Ganguli, Statistical mechanics of optimal convex inference in high dimensions, Physical Review X, 6, 031034, 2016.
M. Advani and S. Ganguli, An equivalence between high dimensional Bayes optimal inference and M-estimation, NeurIPS, 2016.
A.K. Lampinen and S. Ganguli, An analytic theory of generalization dynamics and transfer learning in deep linear networks, International Conference on Learning Representations (ICLR), 2019.
H. Tanaka, A. Nayebi, N. Maheswaranathan, L.M. McIntosh, S. Baccus, S. Ganguli, From deep learning to mechanistic understanding in neuroscience: the structure of retinal prediction, NeurIPS 2019.
S. Deny, J. Lindsey, S. Ganguli, S. Ocko, The emergence of multiple retinal cell types through efficient coding of natural movies, Neural Information Processing Systems (NeurIPS) 2018.
B. Sorscher, G. Mel, S. Ganguli, S. Ocko, A unified theory for the origin of grid cells through the lens of pattern formation, NeurIPS 2019.
Y. Bahri, J. Kadmon, J. Pennington, S. Schoenholz, J. Sohl-Dickstein, and S. Ganguli, Statistical mechanics of deep learning, Annual Reviews of Condensed Matter Physics, 2020.
S.E. Harvey, S. Lahiri, and S. Ganguli, A universal energy accuracy tradeoff in nonequilibrium cellular sensing, https://arxiv.org/abs/2002.1056711/11/2020 Kevin Buzzard (Imperial College London) Title: Teaching proofs to computers Abstract: A mathematical proof is a sequence of logical statements in a precise language, obeying some well-defined rules. In that sense it is very much like a computer program. Various computer tools have appeared over the last 50 years which take advantage of this analogy by turning the mathematical puzzle of constructing a proof of a theorem into a computer game. The newest tools are now capable of understanding some parts of modern research mathematics. In spite of this, these tools are not used in mathematics departments, perhaps because they are not yet capable of telling mathematicians *something new*.
I will give an overview of the Lean theorem prover, showing what it can currently do. I will also talk about one of our goals: using Lean to make practical tools which will be helpful for future researchers in pure mathematics.11/18/2020 Jose A. Scheinkman (Columbia) Title: Re-pricing avalanches Abstract: Monthly aggregate price changes exhibit chronic fluctuations but the aggregate shocks that drive these fluctuations are often elusive. Macroeconomic models often add stochastic macro-level shocks such as technology shocks or monetary policy shocks to produce these aggregate fluctuations. In this paper, we show that a state-dependent pricing model with a large but finite number of firms is capable of generating large fluctuations in the number of firms that adjust prices in response to an idiosyncratic shock to a firm’s cost of price adjustment. These fluctuations, in turn, cause fluctuations in aggregate price changes even in the absence of aggregate shocks. (Joint work with Makoto Nirei.)
11/25/2020 10:45am
Eric J. Heller (Harvard) Title: Branched Flow Abstract: In classical and quantum phase space flow, there exists a regime of great physical relevance that is belatedly but rapidly generating a new field. In evolution under smooth, random, weakly deflecting but persistent perturbations, a remarkable regime develops, called branched flow. Lying between the first cusp catastrophes at the outset, leading to fully chaotic statistical flow much later, lies the visually beautiful regime of branched flow. It applies to tsunami wave propagation, freak wave formation, light propagation, cosmic microwaves arriving from pulsars, electron flow in metals and devices, sound propagation in the atmosphere and oceans, the large scale structure of the universe, and much more. The mathematical structure of this flow is only partially understood, involving exponential instability coexisting with “accidental” stability. The flow is qualitatively universal, but this has not been quantified. Many questions arise, including the scale(s) of the random medium, and the time evolution of manifolds and “fuzzy” manifolds in phase space. The classical-quantum (ray-wave) correspondence in this flow is only partially understood. This talk will be an introduction to the phenomenon, both visual and mathematical, emphasizing unanswered questions
12/2/2020 Douglas Arnold (U of Minnesota) Title: Preserving geometry in numerical discretization Abstract: An important design principle for numerical methods for differential equations is that the discretizations preserve key geometric, topological, and algebraic structures of the original differential system. For ordinary differential equations, such geometric integrators were developed at the end of the last century, enabling stunning computations in celestial mechanics and other applications that would have been impossible without them. Since then, structure-preserving discretizations have been developed for partial differential equations. One of the prime examples has been the finite element exterior calculus or FEEC, in which the structures to preserve are related to Hilbert complexes underlying the PDEs, the de Rham complex being a canonical example. FEEC has led to highly successful new numerical methods for problems in fluid mechanics, electromagnetism, and other applications which relate to the de Rham complex. More recently, new tools have been developed which extend the applications of FEEC far beyond the de Rham complex, leading to progress in discretizations of problems from solid mechanics, materials science, and general relativity.
12/9/2020 Manuel Blum and Lenore Blum (Carnegie Mellon) Title: What can Theoretical Computer Science Contribute to the Discussion of Consciousness? Abstract: The quest to understand consciousness, once the purview of philosophers and theologians, is now actively pursued by scientists of many stripes. We study consciousness from the perspective of theoretical computer science. This is done by formalizing the Global Workspace Theory (GWT) originated by cognitive neuroscientist Bernard Baars and further developed by him, Stanislas Dehaene, and others. We give a precise formal definition of a Conscious Turing Machine (CTM), also called Conscious AI, in the spirit of Alan Turing’s simple yet powerful definition of a computer. We are not looking for a complex model of the brain nor of cognition but for a simple model of (the admittedly complex concept of) consciousness.
After formally defining CTM, we give a formal definition of consciousness in CTM. We then suggest why the CTM has the feeling of consciousness. The reasonableness of the definitions and explanations can be judged by how well they agree with commonly accepted intuitive concepts of human consciousness, the range of related concepts that the model explains easily and naturally, and the extent of the theory’s agreement with scientific evidenceJDG 2017 Conference, April 28 – May 2, 2017
In celebration of the Journal of Differential Geometry’s 50th anniversary, the Harvard Math Department will be hosting the Tenth Conference on Geometry and Topology (JDG 2017) from April 28 – May 2, 2017.
Registration and additional information on the conference can be found at http://abel.harvard.edu/jdg/index.html.
Confirmed Speakers
- Mina Aganagic, UC Berkeley
- Denis Auroux, UC Berkeley
- Caucher Birkar, University of Cambridge
- Huai-Dong Cao, Lehigh University
- Tristan Collins, Harvard University
- Camillo De Lellis, ETH Zurich
- Jean-Pierre Demailly, Grenoble Alpes University
- Simon Donaldson, Stony Brook University
- Dan Freed, University of Texas at Austin
- Kenji Fukaya, Stony Brook University
- David Gabai, Princeton University
- Larry Guth, Massachusetts Institute of Technology
- Richard Hamilton, Columbia University
- Yujiro Kawamata, University of Tokyo
- Frances Kirwan, Oxford University
- Blaine Lawson, Stony Brook University
- Jun Li, Stanford University
- Si Li, Tsinghua University
- Bong Lian, Brandeis University
- Chiu-Chu Melissa Liu, Columbia University
- Ciprian Manolescu, University of California, Los Angeles
- Fernando Marques, Princeton University
- William Meeks, University of Massachusetts Amherst
- William Minicozzi, Massachusetts Institute of Technology
- John Pardon, Princeton University
- Duong Phong, Columbia University
- Alena Pirutka, Courant Institute of New York University
- Richard Schoen, University of California, Irvine
- Artan Sheshmani, QGM Aarhus University/Harvard University
- Cliff Taubes, Harvard University
- Cumrun Vafa, Harvard University
- Mu-Tao Wang, Columbia University
- Shing-Tung Yau, Harvard University
- Steve Zelditch, Northwestern Univeristy
* This event is co-sponsored by Lehigh University and partially supported by the National Science Foundation.
Electric-magnetic duality and the Geometric Langlands duality
Title: Electric-magnetic duality and the Geometric Langlands duality
Abstract: I will give a pedagogical review of the connection between electric-magnetic duality and the Geometric Langlands duality.
Working Conference on Materials and Data Analysis, March 27-30, 2017
The Center of Mathematical Sciences and Applications will be hosting a 5-day working Conference on Materials and Data Analysis and related areas, March 27-30, 2017. The conference will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
Photos of the event can be found on CMSA’s Blog.
Participants:
- Ryan P. Adams, Harvard University
- Jörg Behler, University of Göttingen
- Kieron Burke, University of California, Irvine
- Lucy Colwell, University of Cambridge
- Gábor Csányi, University of Cambridge
- Ekin Doğuş Çubuk, Stanford University
- Leslie Greengard, Courant Institute of Mathematical Sciences, New York University
- Petros Koumoutsakos, Radcliffe Institute for Advanced Study, Harvard University
- Govind Menon, Brown University
- Evan Reed, Stanford University
- Patrick Riley, Google
- Matthias Rupp, Fitz Haber Institute of the Max Planck Society
- Sadasivan Shankar, Harvard University
- Dennis Sheberla, Harvard University
Schedule:
Monday, March 27
Time Speaker Title 8:30am – 9:00am Breakfast 9:00am – 10:00am Kieron Burke, University of California, Irvine Background in DFT and electronic structure calculations 10:00am – 11:00am Kieron Burke, University of California, Irvine The density functionals machines can learn
11:00am – 12:00pm Sadasivan Shankar, Harvard University A few key principles for applying Machine Learning to Materials (or Complex Systems) — Scientific and Engineering Perspectives Tuesday, March 28
Time Speaker Title 8:30am – 9:00am Breakfast 9:00am – 10:00am Ryan Adams, Harvard TBA 10:00am – 11:00am Gábor Csányi, University of Cambridge Interatomic potentials using machine learning: accuracy, transferability and chemical diversity
11:00am – 1:00pm Lunch Break 1:00pm – 2:00pm Evan Reed, Stanford University TBA Wednesday, March 29
Time Speaker Title 8:30am – 9:00am Breakfast 9:00am – 10:00am Patrick Riley, Google The Message Passing Neural Network framework and its application to molecular property prediction 10:00am – 11:00am Jörg Behler, University of Göttingen TBA 11:00am – 12:00pm Ekin Doğuş Çubuk, Stanford Univers TBA 4:00pm Leslie Greengard, Courant Institute Inverse problems in acoustic scattering and cryo-electron microscopy CMSA Colloquium
Thursday, March 30
Time Speaker Title 8:30am – 9:00am Breakfast 9:00am – 10:00am Matthias Rupp, Fitz Haber Institute of the Max Planck Society TBA 10:00am – 11:00am Petros Koumoutsakos, Radcliffe Institute for Advanced Study, Harvard TBA 11:00am – 1:00pm Lunch Break 1:00pm – 2:00pm Dennis Sheberla, Harvard University Rapid discovery of functional molecules by a high-throughput virtual screening Workshop on Discrete and Topological Models for Effective Field Theories, January 9-13, 2017
The Center of Mathematical Sciences and Applications will be hosting a Workshop on “Discrete and Topological Models for Effective Field Theories,” January 9-13, 2017. The workshop will be hosted in G02 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
Titles, abstracts and schedule will be provided nearer to the event.
Participants:
Dan Freed, UT Austin
Anton Kapustin, California Institute of Technology
Alexei Y. Kitaev, California Institute of Technology
Greg Moore, Rutgers University
Constantin Teleman, University of Oxford
Organizers:
Mike Hopkins, Shing-Tung Yau
* This event is sponsored by CMSA Harvard University.
Working Conference on Covariance Analysis in Biology, May 1-4, 2017
The Center of Mathematical Sciences and Applications will be hosting a working Conference on Covariance Analysis in Biology, May 1-4, 2017. The conference will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
This event is open and free. If you would like to attend, please register here to help us keep a headcount. A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Speakers:
Orr Ashenberg, Fred Hutchinson Cancer Research Center
John Barton, Massachusetts Institute of Technology
Simona Cocco, Laboratoire de Physique Statistique de l’ENS
Sean Eddy, Harvard University
Efthimios Kaxiras, Harvard University
Michael Laub, Massachusetts Institute of Technology
Debora S. Marks, Harvard University
Govind Menon, Brown University
Rémi Monasson, Laboratoire de Physique Théorique de l’ENS
Andrew Murray, Harvard University
Ilya Nemenman, Emory College
Chris Sander, Dana-Farber Cancer Institute, Harvard Medical School
Dave Thirumalai, University of Texas at Austin
Martin Weigt, IBPS, Université Pierre et Marie Curie
Matthieu Wyart, EPFL
More speakers will be confirmed soon.
Schedule:
(Please click here for a downloadable version of the schedule.)
Please note that the schedule for both days is currently tentative and is subject to change.
May 1, Monday
Time Speaker Topic 9:00-10:00am Sean Eddy TBA 10:00-11:00am Mike Laub TBA 11:00am-12:00pm Ilya Nemenman TBA May 2, TuesdayTime Speaker Topic 9:00-10:00am Orr Ashenberg TBA 10:00-11:00am Debora Marks TBA 11:00am-12:00pm Martin Weigt TBA 4:30pm-5:30pm Simona Cocco CMSA Colloquia May 3, WednesdayTime Speaker Topic 9:00-10:00am Andrew Murray TBA 10:00-11:00am Matthieu Wyart TBA 11:00am-12:00pm Rémi Monasson TBA May 4, ThursdayTime Speaker Topic 9:00-10:00am David Thirumalai TBA 10:00-11:00am Chris Sander TBA 11:00am-12:00pm John Barton TBA Organizers:
Michael Brenner, Lucy Colwell, Elena Rivas, Eugene Shakhnovich
* This event is sponsored by CMSA Harvard University.
A Celebration of Symplectic Geometry: 15 Years of JSG, June 5-6, 2017
In celebration of the Journal of Symplectic Geometry’s 15th anniversary, the Center of Mathematical Sciences and Applications will be hosting A Celebration of Symplectic Geometry: 15 Years of JSG on June 5-6, 2017.
To register for this event, please click here.
Confirmed speakers:
- Roger Casals, MIT
- Chen He, Northeastern University
- Yael Karshon, University of Toronto
- Ailsa Keating, Institute of Advanced Study
- Eckhard Meinrenken, University of Toronto
- Ana Rita Pires, Fordham University
- Sobhan Seyfaddini, Institute of Advanced Study
- Alejandro Uribe, University of Michigan
- Jonathan Weitsman, Northeastern University
The conference is co-organized by Denis Auroux and Victor Guillemin. Additional information on the conference will be announced closer to the event.
For a list of lodging options convenient to the Center, please see our recommended lodgings page.
Schedule:
The schedule for both days is currently tentative and is subject to change. A pdf version of the schedule can also be downloaded here.
June 5, Monday (Full day)
Time Speaker Topic 8:30am – 9:0am Breakfast 9:00am – 10:00am Jonathan Weitsman Title: On the geometric quantization of (some) Poisson manifolds 10:30am – 11:30am Eckhard Meinrenken Title: On Hamiltonian loop group spaces Abstract: Let G be a compact Lie group. We explain a construction of an LG-equivariant spinor module over any Hamiltonian loop group space with proper moment map. It may be regarded as its `canonical spin-c structure’. We show how to reduce to finite dimensions, resulting in actual spin-s structure on transversals, as well as twisted spin-c structures for the associated quasi-hamiltonian space. This is based on joint work with Yiannis Loizides and Yanli Song.
11:30am – 1:30pm Break 1:30pm – 2:30pm Ana Rita Pires Title: Infinite staircases in symplectic embedding problems Abstract: McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase related to the odd index Fibonacci numbers. Infinite staircases have been shown to exist also in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid E(2,3). I will describe how we use ECH capacities, lattice point counts and Ehrhart theory to show that infinite staircases exist for these and a few other target manifolds, as well as to conjecture that these are the only such target manifolds. This is a joint work with Cristofaro-Gardiner, Holm and Mandini.
3:00pm – 4:00pm Sobhan Seyfaddini Title: Rigidity of conjugacy classes in groups of area-preserving homeomorphisms Abstract: Motivated by understanding the algebraic structure of groups of area-preserving homeomorphims F. Beguin, S. Crvoisier, and F. Le Roux were lead to the following question: Can the conjugacy class of a Hamiltonian homeomorphism be dense? We will show that one can rule out existence of dense conjugacy classes by simply counting fixed points. This is joint work with Le Roux and Viterbo.
4:30pm – 5:30pm Roger Casals Title: Differential Algebra of Cubic Graphs
Abstract: In this talk we will associate a combinatorial dg-algebra to a cubic planar graph. This algebra is defined by counting binary sequences, which we introduce, and we shall provide explicit computations and examples. From there we study the Legendrian surfaces behind these constructions, including Legendrian surgeries, the count of Morse flow trees involved in contact homology, and the relation to microlocal sheaves. Time permitting, I will explain a connection to spectral networks.VideoJune 6, Tuesday (Full day)
Time Speaker Topic 8:30am – 9:00am Breakfast 9:00am – 10:00am Alejandro Uribe Title: Semi-classical wave functions associated with isotropic submanifolds of phase space Abstract: After reviewing fundamental ideas on the quantum-classical correspondence, I will describe how to associate spaces of semi-classical wave functions to isotropic submanifolds of phase space satisfying a Bohr-Sommerfeld condition. Such functions have symbols that are symplectic spinors, and they satisfy a symbol calculus under the action of quantum observables. This is the semi-classical version of the Hermite distributions of Boutet the Monvel and Guillemin, and it is joint work with Victor Guillemin and Zuoqin Wang. I will inlcude applications and open questions.
10:30am – 11:30am Alisa Keating Title: Symplectomorphisms of exotic discs Abstract: It is a theorem of Gromov that the group of compactly supported symplectomorphisms of R^4, equipped with the standard symplectic form, is contractible. While nothing is known in higher dimensions for the standard symplectic form, we show that for some exotic symplectic forms on R^{4n}, for all but finitely n, there exist compactly supported symplectomorphisms that are smoothly non-trivial. The principal ingredients are constructions of Milnor and Munkres, a symplectic and contact version of the Gromoll filtration, and Borman, Eliashberg and Murphy’s work on existence of over-twisted contact structures. Joint work with Roger Casals and Ivan Smith.
11:30am – 1:30pm Break 1:30pm – 2:30pm Chen He Title: Morse theory on b-symplectic manifolds Abstract: b-symplectic (or log-symplectic) manifolds are Poisson manifolds equipped with symplectic forms of logarithmic singularity. Following Guillemin, Miranda, Pires and Scott’s introduction of Hamiltonian group actions on b-symplectic manifolds, we will survey those classical results of Hamiltonian geometry to the b-symplectic case.
3:00pm – 4:00pm Yael Karshon Title: Geometric quantization with metaplectic-c structures Abstract: I will present a variant of the Kostant-Souriau geometric quantization procedure that uses metaplectic-c structures to incorporate the “half form correction” into the prequantization stage. This goes back to the late 1970s but it is not widely known and it has the potential to generalize and improve upon recent works on geometric quantization.
2017 Big Data Conference
1 Oxford Street, Cambridge MA 02138The Center of Mathematical Sciences and Applications will be hosting a conference on Big Data from August 18 – 19, 2017, in Hall D of the Science Center at Harvard University.
The Big Data Conference features many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics. This is the third conference on Big Data the Center will host as part of our annual events, and is co-organized by Richard Freeman, Scott Kominers, Jun Liu, Horng-Tzer Yau and Shing-Tung Yau.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants.
Confirmed Speakers:
- Mohammad Akbarpour, Stanford University
- Albert-László Barabási, Northeastern University
- Noureddine El Karoui, University of California, Berkeley
- Ravi Jagadeesan, Harvard University
- Lucas Janson, Harvard University
- Tracy Ke, University of Chicago
- Tze Leung Lai, Stanford University
- Annie Liang, University of Pennsylvania
- Marena Lin, Harvard University
- Nikhil Naik, Harvard University
- Alex Peysakhovich, Facebook
- Natesh Pillai, Harvard University
- Jann Spiess, Harvard University
- Bradly Stadie, Open AI, University of California, Berkeley
- Zak Stone, Google
- Hau-Tieng Wu, University of Toronto
- Sifan Zhou, Xiamen University
Following the conference, there will be a two-day workshop from August 20-21. The workshop is organized by Scott Kominers, and will feature:
- Jörn Boehnke, Harvard University
- Nikhil Naik, Harvard University
- Bradly Stadie, Open AI, University of California, Berkeley
Conference Schedule
A PDF version of the schedule below can also be downloaded here.
August 18, Friday (Full day)
Time Speaker Topic 8:30 am – 9:00 am Breakfast 9:00 am – 9:40 am Mohammad Akbarpour Title: Information aggregation in overlapping generations and the emergence of experts Abstract: We study a model of social learning with “overlapping generations”, where agents meet others and share data about an underlying state over time. We examine under what conditions the society will produce individuals with precise knowledge about the state of the world. There are two information sharing regimes in our model: Under the full information sharing technology, individuals exchange the information about their point estimates of an underlying state, as well as their sources (or the precision of their signals) and update their beliefs by taking a weighted average. Under the limited information sharing technology, agents only observe the information about the point estimates of those they meet, and update their beliefs by taking a weighted average, where weights can depend on the sequence of meetings, as well as the labels. Our main result shows that, unlike most social learning settings, using such linear learning rules do not guide the society (or even a fraction of its members) to learn the truth, and having access to, and exploiting knowledge of the precision of a source signal are essential for efficient social learning (joint with Amin Saberi & Ali Shameli).
9:40 am – 10:20 am Lucas Janson Title: Model-Free Knockoffs For High-Dimensional Controlled Variable Selection Abstract: Many contemporary large-scale applications involve building interpretable models linking a large set of potential covariates to a response in a nonlinear fashion, such as when the response is binary. Although this modeling problem has been extensively studied, it remains unclear how to effectively control the fraction of false discoveries even in high-dimensional logistic regression, not to mention general high-dimensional nonlinear models. To address such a practical problem, we propose a new framework of model-free knockoffs, which reads from a different perspective the knockoff procedure (Barber and Candès, 2015) originally designed for controlling the false discovery rate in linear models. The key innovation of our method is to construct knockoff variables probabilistically instead of geometrically. This enables model-free knockoffs to deal with arbitrary (and unknown) conditional models and any dimensions, including when the dimensionality p exceeds the sample size n, while the original knockoffs procedure is constrained to homoscedastic linear models with n greater than or equal to p. Our approach requires the design matrix be random (independent and identically distributed rows) with a covariate distribution that is known, although we show our procedure to be robust to unknown/estimated distributions. As we require no knowledge/assumptions about the conditional distribution of the response, we effectively shift the burden of knowledge from the response to the covariates, in contrast to the canonical model-based approach which assumes a parametric model for the response but very little about the covariates. To our knowledge, no other procedure solves the controlled variable selection problem in such generality, but in the restricted settings where competitors exist, we demonstrate the superior power of knockoffs through simulations. Finally, we apply our procedure to data from a case-control study of Crohn’s disease in the United Kingdom, making twice as many discoveries as the original analysis of the same data.
10:20 am – 10:50 am Break 10:50 pm – 11:30 pm Noureddine El Karoui Title: Random matrices and high-dimensional statistics: beyond covariance matrices Abstract: Random matrices have played a central role in understanding very important statistical methods linked to covariance matrices (such as Principal Components Analysis, Canonical Correlation Analysis etc…) for several decades. In this talk, I’ll show that one can adopt a random-matrix-inspired point of view to understand the performance of other widely used tools in statistics, such as M-estimators, and very common methods such as the bootstrap. I will focus on the high-dimensional case, which captures well the situation of “moderately” difficult statistical problems, arguably one of the most relevant in practice. In this setting, I will show that random matrix ideas help upend conventional theoretical thinking (for instance about maximum likelihood methods) and highlight very serious practical problems with resampling methods.
11:30 am – 12:10 pm Nikhil Naik Title: Understanding Urban Change with Computer Vision and Street-level Imagery Abstract: Which neighborhoods experience physical improvements? In this work, we introduce a computer vision method to measure changes in the physical appearances of neighborhoods from time-series street-level imagery. We connect changes in the physical appearance of five US cities with economic and demographic data and find three factors that predict neighborhood improvement. First, neighborhoods that are densely populated by college-educated adults are more likely to experience physical improvements. Second, neighborhoods with better initial appearances experience, on average, larger positive improvements. Third, neighborhood improvement correlates positively with physical proximity to the central business district and to other physically attractive neighborhoods. Together, our results illustrate the value of using computer vision methods and street-level imagery to understand the physical dynamics of cities.
(Joint work with Edward L. Glaeser, Cesar A. Hidalgo, Scott Duke Kominers, and Ramesh Raskar.)
12:10 pm – 12:25 pm Video #1 Data Science Lightning Talks 12:25 pm – 1:30 pm Lunch 1:30 pm – 2:10 pm Tracy Ke Title: A new SVD approach to optimal topic estimation Abstract: In the probabilistic topic models, the quantity of interest—a low-rank matrix consisting of topic vectors—is hidden in the text corpus matrix, masked by noise, and Singular Value Decomposition (SVD) is a potentially useful tool for learning such a low-rank matrix. However, the connection between this low-rank matrix and the singular vectors of the text corpus matrix are usually complicated and hard to spell out, so how to use SVD for learning topic models faces challenges.
We overcome the challenge by revealing a surprising insight: there is a low-dimensional simplex structure which can be viewed as a bridge between the low-rank matrix of interest and the SVD of the text corpus matrix, and which allows us to conveniently reconstruct the former using the latter. Such an insight motivates a new SVD-based approach to learning topic models.
For asymptotic analysis, we show that under a popular topic model (Hofmann, 1999), the convergence rate of the l1-error of our method matches that of the minimax lower bound, up to a multi-logarithmic term. In showing these results, we have derived new element-wise bounds on the singular vectors and several large deviation bounds for weakly dependent multinomial data. Our results on the convergence rate and asymptotical minimaxity are new. We have applied our method to two data sets, Associated Process (AP) and Statistics Literature Abstract (SLA), with encouraging results. In particular, there is a clear simplex structure associated with the SVD of the data matrices, which largely validates our discovery.
2:10 pm – 2:50 pm Albert-László Barabási Title: Taming Complexity: From Network Science to Controlling Networks Abstract: The ultimate proof of our understanding of biological or technological systems is reflected in our ability to control them. While control theory offers mathematical tools to steer engineered and natural systems towards a desired state, we lack a framework to control complex self-organized systems. Here we explore the controllability of an arbitrary complex network, identifying the set of driver nodes whose time-dependent control can guide the system’s entire dynamics. We apply these tools to several real networks, unveiling how the network topology determines its controllability. Virtually all technological and biological networks must be able to control their internal processes. Given that, issues related to control deeply shape the topology and the vulnerability of real systems. Consequently unveiling the control principles of real networks, the goal of our research, forces us to address series of fundamental questions pertaining to our understanding of complex systems.
2:50 pm – 3:20 pm Break 3:20 pm – 4:00 pm Marena Lin Title: Optimizing climate variables for human impact studies Abstract: Estimates of the relationship between climate variability and socio-economic outcomes are often limited by the spatial resolution of the data. As studies aim to generalize the connection between climate and socio-economic outcomes across countries, the best available socio-economic data is at the national level (e.g. food production quantities, the incidence of warfare, averages of crime incidence, gender birth ratios). While these statistics may be trusted from government censuses, the appropriate metric for the corresponding climate or weather for a given year in a country is less obvious. For example, how do we estimate the temperatures in a country relevant to national food production and therefore food security? We demonstrate that high-resolution spatiotemporal satellite data for vegetation can be used to estimate the weather variables that may be most relevant to food security and related socio-economic outcomes. In particular, satellite proxies for vegetation over the African continent reflect the seasonal movement of the Intertropical Convergence Zone, a band of intense convection and rainfall. We also show that agricultural sensitivity to climate variability differs significantly between countries. This work is an example of the ways in which in-situ and satellite-based observations are invaluable to both estimates of future climate variability and to continued monitoring of the earth-human system. We discuss the current state of these records and potential challenges to their continuity.
4:00 pm – 4:40 pm Alex Peysakhovich Title: Building a cooperator Abstract: A major goal of modern AI is to construct agents that can perform complex tasks. Much of this work deals with single agent decision problems. However, agents are rarely alone in the world. In this talk I will discuss how to combine ideas from deep reinforcement learning and game theory to construct artificial agents that can communicate, collaborate and cooperate in productive positive sum interactions.
4:40 pm – 5:20 pm Tze Leung Lai Title: Gradient boosting: Its role in big data analytics, underlying mathematical theory, and recent refinements Abstract: We begin with a review of the history of gradient boosting, dating back to the LMS algorithm of Widrow and Hoff in 1960 and culminating in Freund and Schapire’s AdaBoost and Friedman’s gradient boosting and stochastic gradient boosting algorithms in the period 1999-2002 that heralded the big data era. The role played by gradient boosting in big data analytics, particularly with respect to deep learning, is then discussed. We also present some recent work on the mathematical theory of gradient boosting, which has led to some refinements that greatly improves the convergence properties and prediction performance of the methodology.
August 19, Saturday (Full day)
Time Speaker Topic 8:30 am – 9:00 am Breakfast 9:00 am – 9:40 am Natesh Pillai Title: Accelerating MCMC algorithms for Computationally Intensive Models via Local Approximations Abstract: We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach introduces local approximations of these models into the Metropolis–Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler’s exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the exact posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. Our theoretical results reinforce the key observation underlying this article: when the likelihood has some local regularity, the number of model evaluations per Markov chain Monte Carlo (MCMC) step can be greatly reduced without biasing the Monte Carlo average. Numerical experiments demonstrate multiple order-of-magnitude reductions in the number of forward model evaluations used in representative ordinary differential equation (ODE) and partial differential equation (PDE) inference problems, with both synthetic and real data.
9:40 am – 10:20 am Ravi Jagadeesan Title: Designs for estimating the treatment effect in networks with interference Abstract: In this paper we introduce new, easily implementable designs for drawing causal inference from randomized experiments on networks with interference. Inspired by the idea of matching in observational studies, we introduce the notion of considering a treatment assignment as a quasi-coloring” on a graph. Our idea of a perfect quasi-coloring strives to match every treated unit on a given network with a distinct control unit that has identical number of treated and control neighbors. For a wide range of interference functions encountered in applications, we show both by theory and simulations that the classical Neymanian estimator for the direct effect has desirable properties for our designs. This further extends to settings where homophily is present in addition to interference.
10:20 am – 10:50 am Break 10:50 am – 11:30 am Annie Liang Title: The Theory is Predictive, but is it Complete? An Application to Human Generation of Randomness Abstract: When we test a theory using data, it is common to focus on correctness: do the predictions of the theory match what we see in the data? But we also care about completeness: how much of the predictable variation in the data is captured by the theory? This question is difficult to answer, because in general we do not know how much “predictable variation” there is in the problem. In this paper, we consider approaches motivated by machine learning algorithms as a means of constructing a benchmark for the best attainable level of prediction. We illustrate our methods on the task of predicting human-generated random sequences. Relative to a theoretical machine learning algorithm benchmark, we find that existing behavioral models explain roughly 15 percent of the predictable variation in this problem. This fraction is robust across several variations on the problem. We also consider a version of this approach for analyzing field data from domains in which human perception and generation of randomness has been used as a conceptual framework; these include sequential decision-making and repeated zero-sum games. In these domains, our framework for testing the completeness of theories provides a way of assessing their effectiveness over different contexts; we find that despite some differences, the existing theories are fairly stable across our field domains in their performance relative to the benchmark. Overall, our results indicate that (i) there is a significant amount of structure in this problem that existing models have yet to capture and (ii) there are rich domains in which machine learning may provide a viable approach to testing completeness (joint with Jon Kleinberg and Sendhil Mullainathan).
11:30 am – 12:10 pm Zak Stone Title: TensorFlow: Machine Learning for Everyone Abstract: We’ve witnessed extraordinary breakthroughs in machine learning over the past several years. What kinds of things are possible now that weren’t possible before? How are open-source platforms like TensorFlow and hardware platforms like GPUs and Cloud TPUs accelerating machine learning progress? If these tools are new to you, how should you get started? In this session, you’ll hear about all of this and more from Zak Stone, the Product Manager for TensorFlow on the Google Brain team.
12:10 pm – 1:30 pm Lunch 1:30 pm – 2:10 pm Jann Spiess Title: (Machine) Learning to Control in Experiments Abstract: Machine learning focuses on high-quality prediction rather than on (unbiased) parameter estimation, limiting its direct use in typical program evaluation applications. Still, many estimation tasks have implicit prediction components. In this talk, I discuss accounting for controls in treatment effect estimation as a prediction problem. In a canonical linear regression framework with high-dimensional controls, I argue that OLS is dominated by a natural shrinkage estimator even for unbiased estimation when treatment is random; suggest a generalization that relaxes some parametric assumptions; and contrast my results with that for another implicit prediction problem, namely the first stage of an instrumental variables regression.
2:10 pm – 2:50 pm Bradly Stadie Title: Learning to Learn Quickly: One-Shot Imitation and Meta Learning Abstract: Many reinforcement learning algorithms are bottlenecked by data collection costs and the brittleness of their solutions when faced with novel scenarios.
We will discuss two techniques for overcoming these shortcomings. In one-shot imitation, we train a module that encodes a single demonstration of a desired behavior into a vector containing the essence of the demo. This vector can subsequently be utilized to recover the demonstrated behavior. In meta-learning, we optimize a policy under the objective of learning to learn new tasks quickly. We show meta-learning methods can be accelerated with the use of auxiliary objectives. Results are presented on grid worlds, robotics tasks, and video game playing tasks.2:50 pm – 3:20 pm Break 3:20 pm – 4:00 pm Hau-Tieng Wu Title: When Medical Challenges Meet Modern Data Science Abstract: Adaptive acquisition of correct features from massive datasets is at the core of modern data analysis. One particular interest in medicine is the extraction of hidden dynamics from a single observed time series composed of multiple oscillatory signals, which could be viewed as a single-channel blind source separation problem. The mathematical and statistical problems are made challenging by the structure of the signal which consists of non-sinusoidal oscillations with time varying amplitude/frequency, and by the heteroscedastic nature of the noise. In this talk, I will discuss recent progress in solving this kind of problem by combining the cepstrum-based nonlinear time-frequency analysis and manifold learning technique. A particular solution will be given along with its theoretical properties. I will also discuss the application of this method to two medical problems – (1) the extraction of a fetal ECG signal from a single lead maternal abdominal ECG signal; (2) the simultaneous extraction of the instantaneous heart/respiratory rate from a PPG signal during exercise; (3) (optional depending on time) an application to atrial fibrillation signals. If time permits, the clinical trial results will be discussed.
4:00 pm – 4:40 pm Sifan Zhou Title: Citing People Like Me: Homophily, Knowledge Spillovers, and Continuing a Career in Science Abstract: Forward citation is widely used to measure the scientific merits of articles. This research studies millions of journal article citation records in life sciences from MEDLINE and finds that authors of the same gender, the same ethnicity, sharing common collaborators, working in the same institution, or being geographically close are more likely (and quickly) to cite each other than predicted by their proportion among authors working on the same research topics. This phenomenon reveals how social and geographic distances influence the quantity and speed of knowledge spillovers. Given the importance of forward citations in academic evaluation system, citation homophily potentially put authors from minority group at a disadvantage. I then show how it influences scientists’ chances to survive in the academia and continue publishing. Based on joint work with Richard Freeman.
To view photos and video interviews from the conference, please visit the CMSA blog.
Math Science Lectures in Honor of Raoul Bott: Mina Aganagic
1 Oxford Street, Cambridge MA 02138On April 9 and 10, 2019 the CMSA hosted two lectures by Mina Aganagic (UC Berkeley). This was the second annual Math Science Lecture Series held in honor of Raoul Bott.
The lectures took place in Science Center, Hall C
“Two math lessons from string theory”
Lecture 1: April 9, 2019
Title: “Lesson on Integrability”
Abstract: The quantum Knizhnik-Zamolodchikov (qKZ) equation is a difference generalization of the famous Knizhnik-Zamolodchikov (KZ) equation. The problem to explicitly capture the monodromy of the qKZ equation has been open for over 25 years. I will describe the solution to this problem, discovered jointly with Andrei Okounkov. The solution comes from the geometry of Nakajima quiver varieties and has a string theory origin.
Part of the interest in the qKZ monodromy problem is that its solution leads to integrable lattice models, in parallel to how monodromy matrices of the KZ equation lead to knot invariants. Thus, our solution of the problem leads to a new, geometric approach, to integrable lattice models. There are two other approaches to integrable lattice models, due to Nekrasov and Shatashvili and to Costello, Witten and Yamazaki. I’ll describe joint work with Nikita Nekrasov which explains how string theory unifies the three approaches to integrable lattice models.
Lecture 2: April 10, 2019
Title: “Lesson on Knot Categorification”
Abstract: An old problem is to find a unified approach to the knot categorification problem. The new string theory perspective on the qKZ equation I described in the first talk can be used to derive two geometric approaches to the problem.
The first approach is based on a category of B-type branes on resolutions of slices in affine Grassmannians. The second is based on a category of A-branes in a Landau-Ginzburg theory. The relation between them is two dimensional (equivariant) mirror symmetry. String theory also predicts that a third approach to categorification, based on counting solutions to five dimensional Haydys-Witten equations, is equivalent to the first two.
This talk is mostly based on joint work with Andrei Okounkov.
Information about last year’s Math Science Bott lecture can be found here.
Yip Annual Lecture
1 Oxford Street, Cambridge MA 02138On April 18, 2019 Harvard CMSA hosted the inaugural Yip lecture. The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip. This year’s speaker was Peter Galison (Harvard Physics).
The lecture was held from 4:00-5:00pm in Science Center, Hall A.
Duality String Seminar, Thursdays
The Duality String Seminar is held every Thursday at 4:15pm in Jefferson Lab, 453.
For details, please visit the website.
* The Duality String Seminar is sponsored by the Center of Mathematical Sciences and Applications’ Cheng Yu-Tong Fund, for Research at the Interface of Mathematics and Physics.
Why abstraction is the key to intelligence, and what we’re still missing
Abstract: This talk provides a personal perspective on the way forward towards more human-like and more intelligent artificial systems. Traditionally, symbolic and probabilistic methods have dominated the domains of concept formation, abstraction, and automated reasoning. More recently, deep learning-based approaches have led to significant breakthroughs, including successes in games and combinatorial search tasks. However, the resulting systems are still limited in scope and capabilities — they remain brittle, data-hungry, and their generalization capabilities are limited. We will address a set of questions: why is conceptual abstraction essential for intelligence? What is the nature of abstraction, and its relationship to generalization? What kind of abstraction can deep learning models generate, and where do they fail? What are the methods that are currently successful in generating strong conceptual abstraction? Finally, we will consider how to leverage a hybrid approach to reinforce the strength of different approaches while compensating for their respective weaknesses.
The complexity of matrix multiplication approached via algebraic geometry and representation theory.
Abstract: In 1968 V. Strassen discovered the way we usually multiply matrices is not the most efficient possible, and after considerable work by many authors, it is generally conjectured by computer scientists that as the size of matrices becomes large, it becomes almost as easy to multiply them as it is to add them. I will give a brief history of the problem, explain how this conjecture is naturally understood in the framework of classical algebraic geometry and representation theory, and conclude by describing recent advances using more sophisticated tools from algebraic geometry. For most of the talk, no knowledge of algebraic geometry or representation theory will be needed.
Noga Alon Public Talk, 9-7-17
Noga Alon (Tel Aviv University) will be giving a public talk on September 7, 2017,as part of the program on combinatorics and complexity hosted by the CMSA during AY17-18. The talk will be at 5:00pm in Askwith Hall, 13 Appian Way, Cambridge, MA.
Title: Graph Coloring: Local and Global
Abstract: Graph Coloring is arguably the most popular subject in Discrete Mathematics, and its combinatorial, algorithmic and computational aspects have been studied intensively. The most basic notion in the area, the chromatic number of a graph, is an inherently global property. This is demonstrated by the hardness of computation or approximation of this invariant as well as by the existence of graphs with arbitrarily high chromatic number and no short cycles. The investigation of these graphs had a profound impact on Graph Theory and Combinatorics. It combines combinatorial, probabilistic, algebraic and topological techniques with number theoretic tools. I will describe the rich history of the subject focusing on some recent results.
Jennifer Chayes Public Talk, 11-02-17
Jennifer Chayes (Microsoft Research) will be giving a public talk on November 02, 2017,as part of the program on combinatorics and complexity hosted by the CMSA during AY17-18. The talk will be at 5:00pm in Askwith Hall, 13 Appian Way, Cambridge, MA.
Title: Network Science: From the Online World to Cancer Genomics
Abstract: Everywhere we turn these days, we find that networks can be used to describe relevant interactions. In the high tech world, we see the Internet, the World Wide Web, mobile phone networks, and a variety of online social networks. In economics, we are increasingly experiencing both the positive and negative effects of a global networked economy. In epidemiology, we find disease spreading over our ever growing social networks, complicated by mutation of the disease agents. In biomedical research, we are beginning to understand the structure of gene regulatory networks, with the prospect of using this understanding to manage many human diseases. In this talk, I look quite generally at some of the models we are using to describe these networks, processes we are studying on the networks, algorithms we have devised for the networks, and finally, methods we are developing to indirectly infer network structure from measured data. I’ll discuss in some detail particular applications to cancer genomics, applying network algorithms to suggest possible drug targets for certain kinds of cancer.
CMSA Math-Science Literature Lecture: Indistinguishability Obfuscation: How to Hide Secrets within Software
Amit Sahai (UCLA)
Title: Indistinguishability Obfuscation: How to Hide Secrets within Software
Abstract: At least since the initial public proposal of public-key cryptography based on computational hardness conjectures (Diffie and Hellman, 1976), cryptographers have contemplated the possibility of a “one-way compiler” that translates computer programs into “incomprehensible” but equivalent forms. And yet, the search for such a “one-way compiler” remained elusive for decades.
In this talk, we look back at our community’s attempts to formalize the notion of such a compiler, culminating in our 2001 work with Barak, Goldreich, Impagliazzo, Rudich, Vadhan, and Yang, which proposed the notion of indistinguishability obfuscation (iO). Roughly speaking, iO requires that the compiled versions of any two equivalent programs (with the same size and running time) be indistinguishable to any efficient adversary. Leveraging the notion of punctured programming, introduced in our work with Waters in 2013, well over a hundred papers have explored the remarkable power of iO.
We’ll then discuss the intense effort that recently culminated in our 2020 work with Jain and Lin, finally showing how to construct iO in such a way that, for the first time, we can prove the security of our iO scheme based on well-studied computational hardness conjectures in cryptography.
Talk chair: Sergiy Verstyuk
2018 HMS Focused Lecture Series
As part of their CMSA visitation, HMS focused visitors will be giving lectures on various topics related to Homological Mirror Symmetry throughout the Spring 2018 Semester. The lectures will take place on Tuesdays and Thursdays in the CMSA Building, 20 Garden Street, Room G10.
The schedule will be updated below.
Date Speaker Title/Abstract January 23, 25, 30 and February 1 3-5pm
*Room G10*
Ivan Losev (Northeastern)
Title: BGG category O: towards symplectic duality Abstract: We will discuss a very classical topic in the representation theory of semisimple Lie algebras: the Bernstein-Gelfand-Gelfand (BGG) category O. Our aim will be to motivate and state a celebrated result of Beilinson, Ginzburg and Soergel on the Koszul duality for such categories, explaining how to compute characters of simple modules (the Kazhdan-Lusztig theory) along the way. The Koszul duality admits a conjectural generalization (Symplectic duality) that is a Mathematical manifestation of 3D Mirror symmetry. We will discuss that time permitting.
Approximate (optimistic) plan of the lectures:
1) Preliminaries and BGG category O.
2) Kazhdan-Lusztig bases. Beilinson-Bernstein localization theorem.
3) Localization theorem continued. Soergel modules.
4) Koszul algebras and Koszul duality for categories O.
Time permitting: other instances of Symplectic duality.
Prerequisites:
Semi-simple Lie algebras and their finite dimensional representation theory.
Some Algebraic geometry. No prior knowledge of category O/ Geometric
Representation theory is assumed.
February 27, and March 1
3-5pm
Colin Diemer (IHES)
Title: Moduli spaces of Landau-Ginzburg models and (mostly Fano) HMS. Abstract: Mirror symmetry as a general phenomenon is understood to take place near the large complex structure limit resp. large radius limit, and so implicitly involves degenerations of the spaces under consideration. Underlying most mirror theorems is thus a mirror map which gives a local identification of respective A-model and B-model moduli spaces. When dealing with mirror symmetry for Calabi-Yau’s the role of the mirror map is well-appreciated. In these talks I’ll discuss the role of moduli in mirror symmetry of Fano varieties (where the mirror is a Landau-Ginzburg (LG) model). Some topics I expect to cover are a general structure theory of moduli of LG models (follows Katzarkov, Kontsevich, Pantev), the interplay of the topology of LG models with autoequivalence relations in the Calabi-Yau setting, and the relationship between Mori theory in the B-model and degenerations of the LG A-model. For the latter topic we’ll focus on the case of del Pezzo surfaces (due to unpublished work of Pantev) and the toric case (due to the speaker with Katzarkov and G. Kerr). Time permitting, we may make some speculations on the role of LG moduli in the work of Gross-Hacking-Keel (in progress work of the speaker with T. Foster).
March 6 and 8 4-5pm
Adam Jacob (UC Davis)
Title: The deformed Hermitian-Yang-Mills equation Abstract: In this series I will discuss the deformed Hermitian-Yang-Mills equation, which is a complex analogue of the special Lagrangian graph equation of Harvey-Lawson. I will describe its derivation in relation to the semi-flat setup of SYZ mirror symmetry, followed by some basic properties of solutions. Later I will discuss methods for constructing solutions, and relate the solvability to certain geometric obstructions. Both talks will be widely accessible, and cover joint work with T.C. Collins and S.-T. Yau.
March 6, 8, 13, 15 3-4pm
Dmytro Shklyarov (TU Chemnitz)
Title: On categories of matrix factorizations and their homological invariants Abstract: The talks will cover the following topics:
1. Matrix factorizations as D-branes. According to physicists, the matrix factorizations of an isolated hypersurface singularity describe D-branes in the Landau-Ginzburg (LG) B-model associated with the singularity. The talk is devoted to some mathematical implications of this observation. I will start with a review of open-closed topological field theories underlying the LG B-models and then talk about their refinements.
2. Semi-infinite Hodge theory of dg categories. Homological mirror symmetry asserts that the “classical” mirror correspondence relating the number of rational curves in a CY threefold to period integrals of its mirror should follow from the equivalence of the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. The classical mirror correspondence can be upgraded to an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the assertion would be to try to attach similar Hodge-like data to abstract derived categories. I will talk about some recent results in this direction and illustrate the approach in the context of the LG B-models.
3. Hochschild cohomology of LG orbifolds. The scope of applications of the LG mod- els in mirror symmetry is significantly expanded once we include one extra piece of data, namely, finite symmetry groups of singularities. The resulting models are called orbifold LG models or LG orbifolds. LG orbifolds with abelian symmetry groups appear in mir- ror symmetry as mirror partners of varieties of general type, open varieties, or other LG orbifolds. Associated with singularities with symmetries there are equivariant versions of the matrix factorization categories which, just as their non-equivariant cousins, describe D-branes in the corresponding orbifold LG B-models. The Hochschild cohomology of these categories should then be isomorphic to the closed string algebra of the models. I will talk about an explicit description of the Hochschild cohomology of abelian LG orbifolds.
April 10 & 12 3-4pm
Mauricio Romo (IAS)
Title: Gauged Linear Sigma Models, Supersymmetric Localization and Applications Abstract: In this series of lectures I will review various results on connections between gauged linear sigma models (GLSM) and mathematics. I will start with a brief introduction on the basic concepts about GLSMs, and their connections to quantum geometry of Calabi-Yaus (CY). In the first lecture I will focus on nonperturbative results on GLSMs on closed 2-manifolds, which provide a way to extract enumerative invariants and the elliptic genus of some classes of CYs. In the second lecture I will move to nonperturbative results in the case where the worldsheet is a disk, in this case nonperturbative results provide interesting connections with derived categories and stability conditions. We will review those and provide applications to derived functors and local systems associated with CYs. If time allows we will also review some applications to non-CY cases (in physics terms, anomalous GLSMs).
April 17, 19, 26 3-5pm
Andrew Harder (University of Miami)
Title: Perverse sheaves of categories on surfaces Abstract: Perverse sheaves of categories on a Riemann surface S are systems of categories and functors which are encoded by a graphs on S, and which satisfy conditions that resemble the classical characterization of perverse sheaves on a disc.
I’ll review the basic ideas behind Kapranov and Schechtman’s notion of a perverse schober and generalize this to perverse sheaves of categories on a punctured Riemann surface. Then I will give several examples of perverse sheaves of categories in both algebraic geometry, symplectic geometry, and category theory. Finally, I will describe how one should be able to use related ideas to prove homological mirror symmetry for certain noncommutative deformations of projective 3-space.
May 15, 17 1-3pm
Charles Doran (University of Alberta)
Lecture One:
Title: Picard-Fuchs uniformization and Calabi-Yau geometryAbstract:Part 1: We introduce the notion of the Picard-Fuchs equations annihilating periods in families of varieties, with emphasis on Calabi-Yau manifolds. Specializing to the case of K3 surfaces, we explore general results on “Picard-Fuchs uniformization” of the moduli spaces of lattice-polarized K3 surfaces and the interplay with various algebro-geometric normal forms for these surfaces. As an application, we obtain a universal differential-algebraic characterization of Picard rank jump loci in these moduli spaces.Part 2: We next consider families with one natural complex structure modulus, (e.g., elliptic curves, rank 19 K3 surfaces, b_1=4 Calabi-Yau threefolds, …), where the Picard-Fuchs equations are ODEs. What do the Picard-Fuchs ODEs for such families tell us about the geometry of their total spaces? Using Hodge theory and parabolic cohomology, we relate the monodromy of the Picard-Fuchs ODE to the Hodge numbers of the total space. In particular, we produce criteria for when the total space of a family of rank 19 polarized K3 surfaces can be Calabi-Yau.Lecture Two:
Title: Calabi-Yau fibrations: construction and classification
Abstract:Part 1: Codimension one Calabi-Yau submanifolds induce fibrations, with the periods of the total space relating to those of the fibers and the structure of the fibration. We describe a method of iteratively constructing Calabi-Yau manifolds in tandem with their Picard-Fuchs equations. Applications include the tower of mirrors to degree n+1 hypersurfaces in P^n and a tower of Calabi-Yau hypersurfaces encoding the n-sunset Feynman integrals.
Part 2: We develop the necessary theory to both construct and classify threefolds fibered by lattice polarized K3 surfaces. The resulting theory is a complete generalization to threefolds of that of Kodaira for elliptic surfaces. When the total space of the fibration is a Calabi-Yau threefold, we conjecture a unification of CY/CY mirror symmetry and LG/Fano mirror symmetry by mirroring fibrations as Tyurin degenerations. The detailed classification of Calabi-Yau threefolds with certain rank 19 polarized fibrations provides strong evidence for this conjecture by matching geometric characteristics of the fibrations with features of smooth Fano threefolds of Picard rank 1.
2017 Ding Shum Lecture
1 Oxford Street, Cambridge MA 02138Leslie Valiant will be giving the inaugural talk of the Ding Shum Lectures on Tuesday, October 10 at 5:00 pm in Science Center Hall D, Cambridge, MA.
Learning as a Theory of Everything
Abstract: We start from the hypothesis that all the information that resides in living organisms was initially acquired either through learning by an individual or through evolution. Then any unified theory of evolution and learning should be able to characterize the capabilities that humans and other living organisms can possess or acquire. Characterizing these capabilities would tell us about the nature of humans, and would also inform us about feasible targets for automation. With this purpose we review some background in the mathematical theory of learning. We go on to explain how Darwinian evolution can be formulated as a form of learning. We observe that our current mathematical understanding of learning is incomplete in certain important directions, and conclude by indicating one direction in which further progress would likely enable broader phenomena of intelligence and cognition to be realized than is possible at present.
Constructions in combinatorics via neural networks
Abstract: Recently, significant progress has been made in the area of machine learning algorithms, and they have quickly become some of the most exciting tools in a scientist’s toolbox. In particular, recent advances in the field of reinforcement learning have led computers to reach superhuman level play in Atari games and Go, purely through self-play. In this talk I will give a very basic introduction to neural networks and reinforcement learning algorithms. I will also indicate how these methods can be adapted to the ““game” of trying to find a counterexample to a mathematical conjecture, and show some examples where this approach was successful.
New results in Supergravity via ML Technology
Abstract: The infrastructure built to power the Machine Learning revolution has many other uses beyond Deep Learning. Starting from a general architecture-level overview over the lower levels of Google’s TensorFlow machine learning library, we review how this has recently helped us to find all the stable vacua of SO(8) Supergravity in 3+1 dimensions, has allowed major progress on other related questions about M theory, and briefly discuss other applications in field theory and beyond.
Computer-Aided Mathematics and Satisfiability
Abstract: Progress in satisfiability (SAT) solving has made it possible to determine the correctness of complex systems and answer long-standing open questions in mathematics. The SAT solving approach is completely automatic and can produce clever though potentially gigantic proofs. We can have confidence in the correctness of the answers because highly trustworthy systems can validate the underlying proofs regardless of their size.
We demonstrate the effectiveness of the SAT approach by presenting some recent successes, including the solution of the Boolean Pythagorean Triples problem, computing the fifth Schur number, and resolving the remaining case of Keller’s conjecture. Moreover, we constructed and validated a proof for each of these results. The second part of the talk focuses on notorious math challenges for which automated reasoning may well be suitable. In particular, we discuss our progress on applying SAT solving techniques to the chromatic number of the plane (Hadwiger-Nelson problem), optimal schemes for matrix multiplication, an
Why explain mathematics to computers?
Abstract: A growing number of mathematicians are having fun explaining mathematics to computers using proof assistant softwares. This process is called formalization. In this talk I’ll describe what formalization looks like, what kind of things it teaches us, and how it could even turn out to be useful (in our usual sense of “useful”). This will not be a talk about foundations of mathematics, and I won’t assume any prior knowledge about formalization.
Langlands duality for 3 manifolds
Abstract: Langlands duality began as a deep and still mysterious conjecture in number theory, before branching into a similarly deep and mysterious conjecture of Beilinson and Drinfeld concerning the algebraic geometry of Riemann surfaces. In this guise it was given a physical explanation in the framework of 4-dimensional super symmetric quantum field theory by Kapustin and Witten. However to this day the Hilbert space attached to 3-manifolds, and hence the precise form of Langlands duality for them, remains a mystery.
In this talk I will propose that so-called “skein modules” of 3-manifolds give natural candidates for these Hilbert spaces at generic twisting parameter Psi , and I will explain a Langlands duality in this setting, which we have conjectured with Ben-Zvi, Gunningham and Safronov.
Intriguingly, the precise formulation of such a conjecture in the classical limit Psi=0 is still an open question, beyond the scope of the talk.
Fluid Dynamics Seminar
Beginning immediately, until at least April 30, all seminars will take place virtually, through Zoom. Links to connect can be found in the schedule below once they are created.
In the Spring 2019 Semester, the Center of Mathematical Sciences and Applications will be hosting a seminar on Fluid Dynamics. The seminar will take place on Wednesdays from 3:00-4:00pm in CMSA G10.
Spring 2020:
Date Speaker Title/Abstract 2/25/2020 Keaton Burns, MIT Title: Flexible spectral simulations of low-Mach-number astrophysical fluids Abstract: Fluid dynamical processes are key to understanding the formation and evolution of stars and planets. While the astrophysical community has made exceptional progress in simulating highly compressible flows, models of low-Mach-number stellar and planetary flows typically use simplified equations based on numerical techniques for incompressible fluids. In this talk, we will discuss improved numerical models of three low-Mach-number astrophysical phenomena: tidal instabilities in binary neutron stars, waves and convection in massive stars, and ice-ocean interactions in icy moons. We will cover the basic physics of these systems and how ongoing additions to the open-source Dedalus Project are enabling their efficient simulation in spherical domains with spectral accuracy, implicit timestepping, phase-field methods, and complex equations of state.
3/4/2020 G02
3/11/2020 3/18/2020 3/25/2020 4/1/2020 4/8/2020 G02 4/15/2020 4/22/2020 4/29/2020 G02
5/6/2020 5/13/2020 Fall 2019:
Date Speaker Title/Abstract 9/18/2019 Jiawei Zhuang (Harvard) Title: Simulation of 2-D turbulent advection at extreme accuracy with machine learning and differentiable programming Abstract: The computational cost of fluid simulations grows rapidly with grid resolution. With the recent slow-down of Moore’s Law, it can take many decades for 10x higher resolution grids to become affordable. To break this major barrier in high-performance scientific computing, we used a data-driven approach to learn an optimal numerical solver that can retain high-accuracy at much coarser grids. We applied this method to 2-D turbulent advection and achieved 4x effective resolution than traditional high-order flux-limited advection solvers. The machine learning component is tightly integrated with traditional finite-volume schemes and can be trained via an end-to-end differentiable programming framework. The model can achieve near-peak FLOPs on CPUs and accelerators via convolutional filters.
9/25/2019 Yantao Yang (Peking University) Title: Double diffusive convection and thermohaline staircases Abstract: Double diffusive convection (DDC), i.e. the buoyancy-driven flow with fluid density depending on two scalar components, is omnipresent in many natural and engineering environments. In ocean this is especially true since the seawater density is mainly determined by temperature and salinity. In upper water of both (sub-) tropical and polar oceans, DDC causes the intriguing thermohaline staircases, which consist of alternatively stacked convection layers and sharp interfaces with high gradients of temperature and salinity. In this talk, we will focus on the fingering DDC usually found in (sub-)tropical ocean, where the mean temperature and salinity decrease with depth. We numerically investigate the formation and the transport properties of finger structures and thermohaline staircases. Moreover, we show that multiple states exit for the exactly same global condition, and individual finger layers and finger layers within staircases exhibit very different transport behaviors.
10/2/2019 No talk 10/9/2019 Samuel Rudy (MIT) Title: Data-driven methods for discovery of partial differential equations and forecasting Abstract: A critical challenge in many modern scientific disciplines is deriving governing equations and forecasting models from data where derivation from first principals is intractable. The problem of learning dynamics from data is complicated when data is corrupted by noise, when only partial or indirect knowledge of the state is available, when dynamics exhibit parametric dependencies, or when only small volumes of data are available. In this talk I will discuss several methods for constructing models of dynamical systems from data including sparse identification for partial differential equations with or without parametric dependencies and approximation of dynamical systems governing equations using neural networks. Limitations of each approach and future research directions will also be discussed.
10/16/2019 No talk 10/23/2019 Kimee Moore (Harvard) Title: Using magnetic fields to investigate Jupiter’s fluid interior Abstract: The present-day interior structure of a planet is an important reflection of the formation and subsequent thermal evolution of that planet. However, despite decades of spacecraft missions to a variety of target bodies, the interiors of most planets in our Solar System remain poorly constrained. In this talk, I will discuss how actively generated planetary magnetic fields (dynamos) can provide important insights into the interior properties and evolution of fluid planets. Using Jupiter as a case study, I will present new results from the analysis of in situ spacecraft magnetometer data from the NASA Juno Mission (currently in orbit about Jupiter). The spatial morphology of Jupiter’s magnetic field shows surprising hemispheric asymmetry, which may be linked to the dissolution of Jupiter’s rocky core in liquid metallic hydrogen. I also report the first definitive detection of time-variation (secular variation) in a planetary dynamo beyond Earth. This time-variation can be explained by the advection of Jupiter’s magnetic field by the zonal winds, which places a lower bound on the velocity of Jupiter’s winds at depth. These results provide an important complement to other analysis techniques, as gravitational measurements are currently unable to uniquely distinguish between deep and shallow wind scenarios, and between solid and dilute core scenarios. Future analysis will continue to resolve Jupiter’s interior, providing broader insight into the physics of giant planets, with implications for the formation of our Solar System.
10/30/2019 No Talk 11/6/2019 Federico Toschi (Eindhoven University of Technology) Title: Deep learning and reinforcement learning for turbulence Abstract: This talk tells two stories.
Chapter 1: We investigate the capability of a state-of-the-art deep neural model at learning features of turbulent velocity signals. Deep neural network (DNN) models are at the center of the present machine learning revolution. The set of complex tasks in which they over perform human capabilities and best algorithmic solutions grows at an impressive rate and includes, but it is not limited to, image, video and language analysis, automated control, and even life science modeling. Besides, deep learning is receiving increasing attention in connection to a vast set of problems in physics where quantitatively accurate outcomes are expected. We consider turbulent velocity signals, spanning decades in Reynolds numbers, which have been generated via shell models for the turbulent energy cascade. Given the multi-scale nature of the turbulent signals, we focus on the fundamental question of whether a deep neural network (DNN) is capable of learning, after supervised training with very high statistics, feature extractors to address and distinguish intermittent and multi-scale signals. Can the DNN measure the Reynolds number of the signals? Which feature is the DNN learning?
Chapter 2: Thermally driven turbulent flows are common in nature and in industrial applications. The presence of a (turbulent) flow can greatly enhance the heat transfer with respect to its conductive value. It is therefore extremely important -in fundamental and applied perspective- to understand if and how it is possible to control the heat transfer in thermally driven flows. In this work, we aim at maintaining a Rayleigh–Bénard convection (RBC) cell in its conductive state beyond the critical Rayleigh number for the onset of convection. We specifically consider controls based on local modifications of the boundary temperature (fluctuations). We take advantage of recent developments in Artificial Intelligence and Reinforcement Learning (RL) to find -automatically- efficient non-linear control strategies. We train RL agents via parallel, GPU-based, 2D lattice Boltzmann simulations. Trained RL agents are capable of increasing the critical Rayleigh number of a factor 3 in comparison with state-of-the-art linear control approaches. Moreover, we observe that control agents are able to significantly reduce the convective flow also when the conductive state is unobtainable. This is achieved by finding and inducing complex flow fields.
11/13/2019 2:10pm
G02
Martin Lellep (Philipps University of Marburg, Germany) Title: Predictions of relaminarisation in turbulent shear flows using deep learning Abstract: Given the increasing performance of deep learning algorithms in tasks such as classification during the last years and the vast amount of data that can be generated in turbulence research, I present one application of deep learning to fluid dynamics in this talk. We train a deep learning machine learning model to classify if turbulent shear flow becomes laminar a certain amount of time steps ahead in the future. Prior to this, we use a 2D toy example to develop an understanding how the performance of the deep learning algorithm depends on hyper parameters and how to understand the errors. The performance of both algorithms is high and therefore opens up further steps towards the interpretation of the results in future work.
11/19/2019 Tuesday
3-4 pm
Pierce Hall 209, 29 Oxford Street
Detlef Lohse (University of Twente) Title: Rayleigh vs. Marangoni Abstract: In this talk I will show several examples of an interesting and surprising competition between buoyancy and Marangoni forces. First, I will introduce the audience to the jumping oil droplet – and its sudden death – in a density stratified liquid consisting of water in the bottom and ethanol in the top : After sinking for about a minute, before reaching the equilibrium the droplet suddenly jumps up thanks to the Marangoni forces. This phenomenon repeats about 30-50 times, before the droplet falls dead all the sudden. We explain this phenomenon and explore the phase space where it occurs. Next, I will focus on the evaporation of multicomponent droplets, for which the richness of phenomena keeps surprising us. I will show and explain several of such phenomena, namely evaporation-triggered segregation thanks to either weak solutal Marangoni flow or thanks to gravitational effects. The dominance of the latter implies that sessile droplets and pending droplets show very different evaporation behavior, even for Bond number << 1. I will also explain the full phase diagram in the Marangoni number vs Rayleigh number phase space, and show where Rayleigh convections rolls prevail, where Marangoni convection rolls prevail, and where they compete.
The research work shown in this talks combines experiments, numerical simulations, and theory. It has been done by and in collaboration with Yanshen Li, Yaxing Li, and Christian Diddens, and many others.
11/20/2019 Time: 3:00-3:35 pm Speaker: Haoran Liu
Title: Applications of Phase Field method: drop impact and multiphase turbulence
Abstract: Will a mosquito survive raindrop collisions? How the bubbles under a ship reduce the drag force? In nature and industry, flows with drops and bubbles exist everywhere. To understand these flows, one of the powerful tools is the direct numerical simulation (DNS). Among all the DNS methods, we choose the Phase Field (PF) method and develop some models based on it to simulate the complicated flows, such as flows with moving contact lines, fluid-structure interaction, ternary fluids and turbulence. In this talk, I will firstly introduce the advantages and disadvantages of PF method. Then, I will show its applications: drop impact on an object, compound droplet dynamics, water entry of an object and multiphase turbulence.
Time: 3:35-4:10 pm
Speaker: Steven Chong
Abstract: For Rayleigh-Bénard under geometrical confinement, under rotation or the double diffusive convection with the second scalar component stabilizing the convective flow, they seem to be the three different canonical models in turbulent flow. However, previous research coincidentally reported the scalar transport enhancement in these systems. The results are counter-intuitive because the higher efficiency of scalar transport is bought about by the slower flow. In this talk, I will show you a fundamental and unified perspective on such the global transport behavior observed in the seemingly different systems. We further show that the same view can be applied to the quasi-static magnetoconvection, and indeed the regime with heat transport enhancement has been found. The beauty of physics is to understand the seemingly unrelated phenomena by a simplified concept. Here we provide a simplified and generic view, and this concept could be potentially extended to other situations where the turbulent flow is subjected to an additional stabilization.
11/27/2019 12/4/2019 12/11/2019 See previous seminar information here.
Topological Aspects of Condensed Matter Seminar
As part of the Program on Topological Aspects of Condensed Matter, a weekly seminar will be held on Mondays from 10:00-11:30pm in CMSA room G10.
Date Speaker Title/Abstract 8/29/2018 Zeng-Cheng Gu Title: Towards a complete classification of symmetry protected topological phases for interacting fermions in three dimensions and a general group supercohomology theory Abstract: Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this talk, I will revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. I will show how to construct very general fixed point SPT wavefunctions for interacting fermion systems. I will also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions.
9/10/2018 Dominic Else, MIT Title: Phases and topology in periodically driven (Floquet) systems Abstract: I will give a pedagogical overview of new topological phenomena that occur in systems that are driven periodically in time (Floquet systems). As a warm-up, I will review new topological invariants in free-fermion Floquet systems. Then, I will discuss the richer physics that occurs in interacting Floquet phases, stabilized in systems with strong quenched disorder by many-body-localization (MBL). Finally, time permitting, I will explain how to realize interacting topological phenomena in a metastable (“pre-thermal”) regime of a clean system.
9/17/2018 Adrian Po, MIT Title: A modern solution to the old problem of symmetries in band theory Abstract: There are 230 space groups and 1,651 magnetic space groups in three dimensions. Thankfully, these are finite numbers, and one might go about solving all the possible ways free electrons represent them. This is a central question in the nine-decade-old band theory, which is long-thought to be solvable if only one had the time and patience to crank through all the cases. In this talk, I would describe how this problem can be solved efficiently from the modern perspective of band topology. As a by-product, we will describe a simple method to detect topologically nontrivial band insulators using only symmetry eigenvalues, which offers great computational advantage compared to the traditional, wave-function-based definitions of topological band invariants.
9/24/2018 Maxim Metlitski Title: Surface Topological Order and a new ‘t Hooft Anomaly of Interaction Enabled 3+1D Fermion SPTs Abstract: Symmetry protected topological (SPT) phases have attracted a lot of attention in recent years. A key property of SPTs is the presence of non-trivial surface states. While for 1+1D and 2+1D SPTs the boundary must be either symmetry broken or gapless, some 3+1D SPTs admit symmetric gapped surface states that support anyon excitation (intrinsic topological order). In all cases, the boundary of an SPT is anomalous – it cannot be recreated without the bulk; furthermore, the anomaly must “match” the bulk. I will review this bulk-boundary correspondence for 3d SPT phases of bosons with topologically ordered boundaries where it is fairly well understood. I will then proceed to describe recent advances in the understanding of strongly interacting 3+1D SPT phases of fermions and their topologically ordered surface states.
10/1/2018 Cancelled 10/9/2018 Tuesday
3:00-4:30pm
Sagar Vijay Title: Fracton Phases of Matter Abstract: Fracton phases are new kinds of highly-entangled quantum matter in three spatial dimensions that are characterized by gapped, point-like excitations (“fractons”) that are strictly immobile at zero temperature, and by degenerate ground-states that are locally indistinguishable. Fracton excitations provide an alternative to Fermi or Bose statistics in three spatial dimensions, and these states of matter are a gateway for exploring mechanisms for quantum information storage, and for studying “slow” dynamical behavior in the absence of disorder. I will review exactly solvable models for these phases, constructions of these states using well-studied two-dimensional topological phases, and a model in which the fracton excitations carry a protected internal degeneracy, which provides a natural generalization of non-Abelian anyons to three spatial dimensions. I will then describe recent advances in categorizing these states of matter using finite-depth unitary transformations.
10/15/2018 Ethan Lake Title: A primer on higher symmetries Abstract: The notion of a higher symmetry, namely a symmetry whose charged objects have a dimension greater than zero, is proving to be very useful for organizing our understanding of gauge theories and topological phases of matter. Just like regular symmetries, higher symmetries can be gauged, spontaneously broken, and can have anomalies. I will review these aspects of higher symmetries and motivate why beyond their conceptual utility, they are often an indispensable tool for making statements about dualities and phase diagrams of theories with gauge fields.
10/22/2018 Room G02
Yin-Chen He, Perimeter Title: Emergent QED3 and QCD3 in condensed matter system Abstract: QED3-Chern-Simons and QCD3-Chern-Simons theories are interesting critical theories in the 2+1 dimension. These theories are described by gapless Dirac fermions interacting with dynamical gauge fields (U(1), SU(N), U(N), etc.) with a possible Chern-Simon term. These theories have fundamental importance as it will flow to the 3D conformal field theories and have interesting dualities in the infrared. Various of condensed matter system are described by these critical theories. I will introduce several examples including the Dirac spin liquid in the frustrated magnets (kagome, triangular lattice), quantum phase transitions in the fractional quantum Hall systems and Kitaev materials.
10/29/2018 Dominic Williamson, Yale Title: Symmetry and topological order in tensor networks Abstract: I will present an overview of how topological states of matter with global symmetries can be described using tensor networks. First reviewing the classification of 1D symmetry-protected topological phases with matrix product states, before moving on to the description of 2D symmetry-enriched topological phases with projected-entangled pair states.
11/13/2018 Tuesday
3:00-4:30pm
Jason Alicea, Caltech Title: Time-crystalline topological superconductors 11/19/2018 X. G. Wen, MIT Title: A classification of 3+1D topological orders Abstract: I will discuss a classification of 3+1D topological orders in terms of fusion 2 category. The 3+1D topological orders can be divided into two classes: the ones without emergent fermions and the ones with emergent fermions. The 3+1D topological orders with emergent fermions can be further divided into two classes: the ones without emergent Majorana zero mode and the ones with emergent Majorana zero mode. I will present pictures to understand those 3+1D topological orders.
12/3/2018 *Room G02*
Claudio Chamon, Boston University Title: Many-body scar states with topological properties in 1D, 2D, and 3D. Abstract: We construct (some) exact excited states of a class of non-integrable quantum many-body Hamiltonians in 1D, 2D and 3D. These high energy many-body “scar” states have area law entanglement entropy, and display properties usually associated to gapped ground states of symmetry protected topological phases or topologically ordered phases of matter, including topological degeneracies.
12/10/2018 Room G02
Anders Sandvik, Boston University and Institute of Physics, CAS, Beijing Title: Quantum Monte Carlo simulations of exotic states in 2D quantum magnets Abstract: Some exotic ground states of 2D quantum magnets can be accessed through sign-free quantum Monte Carlo simulations of certain “designer Hamiltonians”. I will discuss recent examples within the J-Q family of models, where the standard Heisenberg exchange J on the square lattice is supplemented by multi-spin terms Q projecting correlated singlets, such that dimer (columnar valence-bond) order is favored. In addition to a possible deconfined quantum critical point separating the Neel and dimer phases, I will discuss recent work on a modified model where a rather strongly first-order transition between the Neel state and a plaquette-singlet-solid is associated with emergent O(4) symmetry up to length scales of at least 100 lattice spacings. This type of transition may be realized in SrCu2(BO3)2 under pressure. I will also discuss a random-singlet state obtained when randomness is introduced in a system with dimerized ground state. This type of state may be realized in some frustrated disordered quantum magnets.
1/8/2019 Lukasz Fidkowski, Univ. of Washington Title: Non-trivial quantum cellular automata in 3 dimensions Abstract: Motivated by studying the entanglement structure of certain symmetry protected topological phases, we construct a non-trivial quantum cellular automaton in a Hilbert space for a 3d lattice of spin 1/2 degrees of freedom. This is an operator which takes local operators to nearby local operators, but is not locally generated. We discuss implications for the classification of SPT phases in equilibrium and Floquet settings.
3/18/2019 Ari Turner, Technion Title: Trapping Excitations at Phantasmagoric Wave Vectors Abstract: This talk will explain some properties of the fracton state devised by Jeongwan Haah. A fracton state has excitations that are extremely localized–it is impossible for them to move (unlike Anderson localization, e.g.–Anderson localized excitations can move if there is an external field to provide energy). One can understand why in a simple way using “mod 2” Fourier analysis. I will explain this, and also introduce “finite fields”, which are the number systems one needs to define exponentials mod. 2.
4/1/2019 Yi-Zhuang You (UCSD) Title: Emergent Symmetry and Conserved Currents at Deconfined Quantum Critical Points Abstract: Noether’s theorem is one of the fundamental laws of physics, relating continuous symmetries and conserved currents. Here we explore the role of Noether’s theorem at the deconfined quantum critical point (DQCP), which is an exotic quantum phase transition beyond the Landau-Ginzburg-Wilson paradigm. It was expected that a larger continuous symmetry could emerge at the DQCP, which, if true, should lead to conserved current at low energy. By identifying the emergent current fluctuation in the spin excitation spectrum, we can quantitatively study the current-current correlation in large-scale quantum Monte Carlo simulations. Our results reveal the conservation of the emergent current, as signified by the vanishing anomalous dimension of the current operator, and hence provide supporting evidence for the emergent symmetry at the DQCP. We also extend our discussion of emergent conserved current to the recently proposed one-dimensional analog of DQCP and confirm the emergent O(2)xO(2) symmetry in that case. Finally, I will briefly discuss the significance of our findings in a potential realization of DQCP in the Shastry-Sutherland lattice material SrCu2(BO3)2.
4/8/2019 Adam Nahum (Oxford) Title: Emergent statistical mechanics of entanglement in random unitary circuits Abstract: I will talk about quantum-classical mappings for real-time observables in some simple many-body systems (random unitary circuits). Specifically I will discuss how (1) entanglement entropy growth and (2) two-point correlation functions in these systems can be related to partition functions for interacting random walks. If time permits I will mention a phase transition in the entanglement structure of a repeatedly measured quantum state.
4/16/2019 Lyman 425
1:30pm
Xie Chen (Calthech) Title: Foliated Fracton Order Abstract: The quantum information study of quantum codes and quantum memory has led to the discovery of a new class of exactly solvable lattice models called the fracton models. The fracton models are similar to the better understood topological models in that they also support fractional excitations and have stable ground state degeneracy. But it is also clear that the fracton models exist beyond the realm of conventional topological order due to their extensive ground state degeneracy and the restricted motion of their fractional excitations. In this talk, I will present a new framework, which we call the “foliated fracton order”, to capture the nontrivial nature of the order in a large class of fracton models. Such a framework not only clarifies the connection between various different models, but also points to the direction of search for interesting new features.
4/24/2019 10:30am
Michael Freedman (Microsoft Station Q) Title: Quantum cellular automata in higher dimensions Abstract: I’ll discuss Joint work with Matt Hastings on local endomorphisms of the operator algebra. We found these have a cohomological invariant similar to that of an incompressible flow.
4/26/2019 10:30am
Maissam Barkeshli (University of Maryland) Title: Relative anomalies in (2+1)D symmetry enriched topological states Abstract: It has recently been understood that some patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. In this talk I will explain some recent advances in our understanding of how to compute relative anomalies between different symmetry fractionalization classes in (2+1)D topological states. The theory applies to general types of symmetries, including symmetries that permute anyon types and space-time reflection symmetries. This allows us to compute anomalies for more general types of space-time reflection symmetries than previously known methods.
5/3/2019 Yuan-Ming Lu (Ohio State) Title: Spontaneous symmetry breaking from anyon condensation Abstract: In the context of quantum spin liquids, it is long known that the condensation of fractionalized excitations can inevitably break certain physical symmetries. For example, condensing spinons will usually break spin rotation and time reversal symmetries. We generalize these phenomena to the context of a generic continuous quantum phase transition between symmetry enriched topological orders, driven by anyon condensation. We provide two rules to determine whether a symmetry is enforced to break across an anyon condensation transition or not. Using a dimensional reduction scheme, we establish a mapping between these symmetry-breaking anyon-condensation transitions in two spatial dimensions, and deconfined quantum criticality in one spatial dimension.
5/9/2019 10:30am
Michael Zaletel (UC Berkeley) Title: Three-partite entanglement in CFTs and chiral topological orders Abstract: While the entanglement entropy provides an essentially complete description of two-partite entanglement, multi-partite entanglement is far richer, with a concomitant zoo of possible measures. This talk will focus on applications of one such measure, the “entanglement of purification,” in many-body systems. I will first present a holographic prescription for calculating it which we can compare with numerical calculations. Interestingly, we find that a 1+1D CFT on a ring contains a universal number of GHZ states for any tri-partition of the ring. Using this result I’ll conjecture a bulk entanglement diagnostic for 2+1D chiral orders, and solicit the audience’s help in proving or disproving it.
5/28/2019 10:30am
Masaki Oshikawa (U Tokyo) Title: Gauge invariance, polarization, and conductivity Abstract: The large gauge transformation on a quantum many-body system under a periodic boundary condition has had numerous applications including generalizations of Lieb-Schultz-Mattis theorem. It is also deeply related to the electric polarization in insulators. I will discuss an application to a scaling of the fluctuation of the polarization in conductors, and also to general constraints on the electric conductivity.
7/18/2019 Eslam Khalaf (Harvard) Title: Dynamical correlations in anomalous disordered wires
Abstract: In a (multichannel) disordered wire, classical diffusion at short times (large frequencies) gives way to Anderson localization at long times (small frequencies). I study what happens in a disordered wire with topologically protected channels, e.g. a wire with unequal number of left and right movers which is realizable at the edge of a Quantum Hall system. In this case, the classical dynamics are described by diffusion + drift, but it is unclear what the effect of quantum corrections in the long time (small frequency) limit is.The problem is described by a 0+1-dimensional supersymmetric (graded) non-linear sigma model with a topological WZW term and a scalar potential. The computation of the local dynamical correlations of this model is equivalent to finding the ground state (zero mode) of the Laplace-Beltrami operator on a symmetric superspace with specific scalar and vector potentials. Surprisingly, I find that this zero mode has a relatively simple explicit integral representation in the Wigner-Dyson symmetry classes which has no counterpart in the absence of supersymmetry. This leads to an exact mapping between the local correlation functions in this 0+1D theory and observables in a 0+0D chiral random matrix problem.The mapping is used to explicitly compute two simple dynamical observables: the diffusion probability of return and the correlation of local density of states. In the former, we find that the interference effects change the exponential decay expected from drift-diffusion to a power law decay. In the latter, we find that the local density of states exhibits statistical level attraction in contrast to the level repulsion expected in a a standard Anderson insulator. At the end, I discuss possible relationship to the recently developed framework of non-Hermitian topological systems.Spacetime and Quantum Mechanics Seminar
As part of the program on Spacetime and Quantum Mechanics, the CMSA will be hosting a weekly seminar on Thursdays at 2:30pm in room G10.
Date Speaker Title/Abstract 9/12/2019 Pasha Pylyavskyy (University of Minnesota) Title: Vector-relation configurations and plabic graphs 19/18/2019 2:00pm
G02
Francis Brown (University of Oxford) Title: Amplitudes, Polylogs and Moduli Spaces 9/19/2019 Chuck Doran (University of Alberta) Title: Calabi-Yau geometry of the N-loop sunset Feynman integrals Abstract: I will present an overview of the algebraic and transcendental features of the computation of N-loop sunset Feynman integrals.
Starting from the realization of arbitrary Feynman graph hypersurfaces as (generalized) determinantal varieties, we describe the Calabi-Yau subvarieties of permutohedral varieties that arise from the N-loop sunset Feynman graphs and some key features of their geometry and moduli.
These include: (1) an iterated fibration structure which allows one to “bootstrap” both periods and Picard-Fuchs equations from lower N cases; (2) specialization to and interpretation of coincident mass loci (“jump loci”) in moduli; (3) a significant generalization of the Griffiths-Dwork algorithm via “creative telescoping”; and (4) the realization of Calabi-Yau pencils as Landau-Ginzburg models mirror to weak Fano varieties.
Details of each of these will be discussed in later lectures this semester. This is joint work with Pierre Vanhove and Andrey Novoseltsev.
9/26/2019 Tomasz Taylor (Northeastern) Title: Celestial Amplitudes 10/3/2019 Simon Caron-Huot (McGill) Title: Poincare Duals of Feynman Integrals 10/10/2019 3:30pm
Yutin Huang (National Taiwan University) Title: Dualities of Planar Ising Networks and the Positive Orthogonal Grassmannian 10/15/2019 Tuesday
3:30pm
Sergey Fomin (Univ. of Michigan)
Title: “Morsifications and mutations” (joint work with P. Pylyavskyy, E. Shustin, and D. Thurston). 10/18/2019 Friday
G02
Sebastian Franco (The City College of New York) Title: Graded quivers, generalized dimer models, and topic geometry 10/31/2019 Junjie Rao (Albert Einstein Institute) Title: All-loop Mondrian Reduction of 4-particle Amplituhedron at Positive Infinity 11/1/2019 SC 232
1:30pm
George Lusztig (MIT) Title: Total positivity in Springer fibres 11/12/2019 Tuesday
G02
3:30pm
Pierpaolo Mastrolia (University of Padova)
Title: Feynman Integrals and Intersection Theory 11/14/2019 G02
Pierpaolo Mastrolia (University of Padova) Title: Feynman Integrals and Intersection Theory Pt. II 11/21/2019 Cristian Vergu (Niels Bohr Institute) Title: The Octagonal Alphabet 11/26/2019 Stephan Stieberger (IAS) Title: Strings on the Celestial Sphere 12/4/2019 Hadleigh Frost (Oxford) Title: BCJ numerators, $\mathcal{M}_{0,n}$, and ABHY Abstract: We relate the BCJ numerator Jacobi property to the classical fact that the top homology group of $\mathcal{M}_{0,n}$ is isomorphic to a component of the free Lie algebra. We describe ways to get BCJ numerators, and caution that the BCJ Jacobi property doesn’t imply the existence of what has been called a ‘kinematic algebra.’
12/5/2019 David Kosower (IAS) Title: From scattering amplitudes to classical observables 12/10/2019 Ramis Movassagh (MIT) Title: Highly entangled quantum spin chains: Exactly solvable counter-examples to the area law Abstract: In recent years, there has been a surge of activities in proposing “exactly solvable” quantum spin chains with surprising high amount of ground state entanglement–exponentially more than the critical systems that have $\log(n)$ von Neumann entropy. We discuss these models from first principles. For a spin chain of length $n$, we prove that the ground state entanglement entropy scales as $\sqrt(n)$ and in some cases even extensive (i.e., as $n$) despite the underlying Hamiltonian being: (1) Local (2) Having a unique ground state and (3) Translationally invariant in the bulk. These models have rich connections with combinatorics, random walks, Markov chains, and universality of Brownian excursions. Lastly, we develop techniques for proving the gap. As a consequence, the gap of Motzkin and Fredkin spin chains are proved to vanish as 1/n^c with c>2; this rules out the possibility of these models to be relativistic conformal field theories in the continuum limit. Time permitting we will discuss more recent developments in this direction and ‘generic’ aspects of local spin chains.
Current Developments In Mathematics 2018
Current Developments in Mathematics 2018 Conference.
Friday, Nov. 16, 2018 2:15 pm – 6:00 pm
Saturday, Nov. 17, 2018 9:00 am – 5:00 pm
Harvard University Science Center, Hall B
Visit the conference page here
Workshop on Additive Combinatorics, Oct. 2-6, 2017
The workshop on additive combinatorics will take place October 2-6, 2017 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
Additive combinatorics is a mathematical area bordering on number theory, discrete mathematics, harmonic analysis and ergodic theory. It has achieved a number of successes in pure mathematics in the last two decades in quite diverse directions, such as:
- The first sensible bounds for Szemerédi’s theorem on progressions (Gowers);
- Linear patterns in the primes (Green, Tao, Ziegler);
- Construction of expanding sets in groups and expander graphs (Bourgain, Gamburd);
- The Kakeya Problem in Euclidean harmonic analysis (Bourgain, Katz, Tao).
Ideas and techniques from additive combinatorics have also had an impact in theoretical computer science, for example
- Constructions of pseudorandom objects (eg. extractors and expanders);
- Constructions of extremal objects (eg. BCH codes);
- Property testing (eg. testing linearity);
- Algebraic algorithms (eg. matrix multiplication).
The main focus of this workshop will be to bring together researchers involved in additive combinatorics, with a particular inclination towards the links with theoretical computer science. Thus it is expected that a major focus will be additive combinatorics on the boolean cube (Z/2Z)^n , which is the object where the exchange of ideas between pure additive combinatorics and theoretical computer science is most fruitful. Another major focus will be the study of pseudorandom phenomena in additive combinatorics, which has been an important contributor to modern methods of generating provably good randomness through deterministic methods. Other likely topics of discussion include the status of major open problems (the polynomial Freiman-Ruzsa conjecture, inverse theorems for the Gowers norms with bounds, explicit correlation bounds against low degree polynomials) as well as the impact of new methods such as the introduction of algebraic techniques by Croot–Pach–Lev and Ellenberg–Gijswijt.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Confirmed participants include:
- Arnab Bhattacharyya (Indian Institute of Science)
- Thomas Bloom (University of Bristol)
- Jop Briët (Centrum Wiskunde & Informatica, Amsterdam)
- Mei-Chu Chang (University of California, Riverside)
- Noam Elkies (Harvard University)
- Asaf Ferber (MIT)
- Jacob Fox (Stanford University)
- Shafi Goldwasser (MIT)
- Elena Grigorescu (Purdue University)
- Hamed Hatami (McGill University)
- Pooya Hatami (Institute for Advanced Study)
- Kaave Hosseini (University of California, San Diego)
- Guy Kindler (Hebrew University of Jerusalem)
- Vsevolod Lev (University of Haifa at Oranim)
- Sean Prendiville (University of Manchester)
- Ronitt Rubinfeld (MIT)
- Will Sawin (ETH Zürich)
- Fernando Shao (Oxford University)
- Olof Sisask (KTH Royal Institute of Technology)
- Madhur Tulsiani (University of Chicago)
- Julia Wolf (University of Bristol)
- Emanuele Viola (Northeastern University)
- Yufei Zhao (MIT)
Co-organizers of this workshop include Ben Green, Swastik Kopparty, Ryan O’Donnell, Tamar Ziegler.
Monday, October 2
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Jacob Fox Tower-type bounds for Roth’s theorem with popular differences Abstract: A famous theorem of Roth states that for any $\alpha > 0$ and $n$ sufficiently large in terms of $\alpha$, any subset of $\{1, \dots, n\}$ with density $\alpha$ contains a 3-term arithmetic progression. Green developed an arithmetic regularity lemma and used it to prove that not only is there one arithmetic progression, but in fact there is some integer $d > 0$ for which the density of 3-term arithmetic progressions with common difference $d$ is at least roughly what is expected in a random set with density $\alpha$. That is, for every $\epsilon > 0$, there is some $n(\epsilon)$ such that for all $n > n(\epsilon)$ and any subset $A$ of $\{1, \dots, n\}$ with density $\alpha$, there is some integer $d > 0$ for which the number of 3-term arithmetic progressions in $A$ with common difference $d$ is at least $(\alpha^3-\epsilon)n$. We prove that $n(\epsilon)$ grows as an exponential tower of 2’s of height on the order of $\log(1/\epsilon)$. We show that the same is true in any abelian group of odd order $n$. These results are the first applications of regularity lemmas for which the tower-type bounds are shown to be necessary.
The first part of the talk by Jacob Fox includes an overview and discusses the upper bound. The second part of the talk by Yufei Zhao focuses on the lower bound construction and proof. These results are all joint work with Huy Tuan Pham.
10:20-11:00am Coffee Break 11:00-11:50am Yufei Zhao Tower-type bounds for Roth’s theorem with popular differences Abstract: Continuation of first talk by Jacob Fox. The first part of the talk by Jacob Fox includes an overview and discusses the upper bound. The second part of the talk by Yufei Zhao focuses on the lower bound construction and proof. These results are all joint work with Huy Tuan Pham.
12:00-1:30pm Lunch 1:30-2:20pm Jop Briët Locally decodable codes and arithmetic progressions in random settings Abstract: This talk is about a common feature of special types of error correcting codes, so-called locally decodable codes (LDCs), and two problems on arithmetic progressions in random settings, random differences in Szemerédi’s theorem and upper tails for arithmetic progressions in a random set in particular. It turns out that all three can be studied in terms of the Gaussian width of a set of vectors given by a collection of certain polynomials. Using a matrix version of the Khintchine inequality and a lemma that turns such polynomials into matrices, we give an alternative proof for the best-known lower bounds on LDCs and improved versions of prior results due to Frantzikinakis et al. and Bhattacharya et al. on arithmetic progressions in the aforementioned random settings.
Joint work with Sivakanth Gopi
2:20-3:00pm Coffee Break 3:00-3:50pm Fernando Shao Large deviations for arithmetic progressions
Abstract: We determine the asymptotics of the log-probability that the number of k-term arithmetic progressions in a random subset of integers exceeds its expectation by a constant factor. This is the arithmetic analog of subgraph counts in a random graph. I will highlight some open problems in additive combinatorics that we encountered in our work, namely concerning the “complexity” of the dual functions of AP-counts.
4:00-6:00pm Welcome Reception Tuesday, October 3
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Emanuele Viola Interleaved group products Authors: Timothy Gowers and Emanuele Viola
Abstract: Let G be the special linear group SL(2,q). We show that if (a1,a2) and (b1,b2) are sampled uniformly from large subsets A and B of G^2 then their interleaved product a1 b1 a2 b2 is nearly uniform over G. This extends a result of Gowers (2008) which corresponds to the independent case where A and B are product sets. We obtain a number of other results. For example, we show that if X is a probability distribution on G^m such that any two coordinates are uniform in G^2, then a pointwise product of s independent copies of X is nearly uniform in G^m, where s depends on m only. Similar statements can be made for other groups as well.
These results have applications in computer science, which is the area where they were first sought by Miles and Viola (2013).
10:20-11:00am Coffee Break 11:00-11:50am Vsevolod Lev On Isoperimetric Stability Abstract: We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-c)|S||A|$ with a real $c\in(0,1]$, then $|A|\ge4^{(1-1/d)c|S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible.
As a corollary, we derive an upper bound for the size of the largest independent subset of the set of popular differences of a finite subset of an abelian group. For groups of exponent $2$ and $3$, our bound translates into a sharp estimate for the additive dimension of the popular difference set.
We also prove, as an auxiliary result, the following estimate of possible independent interest: if $A\subseteq{\mathbb Z}^n$ is a finite, non-empty downset, then, denoting by $w(z)$ the number of non-zero components of the vector $z\in\mathbb{Z}^n$, we have $$ \frac1{|A|} \sum_{a\in A} w(a) \le \frac12\, \log_2 |A|. $$
12:00-1:30pm Lunch 1:30-2:20pm Elena Grigorescu NP-Hardness of Reed-Solomon Decoding and the Prouhet-Tarry-Escott Problem Abstract: I will discuss the complexity of decoding Reed-Solomon codes, and some results establishing NP-hardness for asymptotically smaller decoding radii than the maximum likelihood decoding radius. These results follow from the study of a generalization of the classical Subset Sum problem to higher moments, which may be of independent interest. I will further discuss a connection with the Prouhet-Tarry-Escott problem studied in Number Theory, which turns out to capture a main barrier in extending our techniques to smaller radii.
Joint work with Venkata Gandikota and Badih Ghazi.
2:20-3:00pm Coffee Break 3:00-3:50pm Sean Prendiville Partition regularity of certain non-linear Diophantine equations. Abstract: We survey some results in additive Ramsey theory which remain valid when variables are restricted to sparse sets of arithmetic interest, in particular the partition regularity of a class of non-linear Diophantine equations in many variables.
Wednesday, October 4
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Olof Sisask Bounds on capsets via properties of spectra Abstract: A capset in F_3^n is a subset A containing no three distinct elements x, y, z satisfying x+z=2y. Determining how large capsets can be has been a longstanding problem in additive combinatorics, particularly motivated by the corresponding question for subsets of {1,2,…,N}. While the problem in the former setting has seen spectacular progress recently through the polynomial method of Croot–Lev–Pach and Ellenberg–Gijswijt, such progress has not been forthcoming in the setting of the integers. Motivated by an attempt to make progress in this setting, we shall revisit the approach to bounding the sizes of capsets using Fourier analysis, and in particular the properties of large spectra. This will be a two part talk, in which many of the ideas will be outlined in the first talk, modulo the proof of a structural result for sets with large additive energy. This structural result will be discussed in the second talk, by Thomas Bloom, together with ideas on how one might hope to achieve Behrend-style bounds using this method.
Joint work with Thomas Bloom.
10:20-11:00am Coffee Break 11:00-11:50am Thomas Bloom Bounds on capsets via properties of spectra This is a continuation of the previous talk by Olof Sisask.
12:00-1:30pm Lunch 1:30-2:20pm Hamed Hatami Polynomial method and graph bootstrap percolation Abstract: We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to determine the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel, and the latter provides an alternative and simpler proof for one of their main results. This is based on a joint work with Lianna Hambardzumyan and Yingjie Qian.
2:20-3:00pm Coffee Break 3:00-3:50pm Arnab Bhattacharyya Algorithmic Polynomial Decomposition Abstract: Fix a prime p. Given a positive integer k, a vector of positive integers D = (D_1, …, D_k) and a function G: F_p^k → F_p, we say a function P: F_p^n → F_p admits a (k, D, G)-decomposition if there exist polynomials P_1, …, P_k: F_p^n -> F_p with each deg(P_i) <= D_i such that for all x in F_p^n, P(x) = G(P_1(x), …, P_k(x)). For instance, an n-variate polynomial of total degree d factors nontrivially exactly when it has a (2, (d-1, d-1), prod)-decomposition where prod(a,b) = ab.
When show that for any fixed k, D, G, and fixed bound d, we can decide whether a given polynomial P(x_1, …, x_n) of degree d admits a (k,D,G)-decomposition and if so, find a witnessing decomposition, in poly(n) time. Our approach is based on higher-order Fourier analysis. We will also discuss improved analyses and algorithms for special classes of decompositions.
Joint work with Pooya Hatami, Chetan Gupta and Madhur Tulsiani.
Thursday, October 5
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Madhur Tulsiani Higher-order Fourier analysis and approximate decoding of Reed-Muller codes Abstract: Decomposition theorems proved by Gowers and Wolf provide an appropriate notion of “Fourier transform” for higher-order Fourier analysis. I will discuss some questions and techniques that arise from trying to develop polynomial time algorithms for computing these decompositions.
I will discuss constructive proofs of these decompositions based on boosting, which reduce the problem of computing these decompositions to a certain kind of approximate decoding problem for codes. I will also discuss some earlier and recent works on this decoding problem.
Based on joint works with Arnab Bhattacharyya, Eli Ben-Sasson, Pooya Hatami, Noga Ron-Zewi and Julia Wolf.
10:20-11:00am Coffee Break 11:00-11:50am Julia Wolf Stable arithmetic regularity The arithmetic regularity lemma in the finite-field model, proved by Green in 2005, states that given a subset A of a finite-dimensional vector space over a prime field, there exists a subspace H of bounded codimension such that A is Fourier-uniform with respect to almost all cosets of H. It is known that in general, the growth of the codimension of H is required to be of tower type depending on the degree of uniformity, and that one must allow for a small number of non-uniform cosets.
Our main result is that, under a natural model-theoretic assumption of stability, the tower-type bound and non-uniform cosets in the arithmetic regularity lemma are not necessary. Specifically, we prove an arithmetic regularity lemma for k-stable subsets in which the bound on the codimension of the subspace is a polynomial (depending on k) in the degree of uniformity, and in which there are no non-uniform cosets.
This is joint work with Caroline Terry.
12:00-1:30pm Lunch 1:30-2:20pm Will Sawin Constructions of Additive Matchings
Abstract: I will explain my work, with Robert Kleinberg and David Speyer, constructing large tri-colored sum-free sets in vector spaces over finite fields, and how it shows that some additive combinatorics problems over finite fields are harder than corresponding problems over the integers.
2:20-3:00pm Coffee Break 3:00-3:50pm Mei-Chu Chang Arithmetic progressions in multiplicative groups of finite fields Abstract: Let G be a multiplicative subgroup of the prime field F_p of size |G|> p^{1-\kappa} and r an arbitrarily fixed positive integer. Assuming \kappa=\kappa(r)>0 and p large enough, it is shown that any proportional subset A of G contains non-trivial arithmetic progressions of length r.
Friday, October 6
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Asaf Ferber On a resilience version of the Littlewood-Offord problem Abstract: In this talk we consider a resilience version of the classical Littlewood-Offord problem. That is, consider the sum X=a_1x_1+…a_nx_n, where the a_i-s are non-zero reals and x_i-s are i.i.d. random variables with (x_1=1)= P(x_1=-1)=1/2. Motivated by some problems from random matrices, we consider the question: how many of the x_i-s can we typically allow an adversary to change without making X=0? We solve this problem up to a constant factor and present a few interesting open problems.
Joint with: Afonso Bandeira (NYU) and Matthew Kwan (ETH, Zurich).
10:20-11:00am Coffee Break 11:00-11:50am Kaave Hosseini Protocols for XOR functions and Entropy decrement Abstract: Let f:F_2^n –> {0,1} be a function and suppose the matrix M defined by M(x,y) = f(x+y) is partitioned into k monochromatic rectangles. We show that F_2^n can be partitioned into affine subspaces of co-dimension polylog(k) such that f is constant on each subspace. In other words, up to polynomial factors, deterministic communication complexity and parity decision tree complexity are equivalent.
This relies on a novel technique of entropy decrement combined with Sanders’ Bogolyubov-Ruzsa lemma.
Joint work with Hamed Hatami and Shachar Lovett
12:00-1:30pm Lunch 1:30-2:20pm Guy Kindler From the Grassmann graph to Two-to-Two games
Abstract: In this work we show a relation between the structure of the so called Grassmann graph over Z_2 and the Two-to-Two conjecture in computational complexity. Specifically, we present a structural conjecture concerning the Grassmann graph (together with an observation by Barak et. al., one can view this as a conjecture about the structure of non-expanding sets in that graph) which turns out to imply the Two-to-Two conjecture.
The latter conjecture its the lesser-known and weaker sibling of the Unique-Games conjecture [Khot02], which states that unique games (a.k.a. one-to-one games) are hard to approximate. Indeed, if the Grassmann-Graph conjecture its true, it would also rule out some attempts to refute the Unique-Games conjecture, as these attempts provide potentially efficient algorithms to solve unique games, that would actually also solve two-to-two games if they work at all.
These new connections between the structural properties of the Grassmann graph and complexity theoretic conjectures highlight the Grassmann graph as an interesting and worthy object of study. We may indicate some initial results towards analyzing its structure.
This is joint work with Irit Dinur, Subhash Khot, Dror Minzer, and Muli Safra.
Fluid turbulence and Singularities of the Euler/ Navier Stokes equations
The Workshop on Fluid turbulence and Singularities of the Euler/ Navier Stokes equations will take place on March 13-15, 2019. This is the first of two workshop organized by Michael Brenner, Shmuel Rubinstein, and Tom Hou. The second, Machine Learning for Multiscale Model Reduction, will take place on March 27-29, 2019. Both workshops will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Speakers:
- Claude Bardos, University of Paris
- Jiajie Chen, Caltech
- Peter Constantin, Princeton
- Diego Cordoba, ICMAT
- Tarek Elgindi, UCSD
- Susumu Goto, Osaka
- Alexander Kiselev, Duke University
- Alain Pumir, ENS Lyon
- Shmuel Rubinstein, Harvard SEAS
- Vladimir Sverak, University of Minnesota
- Edriss S. Titi, TAMU
- Vlad Vicol, Courant
- Sijue Wu, University of Michigan
- Andrej Zlatos, UCSD
Blockchain Conference
On January 24-25, 2019 the Center of Mathematical Sciences will be hosting a conference on distributed-ledger (blockchain) technology. The conference is intended to cover a broad range of topics, from abstract mathematical aspects (cryptography, game theory, graph theory, theoretical computer science) to concrete applications (in accounting, government, economics, finance, management, medicine). The talks will take place in Science Center, Hall D.
https://youtu.be/FyKCCutxMYo
Photos
Speakers:
- Joseph Abadi, Princeton University
- Benedikt Bunz, Stanford University
- Jake Cacciapaglia, Nebula Genomics/Harvard Medical School
- Eduardo Castello, Massachusetts Institute of Technology
- Alisa DiCaprio, R3
- Zhiguo He, University of Chicago
- Steven Kou, Boston University
- Anne Lafarre, Tilburg University
- Jacob Leshno, University of Chicago
- Bruce Schneier, Harvard Kennedy School
- David Schwartz, Ripple
- Elaine Shi, Cornell University/Thunder Research
- Hong Wan, NCSU
Workshop on Algebraic Methods in Combinatorics
The workshop on Algebraic Methods in Combinatorics will take place November 13-17, 2017 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
The main focus of the workshop is the application of algebraic method to study problems in combinatorics. In recent years there has been a large number of results in which the use of algebraic technique has resulted in significant improvements to long standing open problems. Such problems include the finite field Kakeya problem, the distinct distance problem of Erdos and, more recently, the cap-set problem. The workshop will include talks on all of the above mentioned problem as well as on recent development in related areas combining combinatorics and algebra.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Confirmed participants include:
- Abdul Basit, Rutgers
- Boris Bukh, Carnegie Mellon University
- Pete L. Clark, University of Georgia
- David Conlon, University of Oxford
- Frank de Zeeuw, EPFL
- Thao Thi Thu Do, MIT
- Noam Elkies, Harvard University
- Jordan Ellenberg, University of Wisconsin
- Dion Gijswijt, Delft Institute of Technology
- Sivankanth Gopi, Princeton University
- Venkatesan Guruswami, Carnegie Mellon University
- Marina Iliopoulou, University of California, Berkeley
- Robert Kleinberg, Cornell University
- Michael Krivelevich, Tel Aviv University
- Vsevelod Lev, University of Haifa at Oranim
- László Miklós Lovász, UCLA
- Ben Lund, Rutgers
- Péter Pach, Budapest University of Technology and Economics
- János Pach, New York University
- Zuzana Patáková, Institute of Science and Technology Austria
- Orit Raz, Institute for Advanced Study
- Oliver Roche-Newton, Johannes Kepler University
- Misha Rudnev, University of Bristol
- Adam Sheffer, California Institute of Technology
- Amir Shpilka, Tel-Aviv University
- Noam Solomon, Harvard CMSA
- Jozsef Solymosi, University of British Columbia
- Benny Sudakov, ETH, Zurich
- Andrew Suk, University of California, San Diego
- Tibor Szabó, Freie Universität Berlin
- Chris Umans, California Institute of Technology
- Avi Wigderson, Princeton University
- Josh Zahl, University of British Columbia
Co-organizers of this workshop include Zeev Dvir, Larry Guth, and Shubhangi Saraf.
Click here for a list of registrants.
Monday, Nov. 13
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Jozsef Solymosi On the unit distance problem
Abstract: Erdos’ Unit Distances conjecture states that the maximum number of unit distances determined by n points in the plane is almost linear, it is O(n^{1+c}) where c goes to zero as n goes to infinity. In this talk I will survey the relevant results and propose some questions which would imply that the maximum number of unit distances is o(n^{4/3}).
10:30-11:00am Coffee Break 11:00-12:00pm Orit Raz Intersection of linear subspaces in R^d and instances of the PIT problem Abstract: In the talk I will tell about a new deterministic, strongly polynomial time algorithm which can be viewed in two ways. The first is as solving a derandomization problem, providing a deterministic algorithm to a new special case of the PIT (Polynomial Identity Testing) problem. The second is as computing the dimension of the span of a collection of flats in high dimensional space. The talk is based on a joint work with Avi Wigderson.
12:00-1:30pm Lunch 1:30-2:30pm Andrew Hoon Suk Ramsey numbers: combinatorial and geometric
Abstract: In this talk, I will discuss several results on determining the tower growth rate of Ramsey numbers arising in combinatorics and in geometry. These results are joint work with David Conlon, Jacob Fox, Dhruv Mubayi, Janos Pach, and Benny Sudakov.
2:30-3:00pm Coffee Break 3:00-4:00pm Josh Zahl Cutting curves into segments and incidence geometry
4:00-6:00pm Welcome Reception Tuesday, Nov. 14
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Péter Pál Pach Polynomials, rank and cap sets
Abstract: In this talk we will look at a new variant of the polynomial method which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like $\mathbb{Z}_4^n$ and $\mathbb{F}_q^n$ are exponentially small (compared to the size of the group). We will discuss lower and upper bounds for the size of the extremal subsets and mention further applications of the method.
10:30-11:00am Coffee Break 11:00-12:00pm Jordan Ellenberg The Degeneration Method
Abstract: In algebraic geometry, a very popular way to study (nice, innocent, nonsingular) varieties is to degenerate them to (weird-looking, badly singular, nonreduced) varieties (which are actually not even varieties but schemes.) I will talk about some results in combinatorics using this approach (joint with Daniel Erman) and some ideas for future applications of the method.
12:00-1:30pm Lunch 1:30-2:30pm Larry Guth The polynomial method in Fourier analysis Abstract: This will be a survey talk about how the polynomial method helps to understand problems in Fourier analysis. We will review some applications of the polynomial method to problems in combinatorial geometry. Then we’ll discuss some problems in Fourier analysis, explain the analogy with combinatorial problems, and discuss how to adapt the polynomial method to the Fourier analysis setting.
2:30-3:00pm
Coffee Break 3:00-4:00pm Open Problem Wednesday, Nov. 15
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Avi Wigderson The “rank method” in arithmetic complexity: Lower bounds and barriers to lower bounds
Abstract: Why is it so hard to find a hard function? No one has a clue! In despair, we turn to excuses called barriers. A barrier is a collection of lower bound techniques, encompassing as much as possible from those in use, together with a proof that these techniques cannot prove any lower bound better than the state-of-art (which is often pathetic, and always very far from what we expect for complexity of random functions).
In the setting of Boolean computation of Boolean functions (where P vs. NP is the central open problem), there are several famous barriers which provide satisfactory excuses, and point to directions in which techniques may be strengthened.
In the setting of Arithmetic computation of polynomials and tensors (where VP vs. VNP is the central open problem) we have no satisfactory barriers, despite some recent interesting attempts.
This talk will describe a new barrier for the Rank Method in arithmetic complexity, which encompass most lower bounds in this field. It also encompass most lower bounds on tensor rank in algebraic geometry (where the the rank method is called Flattening).
I will describe the rank method, explain how it is used to prove lower bounds, and then explain its limits via the new barrier result. As an example, it shows that while the best lower bound on the tensor rank of any explicit 3-dimensional tensor of side n (which is achieved by a rank method) is 2n, no rank method can prove a lower bound which exceeds 8n
(despite the fact that a random such tensor has rank quadratic in n).
No special background knowledge is assumed. The audience is expected to come up with new lower bounds, or else, with new excuses for their absence.
10:30-11:00am Coffee Break 11:00-12:00pm Venkat Guruswami Subspace evasion, list decoding, and dimension expanders
Abstract: A subspace design is a collection of subspaces of F^n (F = finite field) most of which are disjoint from every low-dimensional subspace of F^n. This notion was put forth in the context of algebraic list decoding where it enabled the construction of optimal redundancy list-decodable codes over small alphabets as well as for error-correction in the rank-metric. Explicit subspace designs with near-optimal parameters have been constructed over large fields based on polynomials with structured roots. (Over small fields, a construction via cyclotomic function fields with slightly worse parameters is known.) Both the analysis of the list decoding algorithm as well as the subspace designs crucially rely on the *polynomial method*.
Subspace designs have since enabled progress on linear-algebraic analogs of Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In particular, they yield an explicit construction of constant-degree dimension expanders over large fields. While constructions of such dimension expanders are known over any field, they are based on a reduction to a highly non-trivial form of vertex expanders called monotone expanders. In contrast, the subspace design approach is simpler and works entirely within the linear-algebraic realm. Further, in recent (ongoing) work, their combination with rank-metric codes yields dimension expanders with expansion proportional to the degree.
This talk will survey these developments revolving around subspace designs, their motivation, construction, analysis, and connections.
(Based on several joint works whose co-authors include Chaoping Xing, Swastik Kopparty, Michael Forbes, Nicolas Resch, and Chen Yuan.)
12:00-1:30pm Lunch 1:30-2:30pm David Conlon Finite reflection groups and graph norms
Abstract: For any given graph $H$, we may define a natural corresponding functional $\|.\|_H$. We then say that $H$ is norming if $\|.\|_H$ is a semi-norm. A similar notion $\|.\|_{r(H)}$ is defined by $\| f \|_{r(H)} := \| | f | \|_H$ and $H$ is said to be weakly norming if $\|.\|_{r(H)}$ is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. Using results from the theory of finite reflection groups, we identify a much larger class of weakly norming graphs. This result includes all previous examples of weakly norming graphs and adds many more. We also discuss several applications of our results. In particular, we define and compare a number of generalisations of Gowers’ octahedral norms and we prove some new instances of Sidorenko’s conjecture. Joint work with Joonkyung Lee.
2:30-3:00pm Coffee Break 3:00-4:00pm Laszlo Miklós Lovasz Removal lemmas for triangles and k-cycles.
Abstract: Let p be a fixed prime. A k-cycle in F_p^n is an ordered k-tuple of points that sum to zero; we also call a 3-cycle a triangle. Let N=p^n, (the size of F_p^n). Green proved an arithmetic removal lemma which says that for every k, epsilon>0 and prime p, there is a delta>0 such that if we have a collection of k sets in F_p^n, and the number of k-cycles in their cross product is at most a delta fraction of all possible k-cycles in F_p^n, then we can delete epsilon times N elements from the sets and remove all k-cycles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to our work, the best known bound for any k, due to Fox, showed that 1/delta can be taken to be an exponential tower of twos of height logarithmic in 1/epsilon (for a fixed k).
In this talk, we will discuss recent work on Green’s problem. For triangles, we prove an essentially tight bound for Green’s arithmetic triangle removal lemma in F_p^n, using the recent breakthroughs with the polynomial method. For k-cycles, we also prove a polynomial bound, however, the question of the optimal exponent is still open.
The triangle case is joint work with Jacob Fox, and the k-cycle case with Jacob Fox and Lisa Sauermann.
Thursday, Nov. 16
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Janos Pach Let’s talk about multiple crossings Abstract: Let k>1 be a fixed integer. It is conjectured that any graph on n vertices that can be drawn in the plane without k pairwise crossing edges has O(n) edges. Two edges of a hypergraph cross each other if neither of them contains the other, they have a nonempty intersection, and their union is not the whole vertex set. It is conjectured that any hypergraph on n vertices that contains no k pairwise crossing edges has at most O(n) edges. We discuss the relationship between the above conjectures and explain some partial answers, including a recent result of Kupavskii, Tomon, and the speaker, improving a 40 years old bound of Lomonosov.
10:30-11:00am Coffee Break 11:00-12:00pm Misha Rudnev Few products, many sums
Abstract: This is what I like calling “weak Erd\H os-Szemer\’edi conjecture”, still wide open over the reals and in positive characteristic. The talk will focus on some recent progress, largely based on the ideas of I. D. Shkredov over the past 5-6 years of how to use linear algebra to get the best out of the Szemer\’edi-Trotter theorem for its sum-product applications. One of the new results is strengthening (modulo the log term hidden in the $\lesssim$ symbol) the textbook Elekes inequality
$$
|A|^{10} \ll |A-A|^4|AA|^4
$$
to
$$|A|^{10}\lesssim |A-A|^3|AA|^5.$$
The other is the bound
$$E(H) \lesssim |H|^{2+\frac{9}{20}}$$ for additive energy of sufficiently small multiplicative subgroups in $\mathbb F_p$.
12:00-1:30pm Lunch 1:30-2:30pm Adam Sheffer Geometric Energies: Between Discrete Geometry and Additive Combinatorics
Abstract: We will discuss the rise of geometric variants of the concept of Additive energy. In recent years such variants are becoming more common in the study of Discrete Geometry problems. We will survey this development and then focus on a recent work with Cosmin Pohoata. This work studies geometric variants of additive higher moment energies, and uses those to derive new bounds for several problems in Discrete Geometry.
2:30-3:00pm Coffee Break 3:00-4:00pm Boris Bukh Ranks of matrices with few distinct entries
Abstract: Many applications of linear algebra method to combinatorics rely on the bounds on ranks of matrices with few distinct entries and constant diagonal. In this talk, I will explain some of these application. I will also present a classification of sets L for which no low-rank matrix with entries in L exists.
Friday, Nov. 17
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Benny Sudakov Submodular minimization and set-systems with restricted intersections
Abstract: Submodular function minimization is a fundamental and efficiently solvable problem class in combinatorial optimization with a multitude of applications in various fields. Surprisingly, there is only very little known about constraint types under which it remains efficiently solvable. The arguably most relevant non-trivial constraint class for which polynomial algorithms are known are parity constraints, i.e., optimizing submodular function only over sets of odd (or even) cardinality. Parity constraints capture classical combinatorial optimization problems like the odd-cut problem, and they are a key tool in a recent technique to efficiently solve integer programs with a constraint matrix whose subdeter-minants are bounded by two in absolute value.
We show that efficient submodular function minimization is possible even for a significantly larger class than parity constraints, i.e., over all sets (of any given lattice) of cardinality r mod m, as long as m is a constant prime power. To obtain our results, we combine tools from Combinatorial Optimization, Combinatorics, and Number Theory. In particular, we establish an interesting connection between the correctness of a natural algorithm, and the non-existence of set systems with specific intersection properties.
Joint work with M. Nagele and R. Zenklusen
10:30-11:00am Coffee Break 11:00-12:00pm Robert Kleinberg Explicit sum-of-squares lower bounds via the polynomial method
Abstract: The sum-of-squares (a.k.a. Positivstellensatz) proof system is a powerful method for refuting systems of multivariate polynomial inequalities, i.e. proving that they have no solutions. These refutations themselves involve sum-of-squares (sos) polynomials, and while any unsatisfiable system of inequalities has a sum-of-squares refutation, the sos polynomials involved might have arbitrarily high degree. However, if a system admits a refutation where all polynomials involved have degree at most d, then the refutation can be found by an algorithm with running time polynomial in N^d, where N is the combined number of variables and inequalities in the system.
Low-degree sum-of-squares refutations appear throughout mathematics. For example, the above proof search algorithm captures as a special case many a priori unrelated algorithms from theoretical computer science; one example is Goemans and Williamson’s algorithm to approximate the maximum cut in a graph. Specialized to extremal graph theory, they become equivalent to flag algebras. They have also seen practical use in robotics and optimal control.
Therefore, it is of interest to identify “hard” systems of low-degree polynomial inequalities that have no solutions but also have no low-degree sum-of-squares refutations. Until recently, the only known examples were either not explicit (i.e., known to exist by non-constructive means such as the probabilistic method) or not robust (i.e., a system is constructed which is not refutable by degree d sos polynomials, but becomes refutable when perturbed by an amount tending to zero with d). We present a new family of instances derived from the cap-set problem, and we show a super-constant lower bound on the degree of its sum-of-squares refutations. Our instances are both explicit and robust.
This is joint work with Sam Hopkins.
12:00-1:30pm Lunch Naturalness and muon anomalous magnetic moment
Title: Naturalness and muon anomalous magnetic moment
Abstract: We study a model for explaining the apparent deviation of the muon anomalous magnetic moment, (g-2), from the Standard Model expectation. There are no new scalars and hence no new hierarchy puzzles beyond those associated with the Standard model Higgs; the only new particles that are relevant for (g-2) are vector-like singlet and doublet leptons. Interestingly, this simple model provides a calculable example violating the Wilsonian notion of naturalness: despite the absence of any symmetries prohibiting its generation, the coefficient of the naively leading dimension-six operator for (g−2) vanishes at one-loop. While effective field theorists interpret this either as a surprising UV cancellation of power divergences, or as a delicate cancellation between matching UV and calculable IR corrections to (g−2) from parametrically separated scales, there is a simple explanation in the full theory: the loop integrand is a total derivative of a function vanishing in both the deep UV and IR. The leading contribution to (g−2) arises from dimension-eight operators, and thus the required masses of new fermions are lower than naively expected, with a sizable portion of parameter space already covered by direct searches at the LHC. All of the the viable parameter can be probed by the LHC and planned future colliders.
Exotic quantum matter: From lattice gauge theory to hyperbolic lattices
Title: Exotic quantum matter: From lattice gauge theory to hyperbolic lattices
Abstract: This talk, in two parts, will discuss two (unrelated) instances of exotic quantum matter. In the first part, I will discuss quantum critical points describing possible transitions out of the Dirac spin liquid, towards either symmetry-breaking phases or topologically ordered spin liquids. I will also comment on the role of instanton zero modes for symmetry breaking in parton gauge theories. In the second part, I will propose an extension of Bloch band theory to hyperbolic lattices, such as those recently realized in circuit QED experiments, based on ideas from algebraic geometry and Riemann surface theory.
Cornering the universal shape of fluctuations and entanglement
Title: Cornering the universal shape of fluctuations and entanglement
Abstract: Understanding the fluctuations of observables is one of the main goals in physics. We investigate such fluctuations when a subregion of the full system can be observed, focusing on geometries with corners. We report that the dependence on the opening angle is super-universal: up to a numerical prefactor, this function does not depend on anything, provided the system under study is uniform, isotropic, and correlations do not decay too slowly. The prefactor contains important physical information: we show in particular that it gives access to the long-wavelength limit of the structure factor. We illustrate our findings with several examples: classical fluids, fractional quantum Hall (FQH) states, scale invariant quantum critical theories, and metals. Finally, we discuss connections with the entanglement entropy, including new results for Laughlin FQH states.
Ref: arXiv:2102.06223
Quantum gravity from quantum matter
Title: Quantum gravity from quantum matter
Abstract: We present a model of quantum gravity in which dimension, topology and geometry of spacetime are collective dynamical variables that describe the pattern of entanglement of underlying quantum matter. As spacetimes with arbitrary dimensions can emerge, the gauge symmetry is generalized to a group that includes diffeomorphisms in general dimensions. The gauge symmetry obeys a first-class constraint operator algebra, and is reduced to a generalized hypersurface deformation algebra in states that exhibit classical spacetimes. In the semi-classical limit, we find a saddle-point solution that describes a series of (3+1)-dimensional de Sitter-like spacetimes with the Lorentzian signature bridged by Euclidean spaces in between.
9/23/2021 Interdisciplinary Science Seminar
Title: The number of n-queens configurations
Abstract: The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. The problem has a storied history and was studied by such eminent mathematicians as Gauss and Polya. The problem has also found applications in fields such as algorithm design and circuit development.
Despite much study, until recently very little was known regarding the asymptotics of Q(n). We apply modern methods from probabilistic combinatorics to reduce understanding Q(n) to the study of a particular infinite-dimensional convex optimization problem. The chief implication is that (in an appropriate sense) for a~1.94, Q(n) is approximately (ne^(-a))^n. Furthermore, our methods allow us to study the typical “shape” of n-queens configurations.
10/7/2021 Interdisciplinary Science Seminar
Title: SiRNA Targeting TCRb: A Proposed Therapy for the Treatment of Autoimmunity
Abstract: As of 2018, the United States National Institutes of Health estimate that over half a billion people worldwide are affected by autoimmune disorders. Though these conditions are prevalent, treatment options remain relatively poor, relying primarily on various forms of immunosuppression which carry potentially severe side effects and often lose effectiveness over time. Given this, new forms of therapy are needed. To this end, we have developed methods for the creation of small-interfering RNA (siRNA) for hypervariable regions of the T-cell receptor β-chain gene (TCRb) as a highly targeted, novel means of therapy for the treatment of autoimmune disorders.
This talk will review the general mechanism by which autoimmune diseases occur and discuss the pros and cons of conventional pharmaceutical therapies as they pertain to autoimmune disease treatment. I will then examine the rational and design methodology for the proposed siRNA therapy and how it contrasts with contemporary methods for the treatment of these conditions. Additionally, the talk will compare the efficacy of multiple design strategies for such molecules by comparison over several metrics and discuss how this will be guiding future research.
10/14/2021 Interdisciplinary Science Seminar
Title: D3C: Reducing the Price of Anarchy in Multi-Agent Learning
Abstract: In multi-agent systems the complex interaction of fixed incentives can lead agents to outcomes that are poor (inefficient) not only for the group but also for each individual agent. Price of anarchy is a technical game theoretic definition introduced to quantify the inefficiency arising in these scenarios– it compares the welfare that can be achieved through perfect coordination against that achieved by self-interested agents at a Nash equilibrium. We derive a differentiable upper bound on a price of anarchy that agents can cheaply estimate during learning. Equipped with this estimator agents can adjust their incentives in a way that improves the efficiency incurred at a Nash equilibrium. Agents adjust their incentives by learning to mix their reward (equiv. negative loss) with that of other agents by following the gradient of our derived upper bound. We refer to this approach as D3C. In the case where agent incentives are differentiable D3C resembles the celebrated Win-Stay Lose-Shift strategy from behavioral game theory thereby establishing a connection between the global goal of maximum welfare and an established agent-centric learning rule. In the non-differentiable setting as is common in multiagent reinforcement learning we show the upper bound can be reduced via evolutionary strategies until a compromise is reached in a distributed fashion. We demonstrate that D3C improves outcomes for each agent and the group as a whole on several social dilemmas including a traffic network exhibiting Braess’s paradox a prisoner’s dilemma and several reinforcement learning domains.
10/21/2021 Interdisciplinary Science Seminar
Title: Mathematical resolution of the Liouville conformal field theory.
Abstract: The Liouville conformal field theory is a well-known beautiful quantum field theory in physics describing random surfaces. Only recently a mathematical approach based on a well-defined path integral to this theory has been proposed using probability by David, Kupiainen, Rhodes, Vargas.
Many works since the ’80s in theoretical physics (starting with Belavin-Polyakov-Zamolodchikov) tell us that conformal field theories in dimension 2 are in general « Integrable », the correlations functions are solutions of PDEs and can in principle be computed explicitely by using algebraic tools (vertex operator algebras, representations of Virasoro algebras, the theory of conformal blocks). However, for Liouville Theory this was not done at the mathematical level by algebraic methods.
I’ll explain how to combine probabilistic, analytic and geometric tools to give explicit (although complicated) expressions for all the correlation functions on all Riemann surfaces in terms of certain holomorphic functions of the moduli parameters called conformal blocks, and of the structure constant (3-point function on the sphere). This gives a concrete mathematical proof of the so-called conformal bootstrap and of Segal’s gluing axioms for this CFT. The idea is to break the path integral on a closed surface into path integrals on pairs of pants and reduce all correlation functions to the 3-point correlation function on the Riemann sphere $S^2$. This amounts in particular to prove a spectral resolution of a certain operator acting on $L^2(H^{-s}(S^1))$ where $H^{-s}(S^1)$ is the Sobolev space of order -s<0 equipped with a Gaussian measure, which is viewed as the space of fields, and to construct a certain representation of the Virasoro algebra into unbounded operators acting on this Hilbert space.
This is joint work with A. Kupiainen, R. Rhodes and V. Vargas.
More Exact Results in Gauge Theories: Confinement and Chiral Symmetry Breaking
Title: More Exact Results in Gauge Theories: Confinement and Chiral Symmetry Breaking
Abstract: In this follow-up to Hitoshi Murayama’s talk “Some Exact Results in QCD-like and Chiral Gauge Theories”, I present a detailed analysis of the phases of $SO(N_c)$ gauge theory.
Starting with supersymmetric $SO(N_c)$ with $N_F$ flavors, we extrapolate to the non-supersymmetric limit using anomaly-mediated supersymmetry breaking (AMSB). Interestingly, the abelian Coulomb and free magnetic phases do not survive supersymmetry breaking and collapse to a confining phase. This provided one of the first demonstrations of true confinement with chiral symmetry breaking in a non-SUSY theory.ARCH: Know What Your Machine Doesn’t Know
Speaker: Jie Yang, Delft University of Technology
Title: ARCH: Know What Your Machine Doesn’t Know
Abstract: Despite their impressive performance, machine learning systems remain prohibitively unreliable in safety-, trust-, and ethically sensitive domains. Recent discussions in different sub-fields of AI have reached the consensus of knowledge need in machine learning; few discussions have touched upon the diagnosis of what knowledge is needed. In this talk, I will present our ongoing work on ARCH, a knowledge-driven, human-centered, and reasoning-based tool, for diagnosing the unknowns of a machine learning system. ARCH leverages human intelligence to create domain knowledge required for a given task and to describe the internal behavior of a machine learning system; it infers the missing or incorrect knowledge of the system with the built-in probabilistic, abductive reasoning engine. ARCH is a generic tool that can be applied to machine learning in different contexts. In the talk, I will present several applications in which ARCH is currently being developed and tested, including health, finance, and smart buildings.
Three-particle mechanism for pairing and superconductivity
Title: Three-particle mechanism for pairing and superconductivity
Abstract: I will present a new mechanism and an exact theory of electron pairing due to repulsive interaction in doped insulators. When the kinetic energy is small, the dynamics of adjacent electrons on the lattice is strongly correlated. By developing a controlled kinetic energy expansion, I will show that two doped charges can attract and form a bound state, despite and because of the underlying repulsion. This attraction by repulsion is enabled by the virtual excitation of a third electron in the filled band. This three-particle pairing mechanism leads to a variety of novel phenomena at finite doping, including spin-triplet superconductivity, pair density wave, BCS-BEC crossover and Feshbach resonance involving “trimers”. Possible realizations in moire materials, ZrNCl and WTe2 will be discussed.
[1] V. Crepel and L. Fu, Science Advances 7, eabh2233 (2021)
[2] V. Crepel and L. Fu, arXiv:2103.12060
[3] K. Slagle and L. Fu, Phys. Rev. B 102, 235423 (2020)The Hilbert Space of large N Chern-Simons matter theories
Title: The Hilbert Space of large N Chern-Simons matter theories
Abstract: We demonstrate that all known formulae for the thermal partition function for large N Chern Simons matter theory admit a simple Hilbert Space interpretation. In each case this quantity equals the partition function of an associated ungauged large $N$ matter theory with a particular local Lagrangian with one additional element: the Fock Space of this associated theory is projected down to the subspace of its WZW singlets. This projection, in particular, implies the previously encountered `Bosonic Exclusion Principle’, namely that no single particle state can be occupied by more than $k_B$ particles ($k_B$ is the Chern Simons level). Unlike its Gauss Law counterpart, the WZW constraint does not trivialize in the large volume limit. However thermodynamics does simplify in this limit; the final partition function reduces to a product of partition functions associated with each single particle state. These individual single particle state partition functions are a one parameter generalizations of their free boson and free fermion counterparts, and reduce to the later at extreme values of the ‘t Hooft coupling. At generic values of the rank and the level the occupation statistics of each energy level is given by a $q$ deformation of the usual free formulae of Bose and Fermi statistics.
Strong Coupling Theory of Magic-Angle Graphene: A Pedagogical Introduction
Title: Strong Coupling Theory of Magic-Angle Graphene: A Pedagogical Introduction
Abstract: In this talk, I will review a recently developed strong coupling theory of magic-angle twisted bilayer graphene. An advantage of this approach is that a single formulation can capture both the insulating and superconducting states, and with a few simplifying assumptions, can be treated analytically. I begin by reviewing the electronic structure of magic angle graphene’s flat bands, in a limit that exposes their peculiar band topology and geometry. I will show how similarities between the flat bands and the lowest Landau level can provide valuable insights into the effect of interactions and form the basis for an analytic treatment of the problem. At integer fillings, this approach points to flavor ordered insulators, which can be captured by a sigma-model in its ordered phase. Remarkably, topological textures of the sigma model carry electric charge which enables the same theory to describe the doped phases away from integer filling. I will show how this approach can lead to superconductivity on disordering the sigma model, and estimate the Tc for the superconductor. I will highlight the important role played by an effective super-exchange coupling both in pairing and in setting the effective mass of Cooper pairs. At the end, I will show how this theory provides criteria to predict which multilayer graphene stacks are expected to superconduct including the recently discovered alternating twist trilayer platform.
UV/IR and Effective Field Theory
Speaker: Nima Arkani-Hamed (IAS Princeton)
Title: UV/IR and Effective Field Theory
Tropical disk counts
Abstract: (joint with S. Venugopalan) I will describe version of the Fukaya algebra that appears in a tropical degeneration with the Lagrangian being one of the “tropical fibers”. An example is the count of “twenty-one disks in the cubic surface” (suggested by Sheridan) which is an open analog of the twenty-seven lines. As an application, I will explain why the Floer cohomology of such tropical fibers is well-defined; this is a generalization fo a result of Fukaya-Oh-Ohta-Ono for toric varieties.
RANDOM MATRIX PROGRAM
arge random matrices provide some of the simplest models for large, strongly correlated quantum systems. The statistics of the energy levels of ensembles of such systems are expected to exhibit universality, in the sense that they depend only on the symmetry class of the system. Recent advances have enabled a rigorous understanding of universality in the case of orthogonal, Hermitian, or symplectic matrices with independent entries, resolving a conjecture of Wigner-Dyson-Mehta dating back 50 years. These new developments have exploited techniques from a wide range of mathematical areas in addition to probability, including combinatorics, partial differential equations, and hydrodynamic limits. It is hoped that these new techniques will be useful in the analysis of universal behaviour in matrix ensembles with more complicated structure such as random regular graph models, or 2D matrix ensembles, as well as more physically relevant systems such as band matrices and random Schroedinger-type Hamiltonians. For some of these models, results in the direction of universality have already been obtained.
Here is a partial list of the mathematicians who are participating in this program
TOPOLOGICAL ASPECTS OF CONDENSED MATTER
During Academic year 2018-19, the CMSA will be hosting a Program on Topological Aspects of Condensed Matter. New ideas rooted in topology have recently had a big impact on condensed matter physics, and have highlighted new connections with high energy physics, mathematics and quantum information theory. Additionally, these ideas have found applications in the design of photonic systems and of materials with novel mechanical properties. The aim of this program will be to deepen these connections by foster discussion and seeding new collaborations within and across disciplines.
As part of the Program, the CMSA will be hosting two workshops:
- Workshop on Topology and Quantum Phases of Matter (August 27-28, 2018)
- Workshop on Topological Aspects of Condensed Matter (Septmeber 10-11, 2019)
.
Additionally, a weekly Topology Seminar will be held on Mondays from 10:00-11:30pm in CMSA room G10.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special programName Tentative Visiting Dates 11/12/2018-11/16/2018 Maissam Barkeshli 4/22/2019 – 4/26/2019 Xie Chen 4/15-17/2019 4/19-21/2019 4/24-30/2019 1/7/2019-1/11/2019 8/15/2018-8/30/2018 & 5/9/2019-5/19/2019 10/14/2018-10/27/2018 Anton Kapustin 8/26/2018-8/30/2018 & 3/28/2019-4/5/2019 3/11/2019-3/15/2019 Yuan-Ming Lu 4/29/2019-6/01/2019 4/2/2019- 4/19/2019 4/22/2019-5/22/2019 Chong Wang 10/22/2018-11/16/2018 4/1/2019-4/16/2019 Cenke Xu 8/26/2018-10/1/2018 4/1/2019-4/19/2019 5/1/2019-5/10/2019 Mathematical Biology
During Academic year 2018-19, the CMSA will be hosting a Program on Mathematical Biology.
Just over a century ago, the biologist, mathematician and philologist D’Arcy Thompson wrote “On growth and form”. The book was a visionary synthesis of the geometric biology of form at the time. It also served as a call for mathematical and physical approaches to understanding the evolution and development of shape.
In the century since its publication, we have seen a revolution in biology following the discovery of the genetic code, which has uncovered the molecular and cellular basis for life, combined with the ability to probe the chemical, structural, and dynamical nature of molecules, cells, tissues and organs across scales. In parallel, we have seen a blossoming of our understanding of spatiotemporal patterning in physical systems, and a gradual unveiling of the complexity of physical form. And in mathematics and computation, there has been a revolution in terms of posing and solving problems at the intersection of computational geometry, statistics and inference. So, how far are we from realizing a descriptive, predictive and controllable theory of biological shape?
In Fall 2018, CMSA will focus on a program that aims at recent mathematical advances in describing shape using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems
The CMSA will be hosting three workshops as part of this program. The Workshop on Morphometrics, Morphogenesis and Mathematics will take place on October 22-26.
A workshop on Morphogenesis: Geometry and Physics will take place on December 3-6, 2018.
A workshop on Invariance and Geometry in Sensation, Action and Cognition will take place on April 15-17, 2019.
SPACETIME AND QUANTUM MECHANICS, TOTAL POSITIVITY AND MOTIVES
Recent developments have poised this area to make serious advances in 2019, and we feel that bringing together many of the relevant experts for an intensive semester of discussions and collaboration will trigger some great things to happen. To this end, the organizers will host a small workshop during fall 2019, with between 20-30 participants. They will also invite 10-20 longer-term visitors throughout the semester. Additionally, there will be a seminar held weekly on Thursdays at 2:30pm in CMSA G10.
Organizers:
- Nima Arkani-Hamed (IAS)
- Lauren Williams (Harvard)
- Alexander Postnikov (MIT)
- Thomas Lam (Michigan)
.
Workshops:
Spacetime and Quantum Mechanics Workshop, October 28-30, 2019Here is a partial list of the mathematicians and physicists who have indicated that they will attend part or all of this special program as a visitor:
- Paolo Benincasa, 11/17/2019 – 11/29/2019
- Jacob Bourjaily, 9/1/2019 – 12/15/2019
- Francis Brown, 9/15/2019 – 9/20/2019
- Simon Caron-Huot, 9/30/2019 – 10/04/2019
- Lance Dixon, 9/9/2019 – 9/20/2019
- Charles Doran, 10/19/2019 – 11/1/2019
- James Drummond, 10/14/2019 – 10/18/2019
- Nick Early, 11/18/2019 – 11/22/2019
- Livia Ferro, 10/27/2019 – 11/9/2019
- Sergey Fomin, 10/6/2019 – 10/16/2019
- Sebastian Franco, 10/9/2019 – 10/19/2019
- Hadleigh Frost, 9/15/2019 – 12/20/2019
- Michael Green, 10/05/2019 – 10/13/2019
- Alexander Goncharov, 12/05/2019 – 12/20/2019
- Song He, 9/29/2019 – 11/10/2019
- Xuhua He, 10/30/2019-11/03/2019.
- Enrico Herrmann, 10/27/2019 – 11/9/2019
- Yutin Huang, 9/30/2019 – 10/12/2019
- Steven Karp, 10/11/2019 – 11/03/2019
- Tomasz Lukowski, 10/27/2019 – 11/11/2019
- Andrew McLeod, 10/6/2019 – 10/19/2019 & 11/3/2019 – 11/16/2019
- Sebastian Mizera, 10/28/2019 – 11/1/2019
- Erik Panzer, 9/15/2019 – 9/25/2019
- Matteo Parisi, 10/26/2019 – 11/10/2019
- Julio Parra-Martinez, 10/10/2019 – 05/12/2019
- Pierpaolo Mastrolia, 11/8/2019 – 11/16/2019
- Pasha Pylyavskyy, 9/8/2019 – 9/22/2019 & 10/14/2019 – 11/1/2019
- Junjie Rao, 10/25/2019 – 11/04/2019
- Giulio Salvatori, 9/3/2019 – 12/15/2019
- Michael Shapiro, 10/27/2019 – 11/2/2019
- David Speyer, 10/14/2019 – 10/18/2019
- Hugh Thomas, 10/27/2019 – 11/22/2019
- Jaroslav Trnka, 9/30/2019 – 10/04/2019, 10/28/2019 – 11/01/2019, 11/18/2019 – 11/22/2019
- Cristian Vergu, 11/10/2019 – 11/30/2019
- Matthias Volk, 10/14/2019 – 10/25/2019
- Matthew von Hippel, 11/11/2019 – 11/22/2019
- Pierre Vanhove, 10/22/2019 – 10/31/2019
- Matthias Wilhelm, 10/14/2019 – 10/25/2019
THE SIMONS COLLABORATION IN HOMOLOGICAL MIRROR SYMMETRY
The Simons Collaboration program in Homological Mirror Symmetry at Harvard CMSA and Brandeis University is part of the bigger Simons collaboration program on Homological mirror symmetry (https://schms.math.berkeley.edu) which brings to CMSA experts on algebraic geometry, Symplectic geometry, Arithmetic geometry, Quantum topology and mathematical aspects of high energy physics, specially string theory with the goal of proving the homological mirror symmetry conjecture (HMS) in full generality and explore its applications. Mirror symmetry, which emerged in the late 1980s as an unexpected physical duality between quantum field theories, has been a major source of progress in mathematics. At the 1994 ICM, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves of a smooth projective variety is equivalent to the Fukaya category of a mirror symplectic manifold (or Landau-Ginzburg model). We are happy to announce that the Simons Foundation has agreed to renew funding for the HMS collaboration program for three additional years.
A brief induction of the Brandeis-Harvard CMSA HMS/SYZ research agenda and team members are as follow:
Directors:
Shing-Tung Yau (Harvard University)Born in Canton, China, in 1949, S.-T. Yau grew up in Hong Kong, and studied in the Chinese University of Hong Kong from 1966 to 1969. He did his PhD at UC Berkeley from 1969 to 1971, as a student of S.S. Chern. He spent a year as a postdoc at the Institute for Advanced Study in Princeton, and a year as assistant professor at SUNY at Stony Brook. He joined the faculty at Stanford in 1973. On a Sloan Fellowship, he spent a semester at the Courant Institute in 1975. He visited UCLA the following year, and was offered a professorship at UC Berkeley in 1977. He was there for a year, before returning to Stanford. He was a plenary speaker at the 1978 ICM in Helsinki. The following year, he became a faculty member at the IAS in Princeton. He moved to UCSD in 1984. Yau came to Harvard in 1987, and was appointed the Higgins Professor of Mathematics in 1997. He has been at Harvard ever since. Yau has received numerous prestigious awards and honors throughout his career. He was named a California Scientist of the Year in 1979. In 1981, he received a Oswald Veblen Prize in Geometry and a John J. Carty Award for the Advancement of Science, and was elected a member of the US National Academy of Sciences. In 1982, he received a Fields Medal for “his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex MongeAmpre equations”. He was named Science Digest, America’s 100 Brightest Scientists under 40, in 1984. In 1991, he received a Humboldt Research Award from the Alexander von Humboldt Foundation in Germany. He was awarded a Crafoord Prize in 1994, a US National Medal of Science in 1997, and a China International Scientific and Technological Cooperation Award, for “his outstanding contribution to PRC in aspects of making progress in sciences and technology, training researchers” in 2003. In 2010, he received a Wolf Prize in Mathematics, for “his work in geometric analysis and mathematical physics”. Yau has also received a number of research fellowships, which include a Sloan Fellowship in 1975-1976, a Guggenheim Fellowship in 1982, and a MacArthur Fellowship in 1984-1985. Yau’s research interests include differential and algebraic geometry, topology, and mathematical physics. As a graduate student, he started to work on geometry of manifolds with negative curvature. He later became interested in developing the subject of geometric analysis, and applying the theory of nonlinear partial differential equations to solve problems in geometry, topology, and physics. His work in this direction include constructions of minimal submanifolds, harmonic maps, and canonical metrics on manifolds. The most notable, and probably the most influential of this, was his solution of the Calabi conjecture on Ricci flat metrics, and the existence of Kahler-Einstein metrics. He has also succeeded in applying his theory to solve a number of outstanding conjectures in algebraic geometry, including Chern number inequalities, and the rigidity of complex structures of complex projective spaces. Yau’s solution to the Calabi conjecture has been remarkably influential in mathematical physics over the last 30 years, through the creation of the theory of Calabi-Yau manifolds, a theory central to mirror symmetry. He and a team of outstanding mathematicians trained by him, have developed many important tools and concepts in CY geometry and mirror symmetry, which have led to significant progress in deformation theory, and on outstanding problems in enumerative geometry. Lian, Yau and his postdocs have developed a systematic approach to study and compute period integrals of CY and general type manifolds. Lian, Liu and Yau (independently by Givental) gave a proof of the counting formula of Candelas et al for worldsheet instantons on the quintic threefold. In the course of understanding mirror symmetry, Strominger, Yau, and Zaslow proposed a new geometric construction of mirror symmetry, now known as the SYZ construction. This has inspired a rapid development in CY geometry over the last two decades. In addition to CY geometry and mirror symmetry, Yau has done influential work on nonlinear partial differential equations, generalized geometry, Kahler geometry, and general relativity. His proof of positive mass conjecture is a widely regarded as a cornerstone in the classical theory of general relativity. In addition to publishing well over 350 research papers, Yau has trained more than 60 PhD students in a broad range of fields, and mentored dozens of postdoctoral fellows over the last 40 years.
Professor Bong Lian (Brandeis University)Born in Malaysia in 1962, Bong Lian completed his PhD in physics at Yale University under the direction of G. Zuckerman in 1991. He joined the permanent faculty at Brandeis University in 1995, and has remained there since. Between 1995 and 2013, he had had visiting research positions at numerous places, including the National University of Taiwan, Harvard University, and Tsinghua University. Lian received a J.S. Guggenheim Fellowship in 2003. He was awarded a Chern Prize at the ICCM in Taipei in 2013, for his “influential and fundamental contributions in mathematical physics, in particular in the theory of vertex algebras and mirror symmetry.” He has also been co-Director, since 2014, of the Tsinghua Mathcamp, a summer outreach program launched by him and Yau for mathematically talented teenagers in China. Since 2008, Lian has been the President of the International Science Foundation of Cambridge, a non-profit whose stated mission is “to provide financial and logistical support to scholars and universities, to promote basic research and education in mathematical sciences, especially in the Far East.” Over the last 20 years, he has mentored a number of postdocs and PhD students. His research has been supported by an NSF Focused Research Grant since 2009. Published in well over 60 papers over 25 years, Lian’s mathematical work lies in the interface between representation theory, Calabi-Yau geometry, and string theory. Beginning in the late 80’s, Lian, jointly with Zuckerman, developed the theory of semi-infinite cohomology and applied it to problems in string theory. In 1994, he constructed a new invariant (now known as the Lian- Zuckerman algebra) of a topological vertex algebra, and conjectured the first example of a G algebra in vertex algebra theory. The invariant has later inspired a new construction of quantum groups by I. Frenkel and A. Zeitlin, as semi-infinite cohomology of braided vertex algebras, and led to a more recent discovery of new relationships between Courant algebroids, A-algebras, operads, and deformation theory of BV algebras. In 2010, he and his students Linshaw and Song developed important applications of vertex algebras in equivariant topology. Lian’s work in CY geometry and mirror symmetry began in early 90’s. Using a characteristic p version of higher order Schwarzian equations, Lian and Yau gave an elementary proof that the instanton formula of Candelas et al implies Clemens’s divisibility conjecture for the quintic threefold, for infinitely many degrees. In 1996, Lian (jointly with Hosono and Yau) answered the so-called Large Complex Structure Limit problem in the affirmative in many important cases. Around the same year, they announced their hyperplane conjecture, which gives a general formula for period integrals for a large class of CY manifolds, extending the formula of Candelas et al. Soon after, Lian, Liu and Yau (independently by Givental) gave a proof of the counting formula. In 2003, inspired by mirror symmetry, Lian (jointly with Hosono, Oguiso and Yau) discovered an explicit counting formula for Fourier-Mukai partners, and settled an old problem of Shioda on abelian and K3 surfaces. Between 2009 and 2014, Lian (jointly with Bloch, Chen, Huang, Song, Srinivas, Yau, and Zhu) developed an entirely new approach to study the so-called Riemann-Hilbert problem for period integrals of CY manifolds, and extended it to general type manifolds. The approach leads to an explicit description of differential systems for period integrals with many applications. In particular, he answered an old question in physics on the completeness of Picard-Fuchs systems, and constructed new differential zeros of hypergeometric functions.
Denis Auroux (Harvard University)Denis Auroux’s research concerns symplectic geometry and its applications to mirror symmetry. While his early work primarily concerned the topology of symplectic 4-manifolds, over the past decade Auroux has obtained pioneering results on homological mirror symmetry outside of the Calabi-Yau setting (for Fano varieties, open Riemann surfaces, etc.), and developed an extension of the SYZ approach to non-Calabi-Yau spaces.After obtaining his PhD in 1999 from Ecole Polytechnique (France), Auroux was employed as Chargé de Recherche at CNRS and CLE Moore Instructor at MIT, before joining the faculty at MIT in 2002 (as Assistant Professor from 2002 to 2004, and as Associate Professor from 2004 to 2009, with tenure starting in 2006). He then moved to UC Berkeley as a Full Professor in 2009.
Auroux has published over 30 peer-reviewed articles, including several in top journals, and given 260 invited presentations about his work. He received an Alfred P. Sloan Research Fellowship in 2005, was an invited speaker at the 2010 International Congress of Mathematicians, and in 2014 he was one of the two inaugural recipients of the Poincaré Chair at IHP. He has supervised 10 PhD dissertations, won teaching awards at MIT and Berkeley, and participated in the organization of over 20 workshops and conferences in symplectic geometry and mirror symmetry.
Senior Personnel:Artan Sheshmani (Harvard CMSA)
Artan Sheshmani’s research is focused on enumerative algebraic geometry and mathematical aspects of string theory. He is interested in applying techniques in algebraic geometry, such as, intersection theory, derived category theory, and derived algebraic geometry to construct and compute the deformation invariants of algebraic varieties, in particular Gromov-Witten (GW) or Donaldson-Thomas (DT) invariants. In the past Professor Sheshmani has worked on proving modularity property of certain DT invariants of K3-fibered threefolds (as well as their closely related Pandharipande-Thomas (PT) invariants), local surface threefolds, and general complete intersection Calabi-Yau threefolds. The modularity of DT/PT invariants in this context is predicted in a famous conjecture of string theory called S-duality modularity conjecture, and his joint work has provided the proof to some cases of it, using degenerations, virtual localizations, as well as wallcrossing techniques. Recently, Sheshmani has focused on proving a series of dualities relating the various enumerative invariants over threefolds, notably the GW invariants and invariants that arise in topological gauge theory. In particular in his joint work with Gholampour, Gukov, Liu, Yau he studied DT gauge theory and its reductions to D=4 and D=2 which are equivalent to local theory of surfaces in Calabi-Yau threefolds. Moreover, in a recent joint work with Yau and Diaconescu, he has studied the construction and computation of DT invariants of Calabi-Yau fourfolds via a suitable derived categorical reduction of the theory to the DT theory of threefolds. Currently Sheshmani is interested in a wide range of problems in enumerative geometry of CY varieties in dimensions 3,4,5.
Artan has received his PhD and Master’s degrees in pure mathematics under Sheldon Katz and Thomas Nevins from the University of Illinois at Urbana Champaign (USA) in 2011 and 2008 respectively. He holds a Master’s degree in Solid Mechanics (2004) and two Bachelor’s degrees, in Mechanical Engineering and Civil Engineering from the Sharif University of Technology, Tehran, Iran. Artan has been a tenured Associate Professor of Mathematics with joint affiliation at Harvard CMSA and center for Quantum Geometry of Moduli Spaces (QGM), since 2016. Before that he has held visiting Associate Professor and visiting Assistant Professor positions at MIT.
An Huang (Brandeis University)
The research of An Huang since 2011 has been focused on the interplay between algebraic geometry, the theory of special functions and mirror symmetry. With S. Bloch, B. Lian, V. Srinivas, S.-T. Yau, X. Zhu, he has developed the theory of tautological systems, and has applied it to settle several important problems concerning period integrals in relation to mirror symmetry. With B. Lian and X. Zhu, he has given a precise geometric interpretation of all solutions to GKZ systems associated to Calabi-Yau hypersurfaces in smooth Fano toric varieties. With B. Lian, S.-T. Yau, and C.-L. Yu, he has proved a conjecture of Vlasenko concerning an explicit formula for unit roots of the zeta functions of hypersurfaces, and has further related these roots to p-adic interpolations of complex period integrals. Beginning in 2018, with B. Stoica and S.-T. Yau, he has initiated the study of p-adic strings in curved spacetime, and showed that general relativity is a consequence of the self-consistency of quantum p-adic strings. One of the goals of this study is to understand p-adic A and B models.
An Huang received his PhD in Mathematics from the University of California at Berkeley in 2011. He was a postdoctoral fellow at the Harvard University Mathematics Department, and joined Brandeis University as an Assistant Professor in Mathematics in 2016.
Siu Cheong Lau (Boston University)The research interest of Siu Cheong Lau lies in SYZ mirror symmetry, symplectic and algebraic geometry. His thesis work has successfully constructed the SYZ mirrors for all toric Calabi-Yau manifolds based on quantum corrections by open Gromov-Witten invariants and their wall-crossing phenomenon. In collaboration with N.C. Leung, H.H. Tseng and K. Chan, he derived explicit formulas for the open Gromov-Witten invariants for semi-Fano toric manifolds which have an obstructed moduli theory. It has a beautiful relation with mirror maps and Seidel representations. Recently he works on a local-to-global approach to SYZ mirror symmetry. In joint works with C.H. Cho and H. Hong, he developed a noncommutative local mirror construction for immersed Lagrangians, and a natural gluing method to construct global mirrors. The construction has been realized in various types of geometries including orbifolds, focus-focus singularities and pair-of-pants decompositions of Riemann surfaces.
Siu-Cheong Lau has received the Doctoral Thesis Gold Award (2012) and the Best Paper Silver Award (2017) at the International Congress of Chinese Mathematicians. He was awarded the Simons Collaboration Grant in 2018. He received a Certificate of Teaching Excellence from Harvard University in 2014.
Affiliates:
- Netanel Rubin-Blaier (Cambridge)
- Kwokwai Chan (Chinese University of Hong Kong)
- Mandy Cheung (Harvard University, BP)
- Chuck Doran (University of Alberta)
- Honsol Hong (Yonsei University)
- Shinobu Hosono (Gakushuin University, Japan)
- Conan Leung (Chinese University of Hong Kong)
- Yu-shen Lin (Boston University)
- Hossein Movassati (IMPA Brazil)
- Arnav Tripathhy (Harvard University, BP)
Postdocs:
- Dennis Borisov
- Tsung-Ju Lee
- Dingxin Zhang
- Jingyu Zhao
- Yang Zhou
Jobs:
Postdoctoral Fellowship in Algebraic Geometry
Postdoctoral Fellowship in Mathematical Sciences
To learn about previous programming as part of the Simons Collaboration, click here.
- 0111/01/2021
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Bubble instability of mIIA on AdS_4 x S^6
Abstract: Recently, a set of non-supersymmetric AdS_4 vacua of massive type IIA string theory has been constructed. These vacua are perturbatively stable with respect to the full KK spectrum of type mIIA supergravity and furthermore, they are stable against a variety of non-perturbative decay channels. Hence, at this point, they represent a serious challenge to the AdS swampland conjecture. In my talk, I will review in detail the construction of these vacua as well as introduce a new decay channel, ultimately sealing their fate as being unstable.
- 0211/02/2021
Counting invariant curves on a Calabi-Yau threefold with an involution
Abstract: Gopakumar-Vafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a Calabi-Yau threefold. In this talk, I will give a general overview of two of the sheaf-theoretic approaches to defining these invariants: via stable pairs a la Pandharipande-Thomas (PT) and via perverse sheaves a la Maulik-Toda (MT). I will then outline a parallel theory of Gopakumar-Vafa invariants for a Calabi-Yau threefold X with an involution. They are integers n_beta(g,h) which give a virtual count of curves of genus g in the class beta which are invariant under the involution and whose quotient by the involution has genus h. I will give two definitions of n_beta(g,h) which are conjectured to be equivalent, one in terms of a version of PT theory, and one in terms of a version of MT theory. These invariants can be computed and the conjecture proved in the case where X=SxC where S is an Abelian or K3 surface with a symplectic involution. In these cases, the invariants are given by formulas expressed with Jacobi modular forms. In the case where S is an Abelian surface, the specialization of n_beta(g,h) to h=0 recovers the count of hyperelliptic curves on Abelian surfaces first computed by B-Oberdieck-Pandharipande-Yin. This is joint work with Stephen Pietromonaco.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero
Abstract: We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields. This is a joint work with J. Cao and R. Villaflor. (arXiv:2109.00587)
- 0311/03/2021
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Non-Invertible Duality Defects in 3+1 Dimensions
Speaker: Clay Cordova (U Chicago)
Title: Non-Invertible Duality Defects in 3+1 Dimensions
Abstract: For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions and determine the fusion rule. From modular invariance and a direct analysis of one-form symmetry-protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. By further assuming time-reversal symmetry, we find that the presence of certain duality defects implies that the low-energy phase has to be gapless unless the one-form symmetry is spontaneously broken. We give an explicit realization of this duality defect in the free Maxwell theory where the duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides.
When Computer Algebra Meets Satisfiability: A New Approach to Combinatorial Mathematics
Abstract: Solvers for the Boolean satisfiability (SAT) problem have been increasingly used to resolve problems in mathematics due to their excellent search algorithms. This talk will describe a new method for mathematical search that couples SAT solvers with computer algebra systems (CAS), thereby combining the expressiveness of CASs with the search power of SAT solvers. This paradigm has led to a number of results on long-standing mathematical questions such as the first computer-verifiable resolution of Lam’s problem and the discovery of a new infinite class of Williamson matrices.
- 0411/04/2021
Fusion Category Symmetries in Quantum Field Theory
Speaker: Yifan Wang (NYU)
Title: Fusion Category Symmetries in Quantum Field Theory
Abstract: Topological defects provide a modern perspective on symmetries in quantum field theory. They generalize the familiar inverti
ble symmetries described by groups to non-invertible symmetries described by fusion categories. Such generalized symmetries are ubiquitous in quantum field theory and provide new constraints on renormalization group flows and the IR phase diagram. In this talk I’ll review some recent progress in identifying and understanding fusion category symmetries in 1+1d conformal field theories. Time permitting, I’ll also comment on higher dimensional generalizations.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The stability of charged black holes
Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
11/4/21 CMSA Interdisciplinary Science Seminar
Title: Exploring Invertibility in Image Processing and Restoration
Abstract: Today’s smartphones have enabled numerous stunning visual effects from denoising to beautification, and we can share high-quality JPEG images easily on the internet, but it is still valuable for photographers and researchers to keep the original raw camera data for further post-processing (e.g., retouching) and analysis. However, the huge size of raw data hinders its popularity in practice, so can we almost perfectly restore the raw data from a compressed RGB image and thus avoid storing any raw data? This question leads us to design an invertible image signal processing pipeline. Then we further explore invertibility in other image processing and restoration tasks, including image compression, reversible image conversion (e.g., image-to-video conversion), and embedding novel views in a single JPEG image. We demonstrate that customized invertible neural networks are highly effective in these inherently non-invertible tasks.
- 0511/05/2021
The Greene-Plesser Construction Revisited
Member Seminar
Speaker: Chuck Doran
Title: The Greene-Plesser Construction Revisited
Abstract: The first known construction of mirror pairs of Calabi-Yau manifolds was the Greene-Plesser “quotient and resolve” procedure which applies to pencils of hypersurfaces in projective space. We’ll review this approach, uncover the hints it gives for some more general mirror constructions, and describe a brand-new variant that applies to pencils of hypersurfaces in Grassmannians. This last is joint work with Tom Coates and Elana Kalashnikov (arXiv:2110.0727).
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 0611/06/2021
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 3110/31/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 0111/01/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Bubble instability of mIIA on AdS_4 x S^6
Abstract: Recently, a set of non-supersymmetric AdS_4 vacua of massive type IIA string theory has been constructed. These vacua are perturbatively stable with respect to the full KK spectrum of type mIIA supergravity and furthermore, they are stable against a variety of non-perturbative decay channels. Hence, at this point, they represent a serious challenge to the AdS swampland conjecture. In my talk, I will review in detail the construction of these vacua as well as introduce a new decay channel, ultimately sealing their fate as being unstable.
- 0211/02/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Counting invariant curves on a Calabi-Yau threefold with an involution
Abstract: Gopakumar-Vafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a Calabi-Yau threefold. In this talk, I will give a general overview of two of the sheaf-theoretic approaches to defining these invariants: via stable pairs a la Pandharipande-Thomas (PT) and via perverse sheaves a la Maulik-Toda (MT). I will then outline a parallel theory of Gopakumar-Vafa invariants for a Calabi-Yau threefold X with an involution. They are integers n_beta(g,h) which give a virtual count of curves of genus g in the class beta which are invariant under the involution and whose quotient by the involution has genus h. I will give two definitions of n_beta(g,h) which are conjectured to be equivalent, one in terms of a version of PT theory, and one in terms of a version of MT theory. These invariants can be computed and the conjecture proved in the case where X=SxC where S is an Abelian or K3 surface with a symplectic involution. In these cases, the invariants are given by formulas expressed with Jacobi modular forms. In the case where S is an Abelian surface, the specialization of n_beta(g,h) to h=0 recovers the count of hyperelliptic curves on Abelian surfaces first computed by B-Oberdieck-Pandharipande-Yin. This is joint work with Stephen Pietromonaco.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero
Abstract: We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields. This is a joint work with J. Cao and R. Villaflor. (arXiv:2109.00587)
- 0311/03/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Non-Invertible Duality Defects in 3+1 Dimensions
Speaker: Clay Cordova (U Chicago)
Title: Non-Invertible Duality Defects in 3+1 Dimensions
Abstract: For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions and determine the fusion rule. From modular invariance and a direct analysis of one-form symmetry-protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. By further assuming time-reversal symmetry, we find that the presence of certain duality defects implies that the low-energy phase has to be gapless unless the one-form symmetry is spontaneously broken. We give an explicit realization of this duality defect in the free Maxwell theory where the duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides.
When Computer Algebra Meets Satisfiability: A New Approach to Combinatorial Mathematics
Abstract: Solvers for the Boolean satisfiability (SAT) problem have been increasingly used to resolve problems in mathematics due to their excellent search algorithms. This talk will describe a new method for mathematical search that couples SAT solvers with computer algebra systems (CAS), thereby combining the expressiveness of CASs with the search power of SAT solvers. This paradigm has led to a number of results on long-standing mathematical questions such as the first computer-verifiable resolution of Lam’s problem and the discovery of a new infinite class of Williamson matrices.
- 0411/04/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Fusion Category Symmetries in Quantum Field Theory
Speaker: Yifan Wang (NYU)
Title: Fusion Category Symmetries in Quantum Field Theory
Abstract: Topological defects provide a modern perspective on symmetries in quantum field theory. They generalize the familiar inverti
ble symmetries described by groups to non-invertible symmetries described by fusion categories. Such generalized symmetries are ubiquitous in quantum field theory and provide new constraints on renormalization group flows and the IR phase diagram. In this talk I’ll review some recent progress in identifying and understanding fusion category symmetries in 1+1d conformal field theories. Time permitting, I’ll also comment on higher dimensional generalizations.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The stability of charged black holes
Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
11/4/21 CMSA Interdisciplinary Science Seminar
Title: Exploring Invertibility in Image Processing and Restoration
Abstract: Today’s smartphones have enabled numerous stunning visual effects from denoising to beautification, and we can share high-quality JPEG images easily on the internet, but it is still valuable for photographers and researchers to keep the original raw camera data for further post-processing (e.g., retouching) and analysis. However, the huge size of raw data hinders its popularity in practice, so can we almost perfectly restore the raw data from a compressed RGB image and thus avoid storing any raw data? This question leads us to design an invertible image signal processing pipeline. Then we further explore invertibility in other image processing and restoration tasks, including image compression, reversible image conversion (e.g., image-to-video conversion), and embedding novel views in a single JPEG image. We demonstrate that customized invertible neural networks are highly effective in these inherently non-invertible tasks.
- 0511/05/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
The Greene-Plesser Construction Revisited
Member Seminar
Speaker: Chuck Doran
Title: The Greene-Plesser Construction Revisited
Abstract: The first known construction of mirror pairs of Calabi-Yau manifolds was the Greene-Plesser “quotient and resolve” procedure which applies to pencils of hypersurfaces in projective space. We’ll review this approach, uncover the hints it gives for some more general mirror constructions, and describe a brand-new variant that applies to pencils of hypersurfaces in Grassmannians. This last is joint work with Tom Coates and Elana Kalashnikov (arXiv:2110.0727).
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 0611/06/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 3110/31/2021
Quantum Matter Workshop
Please note: this workshop has been postponed to a later date. Details will be posted to this page when they are available.
Throughout the summer, scheduled speakers for the Quantum Matter Workshop will give talks on Zoom for the Quantum Matter/Condensed Matter seminar.
The CMSA will be hosting our second workshop on Quantum Matter. Both of these workshops are part of our program on Quantum Matter in Mathematics and Physics. The first workshop took place in December 2019, and was extremely successful, attracting participants worldwide. Learn more about the first workshop here.
Organizers: Du Pei, Ryan Thorngren, Juven Wang, Yifan Wang, and Shing-Tung Yau.
Speakers:
- Xiao Chen, Rutgers University
- Bert Halperin, Harvard Physics
- Daniel Harlow, MIT
- Michael Hopkins, Harvard Math
- Chang-Tse Hsieh, Kavli IMPU
- Philip Kim, Harvard Physics
- Ethan Lake, University of Utah
- Hotat Lam, Princeton University
- Mikhail Lukin, Harvard Physics
- Subir Sachdev, Harvard Physics
- Anders Sandvik, Boston University
- Nati Seiberg, IAS
- Husan Shapourian, Harvard
- Xue-Yang Song, Harvard
- Nat Tantivasadakarn, Harvard
- Juven Wang, CMSA
- Yifan Wang, CMSA
- Frank Wilczek, MIT
Big Data Conference 2021
On August 24, 2021, the CMSA hosted our seventh annual Conference on Big Data. The Conference features many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The 2021 Big Data Conference took place virtually on Zoom.
Organizers:
- Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University
- Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business
- Horng-Tzer Yau, Professor of Mathematics, Harvard University
- Sergiy Verstyuk, CMSA, Harvard University
Speakers:
- Andrew Blumberg, University of Texas at Austin
- Moran Koren, Harvard CMSA
- Hima Lakkaraju, Harvard University
- Katrina Ligett, The Hebrew University of Jerusalem
Time (ET; Boston time) Speaker Title/Abstract 9:00AM Conference Organizers Introduction and Welcome 9:10AM – 9:55AM Andrew Blumberg, University of Texas at Austin Title: Robustness and stability for multidimensional persistent homology Abstract: A basic principle in topological data analysis is to study the shape of data by looking at multiscale homological invariants. The idea is to filter the data using a scale parameter that reflects feature size. However, for many data sets, it is very natural to consider multiple filtrations, for example coming from feature scale and density. A key question that arises is how such invariants behave with respect to noise and outliers. This talk will describe a framework for understanding those questions and explore open problems in the area.
10:00AM – 10:45AM Katrina Ligett, The Hebrew University of Jerusalem Title: Privacy as Stability, for Generalization Abstract: Many data analysis pipelines are adaptive: the choice of which analysis to run next depends on the outcome of previous analyses. Common examples include variable selection for regression problems and hyper-parameter optimization in large-scale machine learning problems: in both cases, common practice involves repeatedly evaluating a series of models on the same dataset. Unfortunately, this kind of adaptive re-use of data invalidates many traditional methods of avoiding overfitting and false discovery, and has been blamed in part for the recent flood of non-reproducible findings in the empirical sciences. An exciting line of work beginning with Dwork et al. in 2015 establishes the first formal model and first algorithmic results providing a general approach to mitigating the harms of adaptivity, via a connection to the notion of differential privacy. In this talk, we’ll explore the notion of differential privacy and gain some understanding of how and why it provides protection against adaptivity-driven overfitting. Many interesting questions in this space remain open.
Joint work with: Christopher Jung (UPenn), Seth Neel (Harvard), Aaron Roth (UPenn), Saeed Sharifi-Malvajerdi (UPenn), and Moshe Shenfeld (HUJI). This talk will draw on work that appeared at NeurIPS 2019 and ITCS 2020
10:50AM – 11:35AM Hima Lakkaraju, Harvard University Title: Towards Reliable and Robust Model Explanations Abstract: As machine learning black boxes are increasingly being deployed in domains such as healthcare and criminal justice, there is growing emphasis on building tools and techniques for explaining these black boxes in an interpretable manner. Such explanations are being leveraged by domain experts to diagnose systematic errors and underlying biases of black boxes. In this talk, I will present some of our recent research that sheds light on the vulnerabilities of popular post hoc explanation techniques such as LIME and SHAP, and also introduce novel methods to address some of these vulnerabilities. More specifically, I will first demonstrate that these methods are brittle, unstable, and are vulnerable to a variety of adversarial attacks. Then, I will discuss two solutions to address some of the vulnerabilities of these methods – (i) a framework based on adversarial training that is designed to make post hoc explanations more stable and robust to shifts in the underlying data; (ii) a Bayesian framework that captures the uncertainty associated with post hoc explanations and in turn allows us to generate explanations with user specified levels of confidences. I will conclude the talk by discussing results from real world datasets to both demonstrate the vulnerabilities in post hoc explanation techniques as well as the efficacy of our aforementioned solutions.
11:40AM – 12:25PM Moran Koren, Harvard CMSA Title: A Gatekeeper’s Conundrum Abstract: Many selection processes contain a “gatekeeper”. The gatekeeper’s goal is to examine an applicant’s suitability to a proposed position before both parties endure substantial costs. Intuitively, the introduction of a gatekeeper should reduce selection costs as unlikely applicants are sifted out. However, we show that this is not always the case as the gatekeeper’s introduction inadvertently reduces the applicant’s expected costs and thus interferes with her self-selection. We study the conditions under which the gatekeeper’s presence improves the system’s efficiency and those conditions under which the gatekeeper’s presence induces inefficiency. Additionally, we show that the gatekeeper can sometimes improve selection correctness by behaving strategically (i.e., ignore her private information with some probability).
12:25PM Conference Organizers Closing Remarks Some remarks on contact Calabi-Yau 7-manifolds
Abstract: In geometry and physics it has proved useful to relate G2 and Calabi-Yau geometry via circle bundles. Contact Calabi-Yau 7-manifolds are, in the simplest cases, such circle bundles over Calabi-Yau 3-orbifolds. These 7-manifolds provide testing grounds for the study of geometric flows which seek to find torsion-free G2-structures (and thus Ricci flat metrics with exceptional holonomy). They also give useful backgrounds to examine the heterotic G2 system (also known as the G2-Hull-Strominger system), which is a coupled set of PDEs arising from physics that involves the G2-structure and gauge theory on the 7-manifold. I will report on recent progress on both of these directions in the study of contact Calabi-Yau 7-manifolds, which is joint work with H. Sá Earp and J. Saavedra.
CMSA Math-Science Literature Lecture: Kunihiko Kodaira and complex manifolds
Yujiro Kawamata (University of Tokyo)
Title: Kunihiko Kodaira and complex manifolds
Abstract: Kodaira’s motivation was to generalize the theory of Riemann surfaces in Weyl’s book to higher dimensions. After quickly recalling the chronology of Kodaira, I will review some of Kodaira’s works in three sections on topics of harmonic analysis, deformation theory and compact complex surfaces. Each topic corresponds to a volume of Kodaira’s collected works in three volumes, of which I will cover only tiny parts.
Talk chair: Baohua Fu
Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants for algebraic surfaces
During the Spring 2021 Semester Artan Sheshmani (CMSA/ I.M. A.U.) will be teaching a CMSA special lecture series on Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants for algebraic surfaces.
In order to attend this series, please fill out this form.
The lectures will be held Mondays from 8:00 – 9:30 AM ET and Wednesdays from 8:00 – 9:00 AM ET beginning January 25 on Zoom.
You can watch Prof. Sheshmani describe the series here.
CMSA Math-Science Literature Lecture: Michael Atiyah: Geometry and Physics
Nigel Hitchin (University of Oxford)
Title: Michael Atiyah: Geometry and Physics
Abstract: In mid-career, as an internationally renowned mathematician, Michael Atiyah discovered that some problems in physics responded to current work in algebraic geometry and this set him on a path to develop an active interface between mathematics and physics which was formative in the links which are so active today. The talk will focus, in a fairly basic fashion, on some examples of this interaction, which involved both applying physical ideas to solve mathematical problems and introducing mathematical ideas to physicists.
Talk chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: Is relativity compatible with quantum theory?
Arthur Jaffe (Harvard University)
Title: Is relativity compatible with quantum theory?
Abstract: We review the background, mathematical progress, and open questions in the effort to determine whether one can combine quantum mechanics, special relativity, and interaction together into one mathematical theory. This field of mathematics is known as “constructive quantum field theory.” Physicists believe that such a theory describes experimental measurements made over a 70 year period and now refined to 13-decimal-point precision—the most accurate experiments ever performed.
Talk chair: Zhengwei Liu
CMSA Math-Science Literature Lecture: Noncommutative Geometry, the Spectral Aspect
Alain Connes (Collège de France)
Title: Noncommutative Geometry, the Spectral Aspect
Abstract: This talk will be a survey of the spectral side of noncommutative geometry, presenting the new paradigm of spectral triples and showing its relevance for the fine structure of space-time, its large scale structure and also in number theory in connection with the zeros of the Riemann zeta function.
Talk chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: Homotopy spectra and Diophantine equations
Yuri Manin (Max Planck Institute for Mathematics)
Title: Homotopy spectra and Diophantine equations
Abstract: For a long stretch of time in the history of mathematics, Number Theory and Topology formed vast, but disjoint domains of mathematical knowledge. Origins of number theory can be traced back to the Babylonian clay tablet Plimpton 322 (about 1800 BC) that contained a list of integer solutions of the “Diophantine” equation $a^2+b^2=c^2$: archetypal theme of number theory, named after Diophantus of Alexandria (about 250 BC). Topology was born much later, but arguably, its cousin — modern measure theory, — goes back to Archimedes, author of Psammites (“Sand Reckoner”), who was approximately a contemporary of Diophantus. In modern language, Archimedes measures the volume of observable universe by counting the number of small grains of sand necessary to fill this volume. Of course, many qualitative geometric models and quantitative estimates of the relevant distances precede his calculations. Moreover, since the estimated numbers of grains of sand are quite large (about $10^{64}$), Archimedes had to invent and describe a system of notation for large numbers going far outside the possibilities of any of the standard ancient systems. The construction of the first bridge between number theory and topology was accomplished only about fifty years ago: it is the theory of spectra in stable homotopy theory. In particular, it connects $Z$, the initial object in the theory of commutative rings, with the sphere spectrum $S$. This connection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory. In this talk based upon the authors’ (Yu. Manin and M. Marcolli) joint research project, I suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds.
Talk chair: Michael Hopkins
CMSA Math-Science Literature Lecture: Log Calabi-Yau fibrations
Caucher Birkar (University of Cambridge)
Title: Log Calabi-Yau fibrations
Abstract: Fano and Calabi-Yau varieties play a fundamental role in algebraic geometry, differential geometry, arithmetic geometry, mathematical physics, etc. The notion of log Calabi-Yau fibration unifies Fano and Calabi-Yau varieties, their fibrations, as well as their local birational counterparts such as flips and singularities. Such fibrations can be examined from many different perspectives. The purpose of this talk is to introduce the theory of log Calabi-Yau fibrations, to remind some known results, and to state some open problems.
CMSA Math-Science Literature Lecture: Classical and quantum integrable systems in enumerative geometry
Andrei Okounkov (Columbia University)
Title: Classical and quantum integrable systems in enumerative geometry
Abstract: For more than a quarter of a century, thanks to the ideas and questions originating in modern high-energy physics, there has been a very fruitful interplay between enumerative geometry and integrable system, both classical and quantum. While it is impossible to summarize even the most important aspects of this interplay in one talk, I will try to highlight a few logical points with the goal to explain the place and the role of certain more recent developments.
Talk chair: Cumrun Vafa
Workshop on Quantum Information
The Center of Mathematical Sciences and Applications will be hosting a workshop on Quantum Information on April 23-24, 2018. In the days leading up to the conference, the American Mathematical Society will also be hosting a sectional meeting on quantum information on April 21-22. You can find more information here.
The following speakers are confirmed:
- Fernando G.S.L Brandão (CalTech)
- Jacob Biamonte (Skoltech)
- Isaac Chuang (MIT)
- Iris Cong (Harvard)
- Aram Harrow (MIT)
- Ke Li (HIT)
- Mikhail D. Lukin (Harvard)
- Shunlong Luo (AMSS)
- Renato Renner (ETH Zürich)
- Peter Shor (MIT)
CMSA Math-Science Literature Lecture: Knot Invariants From Gauge Theory in Three, Four, and Five Dimensions
Edward Witten (IAS)
Title: Knot Invariants From Gauge Theory in Three, Four, and Five Dimensions
Abstract: I will explain connections between a sequence of theories in two, three, four, and five dimensions and describe how these theories are related to the Jones polynomial of a knot and its categorification.
Talk chair: Cliff Taubes
F-Theory Conference
The CMSA will be hosting an F-Theory workshop September 29-30, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Click here for videos of the talks.
Organizers:
- Paolo Aluffi (Florida State)
- Lara B. Anderson (Virginia Tech)
- Mboyo Esole (Northeastern)
- Shing-Tung Yau (Harvard)
Speakers:
- Mirjam Cvetic, University of Pennsylvania
- Tommaso de Fernex, University of Utah
- James Gray, Virginia Tech
- Jonathan Heckman, University of Pennsylvania
- Monica Kang, Harvard University
- Sándor Kovács, University of Washington
- Anatoly Libgober, UIC
- Matilde Marcolli, Caltech, University of Toronto, and Perimeter Institute
- Washington Taylor, MIT
- Cumrun Vafa, Harvard University
Morphogenesis: Geometry and Physics
Just over a century ago, the biologist, mathematician and philologist D’Arcy Thompson wrote “On growth and form”. The book – a literary masterpiece – is a visionary synthesis of the geometric biology of form. It also served as a call for mathematical and physical approaches to understanding the evolution and development of shape. In the century since its publication, we have seen a revolution in biology following the discovery of the genetic code, which has uncovered the molecular and cellular basis for life, combined with the ability to probe the chemical, structural, and dynamical nature of molecules, cells, tissues and organs across scales. In parallel, we have seen a blossoming of our understanding of spatiotemporal patterning in physical systems, and a gradual unveiling of the complexity of physical form. So, how far are we from realizing the century-old vision that “Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed” ?
To address this requires an appreciation of the enormous ‘morphospace’ in terms of the potential shapes and sizes that living forms take, using the language of mathematics. In parallel, we need to consider the biological processes that determine form in mathematical terms is based on understanding how instabilities and patterns in physical systems might be harnessed by evolution.
In Fall 2018, CMSA will focus on a program that aims at recent mathematical advances in describing shape using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems.
The first workshop will focus on the interface between Morphometrics and Mathematics, while the second will focus on the interface between Morphogenesis and Physics.The workshop is organized by L. Mahadevan (Harvard), O. Pourquie (Harvard), A. Srivastava (Florida).As part of the program on Mathematical Biology a workshop on Morphogenesis: Geometry and Physics will take place on December 3-5, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Videos
Please Register Here
PDF of the Schedule
Speakers:
- Arkhat Abzhanov, Imperial College
- Yohanns Bellaiche, Paris
- Cheng Ming Chuong, USC
- Zev Gartner, UCSF
- Thomas Gregor, Princeton
- Dagmar Iber, Zurich
- Ian Jermyn, Durham University
- Raymond Keller, UVA
- Allon Klein, HMS
- Lisa Manning, Syracuse
- Cristina Marchetti, UCSB
- Sean Megason, HMS
- Elliot Meyerowitz, Caltech
- Michel Milinkovitch, Geneva
- Leonardo Morsut, USC
- Olivier Pourquié, HMS
- Eric Siggia, Rockefeller University
- Ben Simons, Cambridge
- Sebastian Streichan, UCSB
- Aryeh Warmflash, Rice
2019 Big Data Conference
1 Oxford Street, Cambridge MA 02138On August 19-20, 2019 the CMSA will be hosting our fifth annual Conference on Big Data. The Conference will feature many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The talks will take place in Science Center Hall D, 1 Oxford Street.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants.
Videos can be found in this Youtube playlist or in the schedule below.
Workshop on Foundations of Computational Science
On August 29-31, 2019 the Center of Mathematical Sciences and Applications will be hosting a workshop on Foundations of Computational Science. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by David Xianfeng Gu.
Please register here.
Speakers:
- Sarah Adel Bargal, Boston University
- Jianfeng Chen, Harvard
- Tat Seng Chua, National University of Singapore
- Ke Deng, Tsinghua
- David Xianfeng Gu, Stony Brook
- Yike Guo, Imperial College London
- Minlie Huang, Tsinghua
- Scott Kominers, Harvard
- Brian Kulis, Boston University
- Wee Sun Lee, National University of Singapore
- Qianxiao Li, National University of Singapore
- Hanzhong Liu, Tsinghua
- Jun Liu, Harvard
- Xiao-Li Meng, Harvard
- Cengiz Pehlevan, Harvard
- Donald Rubin, Harvard
- Suproteem Sarkar, Harvard
- Zuowei Shen, National University of Singapore
- Yuanchun Shi, Tsinghua
- Justin Solomon, MIT
- Hang Su, Tsinghua
- Maosong Sun, Tsinghua
- Mirac Suzgun, Harvard
- Sergiy Verstyuk, CMSA
- Xiaoqin Wang, Tsinghua
- Bin Xu, Tsinghua
- Jun Zhu, Tsinghua
- Wenwu Zhu, Tsinghua
Videos of the talks are contained in the Youtube playlist below. They can also be found through links in the schedule.
Angular momentum in general relativity
Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960′, due to the discovery of “supertranslation ambiguity”: the angular momentums recorded by two distant observers of the same system may not be the same. In this talk, I shall show how the mathematical theory of optimal isometric embedding and quasilocal angular momentum identifies a correction term, and leads to a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau.
From Algebraic Geometry to Vision and AI: A Symposium Celebrating the Mathematical Work of David Mumford
On August 18 and 20, 2018, the Center of Mathematic Sciences and Applications and the Harvard University Mathematics Department hosted a conference on From Algebraic Geometry to Vision and AI: A Symposium Celebrating the Mathematical Work of David Mumford. The talks took place in Science Center, Hall B.
Saturday, August 18th: A day of talks on Vision, AI and brain sciencesMonday, August 20th: a day of talks on MathSpeakers:
- Stuart Geman, Brown
- Janos Kollar, Princeton
- Tai Sing Lee, CMU
- Emanuele Macri, Northeastern
- Jitendra Malik, Berkeley / FAIR
- Peter Michor, University of Vienna
- Michael Miller, Johns Hopkins
- Aaron Pixton, MIT
- Jayant Shah, Northeastern
- Josh Tenenbaum, MIT
- Burt Totaro, UCLA
- Avi Wigderson, IAS
- Ying Nian Wu, UCLA
- Laurent Younes, Johns Hopkins
- Song-Chun Zhu, UCLA
Organizers:
- Ching-Li Chai, University of Pennsylvania
- David Gu, Stony Brook University
- Amnon Neeman, Australian National University
- Mark Nitzberg, University of California at Berkeley
- Yang Wang, Hong Kong University of Science and Technology
- Shing-Tung Yau, Harvard University
- Song-Chun Zhu, University of California, Los Angeles
Publication:
Pure and Applied Mathematics Quarterly
Special Issue: In Honor of David Mumford
Guest Editors: Ching-Li Chai, Amnon Neeman
Geometric Analysis Approach to AI Workshop
Due to inclement weather on Sunday, the second half of the workshop has been moved forward one day. Sunday and Monday’s talks will now take place on Monday and Tuesday.
On January 18-21, 2019 the Center of Mathematical Sciences and Applications will be hosting a workshop on the Geometric Analysis Approach to AI.
This workshop will focus on the theoretic foundations of AI, especially various methods in Deep Learning. The topics will cover the relationship between deep learning and optimal transportation theory, DL and information geometry, DL Learning and information bottle neck and renormalization theory, DL and manifold embedding and so on. Furthermore, the recent advancements, novel methods, and real world applications of Deep Learning will also be reported and discussed.
The workshop will take place from January 18th to January 23rd, 2019. In the first four days, from January 18th to January 21, the speakers will give short courses; On the 22nd and 23rd, the speakers will give conference representations. This workshop is organized by Xianfeng Gu and Shing-Tung Yau.
The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Please register here
Speakers:
- Sarah Adel Bargal, Boston University
- Guy Bresler, MIT
- Tina Eliassi-Rad, Northeastern
- Yun Raymond Fu, Northeastern
- Brian Kulis, Boston University
- Na Lei, Dalian University of Technology
- Yi Ma, UC Berkeley
- Minh Hoai Nguyen, Stony Brook
- Francesco Orabona, Boston University
- Cengiz Pehlevan, Harvard SEAS
- Tomaso Poggio, MIT
- Zhiwei Qin, DiDi Research America
- Kate Saenko, Boston University
- Dimitris Samaras, Stony Brook
- Johannes Schmidt-Hieber, University of Twente
- Steven Skiena, Stony Brook
- Vivienne Sze, MIT
- Naftali Tishby, ICNC
- Jiajun Wu, MIT
- Ying Nian Wu, UCLA
- Gangqiang Xia, Morgan Stanley
- Eric Xing, Carnegie Mellon
- Donghui Yan, UMass Dartmouth
- Alan Yuille, Johns Hopkins
- Juhua Zhu, Argus
Workshop on Aspects of General Relativity
The Center of Mathematical Sciences and Applications will be hosting a workshop on General Relativity from May 23 – 24, 2016. The workshop will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138. The workshop will start on Monday, May 23 at 9am and end on Tuesday, May 24 at 4pm.
Speakers:
- Po-Ning Chen, Columbia University
- Piotr T. Chruściel, University of Vienna
- Justin Corvino, Lafayette College
- Greg Galloway, University of Miami
- James Guillochon, Harvard University
- Lan-Hsuan Huang, University of Connecticut
- Dan Kapec, Harvard University
- Dan Lee, CUNY
- Alex Lupsasca, Harvard University
- Pengzi Miao, University of Miami
- Prahar Mitra, Harvard University
- Lorenzo Sironi, Harvard University
- Jared Speck, MIT
- Mu-Tao Wang, Columbia University
Please click Workshop Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Please click here for registration – Registration is capped at 70 participants.
Schedule:
May 23 – Day 1 8:30am Breakfast 8:55am Opening remarks 9:00am – 9:45am Greg Galloway, “Some remarks on photon spheres and their uniqueness“ 9:45am – 10:30am Prahar Mitra, “BMS supertranslations and Weinberg’s soft graviton theorem“ 10:30am – 11:00am Break 11:00am – 11:45am Dan Kapec, “Area, Entanglement Entropy and Supertranslations at Null Infinity“ 11:45am – 12:30pm Piotr T. Chruściel, “The cosmological constant and the energy of gravitational radiation” 12:30pm – 2:00pm Lunch 2:00pm – 2:45pm James Guillochon, “Tidal disruptions of stars by supermassive black holes: dynamics, light, and relics” 2:45pm – 3:30pm Mu-Tao Wang, “Quasi local conserved quantities in general relativity“ 3:30pm – 4:00pm Break 4:00pm – 4:45pm Po-Ning Chen, “Quasi local energy in presence of gravitational radiations” 4:45pm – 5:30pm Pengzi Miao, “Total mean curvature, scalar curvature, and a variational analog of Brown York mass“ May 24 – Day 2 8:45am Breakfast 9:00am – 9:45am Justin Corvino, “Scalar curvature deformation and the Bartnik mass“ 9:45am – 10:30am Lan-Hsuan Huang, “Constraint Manifolds with the Dominant Energy Condition“ 10:30am – 11:00am Break 11:00am – 11:45am Dan Lee, “Lower semicontinuity of Huisken’s isoperimetric mass“ 11:45am – 12:30pm Jared Speck, “Shock Formation in Solutions to the Compressible Euler Equations“ 12:30pm – 2:00pm Lunch 2:00pm – 2:45pm Lorenzo Sironi, “Electron Heating and Acceleration in the Vicinity of Supermassive Black Holes“ 2:45pm – 3:30pm Alex Lupsasca, “Near Horizon Extreme Kerr Magnetospheres“ * Click titles for talk videos. All videos are also available on “Harvard CMSA” channel on Youtube, grouped into playlist “Workshop on Aspects on General Relativity“.
* This event is sponsored by National Science Foundation (NSF) and CMSA Harvard University.
Organizers: Piotr T. Chruściel and Shing-Tung Yau
Big Data Conference 2018
1 Oxford Street, Cambridge MA 02138On August 23-24, 2018 the CMSA will be hosting our fourth annual Conference on Big Data. The Conference will feature many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
The talks will take place in Science Center Hall B, 1 Oxford Street.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants.
Confirmed Speakers:
- Mohammad Akbarpour, Stanford
- Emily Breza, Harvard
- Francesca Dominici, Harvard
- Chiara Farronato, Harvard
- Kobi Gal, Ben Gurion
- Jonah Kallenbach, Reverie Labs
- Samuel Kou, Harvard
- Laura Kreidberg, Harvard
- Danielle Li, MIT
- Libby Mishkin, Uber
- Josh Speagle, Harvard
- William Stein, University of Washington
- Alex Teyltelboym, University of Oxford
- Sergiy Verstyuk, CMSA/Harvard
Organizers:
- Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University
- Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business
- Richard Freeman, Herbert Ascherman Professor of Economics, Harvard University
- Jun Liu, Professor of Statistics, Harvard University
- Horng-Tzer Yau, Professor of Mathematics, Harvard University
Workshop on Morphometrics, Morphogenesis and Mathematics
In Fall 2018, the CMSA will host a Program on Mathematical Biology, which aims to describe recent mathematical advances in using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems.
The plethora of natural shapes that surround us at every scale is both bewildering and astounding – from the electron micrograph of a polyhedral virus, to the branching pattern of a gnarled tree to the convolutions in the brain. Even at the human scale, the shapes seen in a garden at the scale of a pollen grain, a seed, a sapling, a root, a flower or leaf are so numerous that “it is enough to drive the sanest man mad,” wrote Darwin. Can we classify these shapes and understand their origins quantitatively?
In biology, there is growing interest in and ability to quantify growth and form in the context of the size and shape of bacteria and other protists, to understand how polymeric assemblies grow and shrink (in the cytoskeleton), and how cells divide, change size and shape, and move to organize tissues, change their topology and geometry, and link multiple scales and connect biochemical to mechanical aspects of these problems, all in a self-regulated setting.
To understand these questions, we need to describe shape (biomathematics), predict shape (biophysics), and design shape (bioengineering).
For example, in mathematics there are some beautiful links to Nash’s embedding theorem, connections to quasi-conformal geometry, Ricci flows and geometric PDE, to Gromov’s h principle, to geometrical singularities and singular geometries, discrete and computational differential geometry, to stochastic geometry and shape characterization (a la Grenander, Mumford etc.). A nice question here is to use the large datasets (in 4D) and analyze them using ideas from statistical geometry (a la Taylor, Adler) to look for similarities and differences across species during development, and across evolution.
In physics, there are questions of generalizing classical theories to include activity, break the usual Galilean invariance, as well as isotropy, frame indifference, homogeneity, and create both agent (cell)-based and continuum theories for ordered, active machines, linking statistical to continuum mechanics, and understanding the instabilities and patterns that arise. Active generalizations of liquid crystals, polar materials, polymers etc. are only just beginning to be explored and there are some nice physical analogs of biological growth/form that are yet to be studied.
The CMSA will be hosting a Workshop on Morphometrics, Morphogenesis and Mathematics from October 22-24 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
The workshop is organized by L. Mahadevan (Harvard), O. Pourquie (Harvard), A. Srivastava (Florida).
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Videos of the talks
Confirmed Speakers:
- Arkhat Abzhanov, Imperial College
- Siobhan Braybrook, UCLA
- Cassandra Extavour, Harvard
- Anjali Goswami, University College London
- David Gu, Stony Brook
- Jukka Jernvall, Helsinki
- Eric Klassen, Florida State
- Sayan Mukherjee, Duke
- Peter Olver, U Minnesota
- Nipam Patel, Berkeley
- Stephanie Pierce, Harvard
- Karen Sears, UCLA
- Alain Trouve, ENS-Cachan, France
- Laurent Younes, Johns Hopkins
2015 Conference on Big Data
1 Oxford Street, Cambridge MA 02138The Center of Mathematical Sciences and Applications will be having a conference on Big Data August 24-26, 2015, in Science Center Hall B at Harvard University. This conference will feature many speakers from the Harvard Community as well as many scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
For more info, please contact Sarah LaBauve at slabauve@math.harvard.edu.
Registration for the conference is now closed.
Please click here for a downloadable version of this schedule.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found here.
Monday, August 24
Time Speaker Title 8:45am Meet and Greet 9:00am Sendhil Mullainathan Prediction Problems in Social Science: Applications of Machine Learning to Policy and Behavioral Economics 9:45am Mike Luca Designing Disclosure for the Digital Age 10:30 Break 10:45 Jianqing Fan Big Data Big Assumption: Spurious discoveries and endogeneity 11:30am Daniel Goroff Privacy and Reproducibility in Data Science 12:15pm Break for Lunch 2:00pm Ryan Adams Exact Markov Chain Monte Carlo with Large Data 2:45pm David Dunson Scalable Bayes: Simple algorithms with guarantees 3:30pm Break 3:45pm Michael Jordan Computational thinking, inferential thinking and Big Data 4:30pm Joel Tropp Applied Random Matrix Theory 5:15pm David Woodruff Input Sparsity and Hardness for Robust Subspace Approximation Tuesday, August 25
Time Speaker Title 8:45am Meet and Greet 9:00am Gunnar Carlsson Persistent homology for qualitative analysis and feature generation 9:45am Andrea Montanari Semidefinite Programming Relaxations for Graph and Matrix Estimation: Algorithms and Phase Transitions 10:30am Break 10:45am Susan Athey Machine Learning and Causal Inference for Policy Evaluation 11:30am Denis Nekipelov Robust Empirical Evaluation of Large Competitive Markets 12:15pm Break for Lunch 2:00pm Lucy Colwell Using evolutionary sequence variation to make inferences about protein structure and function: Modeling with Random Matrix Theory 2:45pm Simona Cocco Inverse Statistical Physics approaches for the modeling of protein families 3:30pm Break 3:45pm Remi Monasson Inference of top components of correlation matrices with prior informations 4:30pm Sayan Mukherjee Random walks on simplicial complexes and higher order notions of spectral clustering A Banquet from 7:00 – 8:30pm will follow Tuesday’s talks. This event is by invitation only.
Wednesday, August 26
Time Speaker Title 8:45am Meet and Greet 9:00am Ankur Moitra Beyond Matrix Completion 9:45am Florent Krzakala Optimal compressed sensing with spatial coupling and message passing 10:30am Break 10:45am Piotr Indyk Fast Algorithms for Structured Sparsity 11:30am Guido Imbens Exact p-values for network inference 12:15pm Break for lunch 2:00pm Edo Airoldi Some fundamental ideas for causal inference on large networks 2:45pm Ronitt Rubinfeld Something for almost nothing: sublinear time approximation algorithms 3:30pm Break 3:45pm Lenka Zdeborova Clustering of sparse networks: Phase transitions and optimal algorithms 4:30pm Jelani Nelson Dimensionality reductions via sparse matrices The number of n-queens configurations
Speaker: Michael Simkin, Harvard CMSA
Title: The number of n-queens configurations
Abstract: The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. The problem has a storied history and was studied by such eminent mathematicians as Gauss and Polya. The problem has also found applications in fields such as algorithm design and circuit development.
Despite much study, until recently very little was known regarding the asymptotics of Q(n). We apply modern methods from probabilistic combinatorics to reduce understanding Q(n) to the study of a particular infinite-dimensional convex optimization problem. The chief implication is that (in an appropriate sense) for a~1.94, Q(n) is approximately (ne^(-a))^n. Furthermore, our methods allow us to study the typical “shape” of n-queens configurations.
CMSA Math-Science Literature Lecture: Discrepancy Theory and Randomized Controlled Trials
Dan Spielman (Yale University)
Title: Discrepancy Theory and Randomized Controlled Trials
Abstract: Discrepancy theory tells us that it is possible to partition vectors into sets so that each set looks surprisingly similar to every other. By “surprisingly similar” we mean much more similar than a random partition. I will begin by surveying fundamental results in discrepancy theory, including Spencer’s famous existence proofs and Bansal’s recent algorithmic realizations of them. Randomized Controlled Trials are used to test the effectiveness of interventions, like medical treatments. Randomization is used to ensure that the test and control groups are probably similar. When we know nothing about the experimental subjects, uniform random assignment is the best we can do. When we know information about the experimental subjects, called covariates, we can combine the strengths of randomization with the promises of discrepancy theory. This should allow us to obtain more accurate estimates of the effectiveness of treatments, or to conduct trials with fewer experimental subjects. I will introduce the Gram-Schmidt Walk algorithm of Bansal, Dadush, Garg, and Lovett, which produces random solutions to discrepancy problems. I will then explain how Chris Harshaw, Fredrik Sävje, Peng Zhang, and I use this algorithm to improve the design of randomized controlled trials. Our Gram-Schmidt Walk Designs have increased accuracy when the experimental outcomes are correlated with linear functions of the covariates, and are comparable to uniform random assignments in the worst case.
Talk chair: Salil Vadhan
The n-queens problem
Abstract: The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n).
In this talk we’ll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost’ configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.
This is joint work with Peter Keevash.
CMSA Math-Science Literature Lecture: Quantum topology and new types of modularity
Don Zagier (Max Planck Institute for Mathematics and International Centre for Theoretical Physics)
Title: Quantum topology and new types of modularity
Abstract: The talk concerns two fundamental themes of modern 3-dimensional topology and their unexpected connection with a theme coming from number theory. A deep insight of William Thurston in the mid-1970s is that the vast majority of complements of knots in the 3-sphere, or more generally of 3-manifolds, have a unique metric structure as hyperbolic manifolds of constant curvature -1, so that 3-dimensional topology is in some sense not really a branch of topology at all, but of differential geometry. In a different direction, the work of Vaughan Jones and Ed Witten in the late 1980s gave rise to the field of Quantum Topology, in which new types of invariants of knot complements and 3-manifolds are introduced that have their origins in ideas coming from quantum field theory. These two themes then became linked by Kashaev’s famous Volume Conjecture, now some 25 years old, which says that the Kashaev invariant _N of a hyperbolic knot K (this is a quantum invariant defined for each positive integer N and whose values are algebraic numbers) grows exponentially as N tends to infinity with an exponent proportional to the hyperbolic volume of the knot complement. About 10 years ago, I was led by numerical experiments to the discovery that Kashaev’s invariant could be upgraded to an invariant having rational numbers as its argument (with the original invariant being the value at 1/N) and that the Volume Conjecture then became part of a bigger story saying that the new invariant has some sort of strange transformation property under the action x -> (ax+b)/(cx+d) of the modular group SL(2,Z) on the argument. This turned out to be only the beginning of a fascinating and multi-faceted story relating quantum invariants, q-series, modularity, and many other topics. In the talk, which is intended for a general mathematical audience, I would like to recount some parts of this story, which is joint work with Stavros Garoufalidis (and of course involving contributions from many other authors). The “new types of modularity” in the title refer to a specific byproduct of these investigations, namely that there is a generalization of the classical notion of holomorphic modular form – which plays an absolutely central role in modern number theory – to a new class of holomorphic functions in the upper half-plane that no longer satisfy a transformation law under the action of the modular group, but a weaker extendability property instead. This new class, called “holomorphic quantum modular forms”, turns out to contain many other functions of a more number-theoretical nature as well as the original examples coming from quantum invariants.
Talk chair: Mark Kisin
Machine Learning for Multiscale Model Reduction Workshop
The Machine Learning for Multiscale Model Reduction Workshop will take place on March 27-29, 2019. This is the second of two workshops organized by Michael Brenner, Shmuel Rubinstein, and Tom Hou. The first, Fluid turbulence and Singularities of the Euler/ Navier Stokes equations, will take place on March 13-15, 2019. Both workshops will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Speakers:
- Joan Bruna, Courant Institute
- Predrag Cvitanovic, Georgia Tech
- Stephan Hoyer, Google Research
- De Huang, Caltech
- George Karniadakis, Brown University
- Richard Kerswell, Cambridge University
- Stephane Mallat, ENS
- Stanley Osher, UCLA
- Jacob Page, Cambridge University
- Houman Owhadi, Caltech
- Zuowei Shen, National University of Singapore
- Jack Xin, UC Irvine
- Jinchao Xu, Penn State University
- Lexing Ying, Stanford University and Facebook AI Research
- Pengchuan Zhang, Microsoft Research
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Workshop on Coding and Information Theory
The workshop on coding and information theory will take place April 9-13, 2018 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
This workshop will focus on new developments in coding and information theory that sit at the intersection of combinatorics and complexity, and will bring together researchers from several communities — coding theory, information theory, combinatorics, and complexity theory — to exchange ideas and form collaborations to attack these problems.
Squarely in this intersection of combinatorics and complexity, locally testable/correctable codes and list-decodable codes both have deep connections to (and in some cases, direct motivation from) complexity theory and pseudorandomness, and recent progress in these areas has directly exploited and explored connections to combinatorics and graph theory. One goal of this workshop is to push ahead on these and other topics that are in the purview of the year-long program. Another goal is to highlight (a subset of) topics in coding and information theory which are especially ripe for collaboration between these communities. Examples of such topics include polar codes; new results on Reed-Muller codes and their thresholds; coding for distributed storage and for DNA memories; coding for deletions and synchronization errors; storage capacity of graphs; zero-error information theory; bounds on codes using semidefinite programming; tensorization in distributed source and channel coding; and applications of information-theoretic methods in probability and combinatorics. All these topics have attracted a great deal of recent interest in the coding and information theory communities, and have rich connections to combinatorics and complexity which could benefit from further exploration and collaboration.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
Click here for a list of registrants.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Confirmed participants include:
- Emmanuel Abbe, Princeton University
- Simeon Ball, Universitat Politècnica de Catalunya
- Boris Bukh, Carnegie Mellon University
- Mahdi Cheraghchi, Imperial College London
- Sivakanth Gopi, Princeton University
- Elena Grigorescu, University of Purdue
- Hamed Hassani, University of Pennsylvania
- Navin Kashyap, Indian Institute of Science
- Young-Han Kim, University of California, San Diego
- Swastik Kopparty, Rutgers University
- Nati Linial, Hebrew University of Jerusalem
- Shachar Lovett, University of California, San Diego
- William Martin, Worcester Polytechnic Institute
- Arya Mazumdar, University of Massachusetts at Amherst
- Or Meir, University of Haifa
- Olgica Milenkovic, ECE Illinois
- Chandra Nair, Chinese University of Hong Kong
- Yuval Peres, Microsoft Research
- Yury Polyanskiy, Massachusetts Institute of Technology
- Maxim Raginsky, University of Illinois at Urbana-Champaign
- Sankeerth Rao Karingula, UC San Diego
- Ankit Singh Rawat, MIT
- Noga Ron-Zewi, University of Haifa
- Ron Roth, Israel Institute of Technology
- Atri Rudra, State University of New York, Buffalo
- Alex Samorodnitsky, Hebrew University of Jerusalem
- Itzhak Tamo, Tel Aviv University
- Amnon Ta-Shma, Tel Aviv University
- Himanshu Tyagi, Indian Institute of Science
- David Zuckerman, University of Texas at Austin
CMSA Math-Science Literature Lecture: Area-minimizing integral currents and their regularity
Camillo De Lellis (IAS)
Title: Area-minimizing integral currents and their regularity
Abstract: Caccioppoli sets and integral currents (their generalization in higher codimension) were introduced in the late fifties and early sixties to give a general geometric approach to the existence of area-minimizing oriented surfaces spanning a given contour. These concepts started a whole new subject which has had tremendous impacts in several areas of mathematics: superficially through direct applications of the main theorems, but more deeply because of the techniques which have been invented to deal with related analytical and geometrical challenges. In this lecture I will review the basic concepts, the related existence theory of solutions of the Plateau problem, and what is known about their regularity. I will also touch upon several fundamental open problems which still defy our understanding.
Talk Chair: William Minicozzi
CMSA Math-Science Literature Lecture: From Deep Learning to Deep Understanding
Harry Shum (Tsinghua University)
Title: From Deep Learning to Deep Understanding
Abstract: In this talk I will discuss a couple of research directions for robust AI beyond deep neural networks. The first is the need to understand what we are learning, by shifting the focus from targeting effects to understanding causes. The second is the need for a hybrid neural/symbolic approach that leverages both commonsense knowledge and massive amount of data. Specifically, as an example, I will present some latest work at Microsoft Research on building a pre-trained grounded text generator for task-oriented dialog. It is a hybrid architecture that employs a large-scale Transformer-based deep learning model, and symbol manipulation modules such as business databases, knowledge graphs and commonsense rules. Unlike GPT or similar language models learnt from data, it is a multi-turn decision making system which takes user input, updates the belief state, retrieved from the database via symbolic reasoning, and decides how to complete the task with grounded response.
Talk chair: Shing-Tung Yau
CMSA Math-Science Literature Lecture: Hodge structures and the topology of algebraic varieties
Claire Voisin (Collège de France)
Title: Hodge structures and the topology of algebraic varieties
Abstract: We review the major progress made since the 50’s in our understanding of the topology of complex algebraic varieties. Most of the results we will discuss rely on Hodge theory, which has some analytic aspects giving the Hodge and Lefschetz decompositions, and the Hodge-Riemann relations. We will see that a crucial ingredient, the existence of a polarization, is missing in the general Kaehler context. We will also discuss some results and problems related to algebraic cycles and motives.
Talk chair: Joe Harris
Workshop on Optimization in Image Processing
The Center of Mathematical Sciences and Applications will be hosting a workshop on Optimization in Image Processing on June 27 – 30, 2016. This 4-day workshop aims to bring together researchers to exchange and stimulate ideas in imaging sciences, with a special focus on new approaches based on optimization methods. This is a cutting-edge topic with crucial impact in various areas of imaging science including inverse problems, image processing and computer vision. 16 speakers will participate in this event, which we think will be a very stimulating and exciting workshop. The workshop will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
Titles, abstracts and schedule will be provided nearer to the event.
Speakers:
- Antonin Chambolle, CMAP, Ecole Polytechnique
- Raymond Chan, The Chinese University of Hong Kong
- Ke Chen, University of Liverpool
- Patrick Louis Combettes, Université Pierre et Marie Curie
- Mario Figueiredo, Instituto Superior Técnico
- Alfred Hero, University of Michigan
- Ronald Lok Ming Lui, The Chinese University of Hong Kong
- Mila Nikolova, Ecole Normale Superieure Cachan
- Shoham Sabach, Israel Institute of Technology
- Martin Benning, University of Cambridge
- Jin Keun Seo, Yonsei University
- Fiorella Sgallari, University of Bologna
- Gabriele Steidl, Kaiserslautern University of Technology
- Joachim Weickert, Saarland University
- Isao Yamada, Tokyo Institute of Technology
- Wotao Yin, UCLA
Please click Workshop Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Please click here for registration – Registration Deadline: June 7, 2016; Registration is capped at 70 participants.
Schedule:
June 27 – Day 1 9:00am Breakfast 9:20am Opening remarks 9:30am – 10:20am Joachim Weickert, “FSI Schemes: Fast Semi-Iterative Methods for Diffusive or Variational Image Analysis Problems” 10:20am – 10:50am Break 10:50am – 11:40pm Patrick Louis Combettes, “Block-Iterative Asynchronous Variational Image Recovery” 11:40am – 12:30pm Isao Yamada, “Spicing up Convex Optimization for Certain Inverse Problems” 12:30pm – 2:00pm Lunch 2:30pm – 3:20pm Fiorella Sgallari, “Majorization-Minimization for Nonconvex Optimization” 3:20pm – 3:50pm Break 3:50pm – 4:40pm Shoham Sabach, “A Framework for Globally Convergent Methods in Nonsmooth and Nonconvex Problems” June 28 – Day 2 9:00am Breakfast 9:30am – 10:20am Antonin Chambolle, “Acceleration of alternating minimisations” 10:20am – 10:50am Break 10:50am – 11:40am Mario Figueiredo, “ADMM in Image Restoration and Related Problems: Some History and Recent Advances” 11:40am – 12:30pm Ke Chen, “Image Restoration and Registration Based on Total Fractional-Order Variation Regularization” 12:30pm – 2:30pm Lunch 2:30pm – 4:40pm Discussions June 29 – Day 3 9:00am Breakfast 9:30am – 10:20am Alfred Hero, “Continuum relaxations for discrete optimization” 10:20am – 10:50am Break 10:50am – 11:40am Wotao Yin, “Coordinate Update Algorithms for Computational Imaging and Machine Learning” 11:40am – 12:30pm Mila Nikolova, “Limits on noise removal using log-likelihood and regularization” 12:30pm – 2:30pm Lunch 2:30pm – 3:20pm Martin Benning, “Nonlinear spectral decompositions and the inverse scale space method” 3:20pm – 3:50pm Break 3:50pm – 4:40pm Ronald Ming Lui, “TEMPO: Feature-endowed Teichmuller extremal mappings of point cloud for shape classification” June 30 – Day 4 9:00am Breakfast 9:30am – 10:20am Jin Keun Seo, “Mathematical methods for biomedical impedance imaging” 10:20am – 10:50am Break 10:50am – 11:40am Gabriele Steidl, “Iterative Multiplicative Filters for Data Labeling” 11:40am – 12:30pm Raymond Chan, “Point-spread function reconstruction in ground-based astronomy” * This event is sponsored by CMSA Harvard University.
Organizers: Raymond Chan and Shing-Tung Yau
CMSA Math-Science Literature Lecture: Moment maps and the Yang-Mills functional
Frances Kirwan (University of Oxford)
Title: Moment maps and the Yang-Mills functional
Abstract: In the early 1980s Michael Atiyah and Raoul Bott wrote two influential papers, ‘The Yang-Mills equations over Riemann surfaces’ and ‘The moment map and equivariant cohomology’, bringing together ideas ranging from algebraic and symplectic geometry through algebraic topology to mathematical physics and number theory. The aim of this talk is to explain their key insights and some of the new directions towards which these papers led.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
Talk chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: Quantum error correcting codes and fault tolerance
Peter Shor (MIT)
Title: Quantum error correcting codes and fault tolerance
Abstract: We will go over the fundamentals of quantum error correction and fault tolerance and survey some of the recent developments in the field.
Talk chair: Zhengwei LiuWorkshop on Probabilistic and Extremal Combinatorics
The workshop on Probabilistic and Extremal Combinatorics will take place February 5-9, 2018 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
Extremal and Probabilistic Combinatorics are two of the most central branches of modern combinatorial theory. Extremal Combinatorics deals with problems of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects satisfying certain requirements. Such problems are often related to other areas including Computer Science, Information Theory, Number Theory and Geometry. This branch of Combinatorics has developed spectacularly over the last few decades. Probabilistic Combinatorics can be described informally as a (very successful) hybrid between Combinatorics and Probability, whose main object of study is probability distributions on discrete structures.
There are many points of interaction between these fields. There are deep similarities in methodology. Both subjects are mostly asymptotic in nature. Quite a few important results from Extremal Combinatorics have been proven applying probabilistic methods, and vice versa. Such emerging subjects as Extremal Problems in Random Graphs or the theory of graph limits stand explicitly at the intersection of the two fields and indicate their natural symbiosis.
The symposia will focus on the interactions between the above areas. These topics include Extremal Problems for Graphs and Set Systems, Ramsey Theory, Combinatorial Number Theory, Combinatorial Geometry, Random Graphs, Probabilistic Methods and Graph Limits.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Confirmed participants include:
- Jozsef Balogh, University of Illinois, Urbana
- Fan Chung (Graham), University of California, San Diego
- Asaf Ferber, Massachusetts Institute of Technology
- Jacob Fox, Stanford Unviersity
- David Gamarnik, Massachusetts Institute of Technology
- Penny Haxell, University of Waterloo
- Hao Huang, Emory University
- Jeff Kahn, Rutgers University
- Peter Keevash, Oxford University
- Michael Krivelevich, Tel Aviv University
- Daniela Kühn, University of Birmingham
- Shoham Letzer, ITS Zürich
- Shachar Lovett, University of California, San Diego
- Eyal Lubetzky, Courant Institute
- Rob Morris, IMPA
- Bhargav Narayanan, Rutgers University
- Deryk Osthus, University of Birmingham
- Janos Pach, NYU
- Yuval Peres, Microsoft Redmond
- Alexey Pokryovskyi, ETH Zürich
- Wojciech Samotij, Tel Aviv University
- Lisa Sauermann, Stanford University
- Mathias Schacht, University of Hamburg
- Alexander Scott, University of Oxford
- Asaf Shapira, Tel Aviv University
- Jozef Skokan, London School of Economics
- Joel Spencer, New York University
- Angelika Steger, ETH Zurich
- Jacques Verstraete, University of California, San Diego
- Yufei Zhao, Massachusetts Institute of Technology
- David Zuckerman, University of Texas at Austin
Co-organizers of this workshop include Benny Sudakov and David Conlon. More details about this event, including participants, will be updated soon.
CMSA Math-Science Literature Lecture: Isadore Singer’s Work on Analytic Torsion
Edward Witten (IAS)
Title: Isadore Singer’s Work on Analytic Torsion
Abstract: I will review two famous papers of Ray and Singer on analytic torsion written approximately half a century ago. Then I will sketch the influence of analytic torsion in a variety of areas of physics including anomalies, topological field theory, and string theory.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer.
Talk chair: Cumrun Vafa
CMSA Math-Science Literature Lecture: Homological (homotopical) algebra and moduli spaces in Topological Field theories
Kenji Fukaya (Simons Center for Geometry and Physics)
Title: Homological (homotopical) algebra and moduli spaces in Topological Field theories
Abstract: Moduli spaces of various gauge theory equations and of various versions of (pseudo) holomorphic curve equations have played important role in geometry in these 40 years. Started with Floer’s work people start to obtain more sophisticated object such as groups, rings, or categories from (system of) moduli spaces. I would like to survey some of those works and the methods to study family of moduli spaces systematically.
Talk chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: The Atiyah-Singer Index Theorem
Dan Freed (The University of Texas at Austin)
Title: The Atiyah-Singer Index Theorem
Abstract: The story of the index theorem ties together the Gang of Four—Atiyah, Bott, Hirzebruch, and Singer—and lies at the intersection of analysis, geometry, and topology. In the first part of the talk I will recount high points in the early developments. Then I turn to subsequent variations and applications. Throughout I emphasize the role of the Dirac operator.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
Talk chair: Cumrun Vafa
Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer
In 2021, the CMSA hosted a lecture series on the literature of the mathematical sciences. This series highlights significant accomplishments in the intersection between mathematics and the sciences. Speakers include Edward Witten, Lydia Bieri, Simon Donaldson, Michael Freedman, Dan Freed, and many more.
Videos of these talks can be found in this Youtube playlist.
https://youtu.be/vb_JEhUW9t4
In the Spring 2021 semester, the CMSA hosted a sub-program on this series titled A Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer. Below is the schedule for talks in that subprogram
April 6, 2021 | 9:00 – 10:30am ET
Edward Witten (IAS) April 13, 2021 | 9:00 – 10:30am ET
Claire Voisin (College de France) April 20, 2021 | 9:00 – 10:30am ET
Dan Freed (the University of Texas at Austin) April 27, 2021 | 9:00 – 10:30am ET
Frances Kirwan (University of Oxford) 10/12/2021 Combinatorics, Physics and Probability Seminar
Title: On counting algebraically defined graphs
Abstract: For many classes of graphs that arise naturally in discrete geometry (for example intersection graphs of segments or disks in the plane), the edges of these graphs can be defined algebraically using the signs of a finite list of fixed polynomials. We investigate the number of n-vertex graphs in such an algebraically defined class of graphs. Warren’s theorem (a variant of a theorem of Milnor and Thom) implies upper bounds for the number of n-vertex graphs in such graph classes, but all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving a lower bound for this number (under some reasonable assumptions on the fixed list of polynomials), and this lower bound essentially matches the upper bound from Warren’s theorem.
Simons Collaboration Workshop, Jan. 10-13, 2018
The CMSA will be hosting a four-day Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory on January 10-13, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Please click here to register for this event. We have space for up to 30 registrants on a first come, first serve basis.
We may be able to provide some financial support for grad students and postdocs interested in this event. If you are interested in funding, please send a letter of support from your mentor to Hansol Hong at hansol84@gmail.com.Confirmed Participants:
- Mohammed Abouzaid (Columbia University)
- Sergueï Barannikov (Paris Diderot University)
- Cheol-Hyun Cho (Seoul National University)
- Young-Hoon Kiem (Seoul National University)
- Thomas Lam (University of Michigan)
- Siu-Cheong Lau (Boston University)
- Radu Laza (Stony Brook University)
- Si Li (Tsinghua University)
- Kaoru Ono (Kyoto University)
- Tony Pantev (University of Pennsylvania)
- Colleen Robles (Duke University)
- Yan Soibelman (Kansas State University)
- Kazushi Ueda (University of Tokyo)
- Chenglong Yu (Harvard University)
- Eric Zaslow (Northwestern University)
CMSA Math-Science Literature Lecture: On the History of quantum cohomology and homological mirror symmetry
Maxim Kontsevich (IHÉS)
Title: On the History of quantum cohomology and homological mirror symmetry
Abstract: About 30 years ago, string theorists made remarkable discoveries of hidden structures in algebraic geometry. First, the usual cup-product on the cohomology of a complex projective variety admits a canonical multi-parameter deformation to so-called quantum product, satisfying a nice system of differential equations (WDVV equations). The second discovery, even more striking, is Mirror Symmetry, a duality between families of Calabi-Yau varieties acting as a mirror reflection on the Hodge diamond.
Later it was realized that the quantum product belongs to the realm of symplectic geometry, and a half of mirror symmetry (called Homological Mirror Symmetry) is a duality between complex algebraic and symplectic varieties. The search of correct definitions and possible generalizations lead to great advances in many domains, giving mathematicians new glasses, through which they can see familiar objects in a completely new way.
I will review the history of major mathematical advances in the subject of HMS, and the swirl of ideas around it.
Talk chair: Paul Seidel
Quantum Matter Workshop
CMSA, 20 Garden Street, Cambridge, MA 02138 USAOn December 2-4, 2019 the CMSA will be hosting a workshop on Quantum Matter as part of our program on Quantum Matter in Mathematics and Physics. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Pictures can be found here.
Organizers: Juven Wang (CMSA), Xiao-Gang Wen (MIT), and Shing-Tung Yau (Harvard)
Confirmed Speakers:
- Zhen Bi, MIT | Video
- Claudio Chamon, BU | Video
- Trithep Devakul, Princeton | Video
- Anushya Chandran, BU
- Liang Fu, MIT
- Andrey Gromov, Brown | Video
- Daniel Louis Jafferis, Harvard | Video
- Eslam Khalaf, Harvard | Video
- Hong Liu, MIT
- Shang Liu, Harvard | Video
- Emil Prodan, Yeshiva | Video
- Subir Sachdev, Harvard | Video
- Dries Sels, Harvard | Video
- Yuya Tanizaki, NCSU | Video
- Senthil Todadri, MIT | Video
- Juven Wang, CMSA | Video
- Yifan Wang, CMSA | Video
- Xiao-Gang Wen, MIT
- Xueda Wen, MIT | Video
- Xi Yin, Harvard | Video
- Yizhi You, Princeton | Video
- Yunqin Zheng, Princeton | Video
Mini-school on Nonlinear Equations, December 3-4, 2016
The Center of Mathematical Sciences and Applications will be hosting a Mini-school on Nonlinear Equations on December 3-4, 2016. The conference will have speakers and will be hosted at Harvard CMSA Building: Room G10 20 Garden Street, Cambridge, MA 02138.
The mini-school will consist of lectures by experts in geometry and analysis detailing important developments in the theory of nonlinear equations and their applications from the last 20-30 years. The mini-school is aimed at graduate students and young researchers working in geometry, analysis, physics and related fields.
Please click here to register for this event.
Speakers:
- Cliff Taubes (Harvard University)
- Valentino Tosatti (Northwestern University)
- Pengfei Guan (McGill University)
- Jared Speck (MIT)
Schedule:
December 3rd – Day 1 9:00am – 10:30am Cliff Taubes, “Compactness theorems in gauge theories” 10:45am – 12:15pm Valentino Tosatti, “Complex Monge-Ampère Equations” 12:15pm – 1:45pm LUNCH 1:45pm – 3:15pm Pengfei Guan, “Monge-Ampère type equations and related geometric problems” 3:30pm – 5:00pm Jared Speck, “Finite-time degeneration of hyperbolicity without blowup for solutions to quasilinear wave equations” December 4th – Day 2 9:00am – 10:30am Cliff Taubes, “Compactness theorems in gauge theories” 10:45am – 12:15pm Valentino Tosatti, “Complex Monge-Ampère Equations” 12:15pm – 1:45pm LUNCH 1:45pm – 3:15pm Pengfei Guan, “Monge-Ampère type equations and related geometric problems” 3:30pm – 5:00pm Jared Speck, “Finite-time degeneration of hyperbolicity without blowup for solutions to quasilinear wave equations” Please click Mini-School Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
* This event is sponsored by National Science Foundation (NSF) and CMSA Harvard University.
Simons Collaboration on Homological Mirror Symmetry
The Center of Mathematical Sciences and Applications will be hosting a 3-day workshop on Homological Mirror Symmetry and related areas on May 6 – May 8, 2016 at Harvard CMSA Building: Room G10 20 Garden Street, Cambridge, MA 02138
Organizers:
D. Auroux, S.C. Lau, N.C. Leung, Bong Lian, C.C. Liu, S.T. Yau
Speakers:
- Netanel Blaier (MIT)
- Kwokwai Chan (CUHK)
- Bohan Fang (Peking University)
- Amanda Francis (BYU)
- Hansol Hong (CUHK)
- Heather Lee (Purdue University)
- Si Li (Tsinghua University)
- Yu-Shen Lin (Stanford University)
- Alex Perry (Harvard University)
- Hiro Tanaka (Harvard University)
- Sara Tukachinsky (HUJ)
- Michael Viscardi (MIT)
- Eric Zaslow (Northwestern University)
- Jingyu Zhao (Columbia University)
Please click here for the conference Main Website.
Please click Simons Workshop Schedule with Abstract for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Schedule:
May 6 – Day 1 9:00am Breakfast 9:35am Opening remarks 9:45am – 10:45am Si Li, “Quantum master equation, chiral algebra, and integrability” 10:45am – 11:15am Break 11:15am – 12:15pm Sara Tukachinsky, “Point like bounding chains and open WDVV“ 12:15pm – 1:45pm Lunch 1:45pm – 2:45pm Bohan Fang, “Mirror B model for toric Calabi Yau 3 folds“ 2:45pm – 3:00pm Break 3:00pm – 4:00pm Hiro Tanaka, “Toward Fukaya categories over arbitrary coefficients“ 4:00pm – 4:15pm Break 4:15pm – 5:15pm Hansol Hong, “Noncommutative mirror functors“ May 7 – Day 2 9:00am Breakfast 9:45am – 10:45am Eric Zaslow, “Lagrangian fillings what does the sheaf say?“ 10:45am – 11:15am Break 11:15am – 12:15pm Alex Perry, “Derived categories of Gushel Mukai varieties“ 12:15pm – 1:45pm Lunch 1:45pm – 2:45pm Amanda Francis, “A Landau Ginzburg mirror theorem inspired by Borcea Voisin symmetry“ 2:45pm – 3:00pm Break 3:00pm – 4:00pm Heather Lee, “Homological mirror symmetry for open Riemann surfaces from pair of pants decompositions“ 4:00pm – 4:15pm Break 4:15pm – 5:15pm Yu-Shen Lin, “Counting Holomorphic Discs via Tropical Discs on K3 Surfaces“ May 8 – Day 3 9:00am Breakfast 9:45am – 10:45am Kwokwai Chan, “HMS for local CY manifolds via SYZ“ 10:45am – 11:15am Break 11:15am – 12:15pm Netanel Blaier, “The quantum Johnson homomorphism, formality and symplectic isotopy“ 12:15pm – 1:45pm Lunch 1:45pm – 2:45pm Jingyu Zhao, “Periodic symplectic cohomology and the Hodge filtration“ 2:45pm – 3:00pm Break 3:00pm – 4:00pm Michael Viscardi, “Equivariant quantum cohomology and the geometric Satake equivalence“ * Click titles for talk videos. All videos are also available on “Harvard CMSA” channel on Youtube, grouped into playlist “Simons Collaboration on Homological Mirror symmetry“.
This event is sponsored by the Simons Foundation and CMSA Harvard University.
10/5/2021 Combinatorics, Physics and Probability Seminar
Title: Geodesic Geometry on Graphs
Abstract: In a graph G = (V, E) we consider a system of paths S so that for every two vertices u,v in V there is a unique uv path in S connecting them. The path system is said to be consistent if it is closed under taking subpaths, i.e. if P is a path in S then any subpath of P is also in S. Every positive weight function w: E–>R^+ gives rise to a consistent path system in G by taking the paths in S to be geodesics w.r.t. w. In this case, we say w induces S. We say a graph G is metrizable if every consistent path system in G is induced by some such w.
We’ll discuss the concept of graph metrizability, and, in particular, we’ll see that while metrizability is a rare property, there exists infinitely many 2-connected metrizable graphs.
Joint work with Nati Linial.
10/19/2021 Combinatorics, Physics and Probability Seminar
Title: Ising model, total positivity, and criticality
Abstract: The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality.
The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig’s theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes.
I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with P. Pylyavskyy. Building on this connection, I will give a new boundary correlation formula for the critical Ising modelCMSA Math-Science Literature Lecture: Theorems of Torelli type
Eduard Jacob Neven Looijenga (Tsinghua University & Utrecht University)
Title: Theorems of Torelli type
Abstract: Given a closed manifold of even dimension 2n, then Hodge showed around 1950 that a kählerian complex structure on that manifold determines a decomposition of its complex cohomology. This decomposition, which can potentially vary continuously with the complex structure, extracts from a non-linear given, linear data. It can contain a lot of information. When there is essentially no loss of data in this process, we say that the Torelli theorem holds. We review the underlying theory and then survey some cases where this is the case. This will include the classical case n=1, but the emphasis will be on K3 manifolds (n=2) and more generally, on hyperkählerian manifolds. These cases stand out, since one can then also tell which decompositions occur.
Talk chair: Gerard van der Geer
Workshop on Geometry, Imaging, and Computing
On March 24-26, The Center of Mathematical Sciences and Applications will be hosting a workshop on Geometry, Imaging, and Computing, based off the journal of the same name. The workshop will take place in CMSA building, G10.
The organizing committee consists of Yang Wang (HKUST), Ronald Lui (CUHK), David Gu (Stony Brook), and Shing-Tung Yau (Harvard).
Please click here to register for the event.
Confirmed Speakers:
- Jianfeng Cai (HKUST)
- Shikui Chen (Stony Brook)
- Jerome Darbon (Brown University)
- Laurent Demanet (MIT)
- David Gu (Stony Brook)
- Monica Hurdal (Florida State University)
- Rongjie Lai (RPI)
- Yue Lu (Harvard)
- Ronald Lok Ming Lui (CUHK)
- Lakshminarayanan Mahadevan (Harvard)
- Eric Miller (Tufts)
- Ashley Prater (AFOSR)
- Lixin Shen (Syracuse University)
- Allen Tannenbaum (Stony Brook)
- Guowei Wei (Michigan State)
- Stephen Wong (Houston Methodist)
- Jun Zhang (University of Michigan, Ann Arbor)
- Song Zhang (Purdue University)
- Hongkai Zhao (University of California, Irvine)
Mirror symmetry, gauged linear sigma models, matrix factorizations, and related topics
CMSA, 20 Garden Street, Cambridge, MA 02138 USAOn March 4-6, 2020 the CMSA will be hosting a three-day workshop on Mirror symmetry, Gauged linear sigma models, Matrix factorizations, and related topics as part of the Simons Collaboration on Homological Mirror Symmetry. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Speakers:
- Andrei Căldăraru, University of Wisconsin
- David Favero, University of Alberta
- Elana Kalashnikov, Harvard University
- Tsung-Ju Lee, CMSA
- Conan Leung, CUHK
- David Morrison, University of California, Santa Barbara
- Mauricio Romo, YMSC
- Yun Shi, CMSA
- Mark Shoemaker, Colorado State University
- Rachel Webb, University of Michigan
- Chris Woodward, Rutgers University
- Guangbo Xu, Texas A&M University
- Chenglong Yu, University of Pennsylvania
Videos from the workshop are available in the Youtube playlist.
The 2017 Charles River Lectures
The 2017 Charles River Lectures
Charles River with Bench at SunsetJointly organized by Harvard University, Massachusetts Institute of Technology, and Microsoft Research New England, the Charles River Lectures on Probability and Related Topics is a one-day event for the benefit of the greater Boston area mathematics community.
The 2017 lectures will take place 9:15am – 5:30pm on Monday, October 2 at Harvard University in the Harvard Science Center.
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UPDATED LOCATION
Harvard University
Harvard Science Center (Halls C & E)
1 Oxford Street, Cambridge, MA 02138 (Map)
Monday, October 2, 2017
9:15 AM – 5:30 PM
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Please note that registration has closed.
Speakers:
- Paul Bourgade (Courant Institute, NYU)
- Massimiliano Gubinelli (University of Bonn)
- Andrea Montanari (Stanford University)
- Roman Vershynin (University of California, Irvine)
- Ofer Zeitouni (Weizmann Institute)
Agenda:
In Harvard Science Center Hall C:
8:45 am – 9:15 am: Coffee/light breakfast
9:15 am – 10:15 am: Ofer Zeitouni
Title:
Abstract:
10:20 am – 11:20 am: Andrea Montanari
Title:
Abstract:
11:20 am – 11:45 am: Break
11:45 am – 12:45 pm: Paul Bourgade
Title:
Abstract:
1:00 pm – 2:30 pm: Lunch
In Harvard Science Center Hall E:
2:45 pm – 3:45 pm: Roman Vershynin
Title: Deviations of random matrices and applications
Abstract: Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This lecture will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in high dimensional geometry and data science.
3:45 pm – 4:15 pm: Break
4:15 pm – 5:15 pm: Massimiliano Gubinelli
Title:
Abstract:
Poster:
2017 Charles River Lectures Poster
Organizers:
Alexei Borodin, Henry Cohn, Vadim Gorin, Elchanan Mossel, Philippe Rigollet, Scott Sheffield, and H.T. Yau
Workshop on Invariance and Geometry in Sensation, Action and Cognition
As part of the program on Mathematical Biology a workshop on Invariance and Geometry in Sensation, Action and Cognition will take place on April 15-17, 2019.
Legend has it that above the door to Plato’s Academy was inscribed “Μηδείς άγεωµέτρητος είσίτω µον τήν στέγην”, translated as “Let no one ignorant of geometry enter my doors”. While geometry and invariance has always been a cornerstone of mathematics, it has traditionally not been an important part of biology, except in the context of aspects of structural biology. The premise of this meeting is a tantalizing sense that geometry and invariance are also likely to be important in (neuro)biology and cognition. Since all organisms interact with the physical world, this implies that as neural systems extract information using the senses to guide action in the world, they need appropriately invariant representations that are stable, reproducible and capable of being learned. These invariances are a function of the nature and type of signal, its corruption via noise, and the method of storage and use.
This hypothesis suggests many puzzles and questions: What representational geometries are reflected in the brain? Are they learned or innate? What happens to the invariances under realistic assumptions about noise, nonlinearity and finite computational resources? Can cases of mental disorders and consequences of brain damage be characterized as break downs in representational invariances? Can we harness these invariances and sensory contingencies to build more intelligent machines? The aim is to revisit these old neuro-cognitive problems using a series of modern lenses experimentally, theoretically and computationally, with some tutorials on how the mathematics and engineering of invariant representations in machines and algorithms might serve as useful null models.
In addition to talks, there will be a set of tutorial talks on the mathematical description of invariance (P.J. Olver), the computer vision aspects of invariant algorithms (S. Soatto), and the neuroscientific and cognitive aspects of invariance (TBA). The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by L. Mahadevan (Harvard), Talia Konkle (Harvard), Samuel Gershman (Harvard), and Vivek Jayaraman (HHMI).
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Videos
Tentative Speaker List:
- Alessandro Achille, UCLA
- Vijay Balasubramanian, University of Pennsylvania
- Jeannette Bohg, Stanford
- Ed Connor, Johns Hopkins
- Moira Dillon, NYU
- Jacob Feldman, Rutgers
- Ila Fiete, MIT
- Sam Gershman, Harvard
- Gily Ginosar, Weizmann Institute of Science
- Lucia Jacobs, UC Berkeley
- Vivek Jayaraman, HHMI
- Talia Konkle, Harvard
- L. Mahadevan, Harvard
- Michael McCloskey, Johns Hopkins
- Sam Ocko, Stanford
- Peter Olver, University of Minnesota
- Anitha Pasupathy, University of Washington
- Sandro Romani, Janelia
- Stefano Soatto, UCLA
- Tatyana Sharpee, Salk Institute
- Dagmar Sternad, Northeastern
- Elizabeth Torres, Rutgers
Schedule:
Monday, April 15
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 9:15am Welcome and Introduction 9:15 – 10:00am Vivek Jayaraman Title: Insect cognition: Small tales of geometry & invariance Abstract: Decades of field and laboratory experiments have allowed ethologists to discover the remarkable sophistication of insect behavior. Over the past couple of decades, physiologists have been able to peek under the hood to uncover sophistication in insect brain dynamics as well. In my talk, I will describe phenomena that relate to the workshop’s theme of geometry and invariance. I will outline how studying insects —and flies in particular— may enable an understanding of the neural mechanisms underlying these intriguing phenomena.
10:00 – 10:45am Elizabeth Torres Title: Connecting Cognition and Biophysical Motions Through Geometric Invariants and Motion Variability Abstract: In the 1930s Nikolai Bernstein defined the degrees of freedom (DoF) problem. He asked how the brain could control abundant DoF and produce consistent solutions, when the internal space of bodily configurations had much higher dimensions than the space defining the purpose(s) of our actions. His question opened two fundamental problems in the field of motor control. One relates to the uniqueness or consistency of a solution to the DoF problem, while the other refers to the characterization of the diverse patterns of variability that such solution produces.
In this talk I present a general geometric solution to Bernstein’s DoF problem and provide empirical evidence for symmetries and invariances that this solution provides during the coordination of complex naturalistic actions. I further introduce fundamentally different patterns of variability that emerge in deliberate vs. spontaneous movements discovered in my lab while studying athletes and dancers performing interactive actions. I here reformulate the DoF problem from the standpoint of the social brain and recast it considering graph theory and network connectivity analyses amenable to study one of the most poignant developmental disorders of our times: Autism Spectrum Disorders.
I offer a new unifying framework to recast dynamic and complex cognitive and social behaviors of the full organism and to characterize biophysical motion patterns during migration of induced pluripotent stem cell colonies on their way to become neurons.
10:45 – 11:15am Coffee Break 11:15 – 12:00pm Peter Olver Title: Symmetry and invariance in cognition — a mathematical perspective” Abstract: Symmetry recognition and appreciation is fundamental in human cognition. (It is worth speculating as to why this may be so, but that is not my intent.) The goal of these two talks is to survey old and new mathematical perspectives on symmetry and invariance. Applications will arise from art, computer vision, geometry, and beyond, and will include recent work on 2D and 3D jigsaw puzzle assembly and an ongoing collaboration with anthropologists on the analysis and refitting of broken bones. Mathematical prerequisites will be kept to a bare minimum.
12:00 – 12:45pm Stefano Soatto/Alessandro Achille Title: Information in the Weights and Emergent Properties of Deep Neural Networks Abstract: We introduce the notion of information contained in the weights of a Deep Neural Network and show that it can be used to control and describe the training process of DNNs, and can explain how properties, such as invariance to nuisance variability and disentanglement, emerge naturally in the learned representation. Through its dynamics, stochastic gradient descent (SGD) implicitly regularizes the information in the weights, which can then be used to bound the generalization error through the PAC-Bayes bound. Moreover, the information in the weights can be used to defined both a topology and an asymmetric distance in the space of tasks, which can then be used to predict the training time and the performance on a new task given a solution to a pre-training task.
While this information distance models difficulty of transfer in first approximation, we show the existence of non-trivial irreversible dynamics during the initial transient phase of convergence when the network is acquiring information, which makes the approximation fail. This is closely related to critical learning periods in biology, and suggests that studying the initial convergence transient can yield important insight beyond those that can be gleaned from the well-studied asymptotics.
12:45 – 2:00pm Lunch 2:00 – 2:45pm Anitha Pasupathy Title: Invariant and non-invariant representations in mid-level ventral visual cortex My laboratory investigates how visual form is encoded in area V4, a critical mid-level stage of form processing in the macaque monkey. Our goal is to reveal how V4 representations underlie our ability to segment visual scenes and recognize objects. In my talk I will present results from two experiments that highlight the different strategies used by the visual to achieve these goals. First, most V4 neurons exhibit form tuning that is exquisitely invariant to size and position, properties likely important to support invariant object recognition. On the other hand, form tuning in a majority of neurons is also highly dependent on the interior fill. Interestingly, unlike primate V4 neurons, units in a convolutional neural network trained to recognize objects (AlexNet) overwhelmingly exhibit fill-outline invariance. I will argue that this divergence between real and artificial circuits reflects the importance of local contrast in parsing visual scenes and overall scene understanding.
2:45 – 3:30pm Jacob Feldman Title: Bayesian skeleton estimation for shape representation and perceptual organization Abstract: In this talk I will briefly summarize a framework in which shape representation and perceptual organization are reframed as probabilistic estimation problems. The approach centers around the goal of identifying the skeletal model that best “explains” a given shape. A Bayesian solution to this problem requires identifying a prior over shape skeletons, which penalizes complexity, and a likelihood model, which quantifies how well any particular skeleton model fits the data observed in the image. The maximum-posterior skeletal model thus constitutes the most “rational” interpretation of the image data consistent with the given assumptions. This approach can easily be extended and generalized in a number of ways, allowing a number of traditional problems in perceptual organization to be “probabilized.” I will briefly illustrate several such extensions, including (1) figure/ground and grouping (3) 3D shape and (2) shape similarity.
3:30 – 4:00pm Tea Break 4:00 – 4:45pm Moira Dillon Title: Euclid’s Random Walk: Simulation as a tool for geometric reasoning through development Abstract: Formal geometry lies at the foundation of millennia of human achievement in domains such as mathematics, science, and art. While formal geometry’s propositions rely on abstract entities like dimensionless points and infinitely long lines, the points and lines of our everyday world all have dimension and are finite. How, then, do we get to abstract geometric thought? In this talk, I will provide evidence that evolutionarily ancient and developmentally precocious sensitivities to the geometry of our everyday world form the foundation of, but also limit, our mathematical reasoning. I will also suggest that successful geometric reasoning may emerge through development when children abandon incorrect, axiomatic-based strategies and come to rely on dynamic simulations of physical entities. While problems in geometry may seem answerable by immediate inference or by deductive proof, human geometric reasoning may instead rely on noisy, dynamic simulations.
4:45 – 5:30pm Michael McCloskey Title: Axes and Coordinate Systems in Representing Object Shape and Orientation Abstract: I describe a theoretical perspective in which a) object shape is represented in an object-centered reference frame constructed around orthogonal axes; and b) object orientation is represented by mapping the object-centered frame onto an extrinsic (egocentric or environment-centered) frame. I first show that this perspective is motivated by, and sheds light on, object orientation errors observed in neurotypical children and adults, and in a remarkable case of impaired orientation perception. I then suggest that orientation errors can be used to address questions concerning how object axes are defined on the basis of object geometry—for example, what aspects of object geometry (e.g., elongation, symmetry, structural centrality of parts) play a role in defining an object principal axis?
5:30 – 6:30pm Reception Tuesday, April 16
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 9:45am Peter Olver Title: Symmetry and invariance in cognition — a mathematical perspective” Abstract: Symmetry recognition and appreciation is fundamental in human cognition. (It is worth speculating as to why this may be so, but that is not my intent.) The goal of these two talks is to survey old and new mathematical perspectives on symmetry and invariance. Applications will arise from art, computer vision, geometry, and beyond, and will include recent work on 2D and 3D jigsaw puzzle assembly and an ongoing collaboration with anthropologists on the analysis and refitting of broken bones. Mathematical pre
9:45 – 10:30am Stefano Soatto/Alessandro Achille Title: Information in the Weights and Emergent Properties of Deep Neural Networks Abstract: We introduce the notion of information contained in the weights of a Deep Neural Network and show that it can be used to control and describe the training process of DNNs, and can explain how properties, such as invariance to nuisance variability and disentanglement, emerge naturally in the learned representation. Through its dynamics, stochastic gradient descent (SGD) implicitly regularizes the information in the weights, which can then be used to bound the generalization error through the PAC-Bayes bound. Moreover, the information in the weights can be used to defined both a topology and an asymmetric distance in the space of tasks, which can then be used to predict the training time and the performance on a new task given a solution to a pre-training task.
While this information distance models difficulty of transfer in first approximation, we show the existence of non-trivial irreversible dynamics during the initial transient phase of convergence when the network is acquiring information, which makes the approximation fail. This is closely related to critical learning periods in biology, and suggests that studying the initial convergence transient can yield important insight beyond those that can be gleaned from the well-studied asymptotics.
10:30 – 11:00am Coffee Break 11:00 – 11:45am Jeannette Bohg Title: On perceptual representations and how they interact with actions and physical representations Abstract: I will discuss the hypothesis that perception is active and shaped by our task and our expectations on how the world behaves upon physical interaction. Recent approaches in robotics follow this insight that perception is facilitated by physical interaction with the environment. First, interaction creates a rich sensory signal that would otherwise not be present. And second, knowledge of the regularity in the combined space of sensory data and action parameters facilitate the prediction and interpretation of the signal. In this talk, I will present two examples from our previous work where a predictive task facilitates autonomous robot manipulation by biasing the representation of the raw sensory data. I will present results on visual but also haptic data.
11:45 – 12:30pm Dagmar Sternad Title: Exploiting the Geometry of the Solution Space to Reduce Sensitivity to Neuromotor Noise Abstract: Control and coordination of skilled action is frequently examined in isolation as a neuromuscular problem. However, goal-directed actions are guided by information that creates solutions that are defined as a relation between the actor and the environment. We have developed a task-dynamic approach that starts with a physical model of the task and mathematical analysis of the solution spaces for the task. Based on this analysis we can trace how humans develop strategies that meet complex demands by exploiting the geometry of the solution space. Using three interactive tasks – throwing or bouncing a ball and transporting a “cup of coffee” – we show that humans develop skill by: 1) finding noise-tolerant strategies and channeling noise into task-irrelevant dimensions, 2) exploiting solutions with dynamic stability, and 3) optimizing predictability of the object dynamics. These findings are the basis for developing propositions about the controller: complex actions are generated with dynamic primitives, attractors with few invariant types that overcome substantial delays and noise in the neuro-mechanical system.
12:30 – 2:00pm Lunch 2:00 – 2:45pm Sam Ocko Title: Emergent Elasticity in the Neural Code for Space Abstract: To navigate a novel environment, animals must construct an internal map of space by combining information from two distinct sources: self-motion cues and sensory perception of landmarks. How do known aspects of neural circuit dynamics and synaptic plasticity conspire to construct such internal maps, and how are these maps used to maintain representations of an animal’s position within an environment. We demonstrate analytically how a neural attractor model that combines path integration of self-motion with Hebbian plasticity in synaptic weights from landmark cells can self-organize a consistent internal map of space as the animal explores an environment. Intriguingly, the emergence of this map can be understood as an elastic relaxation process between landmark cells mediated by the attractor network during exploration. Moreover, we verify several experimentally testable predictions of our model, including: (1) systematic deformations of grid cells in irregular environments, (2) path-dependent shifts in grid cells towards the most recently encountered landmark, (3) a dynamical phase transition in which grid cells can break free of landmarks in altered virtual reality environments and (4) the creation of topological defects in grid cells. Taken together, our results conceptually link known biophysical aspects of neurons and synapses to an emergent solution of a fundamental computational problem in navigation, while providing a unified account of disparate experimental observations.
2:45 – 3:30pm Tatyana Sharpee Title: Hyperbolic geometry of the olfactory space Abstract: The sense of smell can be used to avoid poisons or estimate a food’s nutrition content because biochemical reactions create many by-products. Thus, the production of a specific poison by a plant or bacteria will be accompanied by the emission of certain sets of volatile compounds. An animal can therefore judge the presence of poisons in the food by how the food smells. This perspective suggests that the nervous system can classify odors based on statistics of their co-occurrence within natural mixtures rather than from the chemical structures of the ligands themselves. We show that this statistical perspective makes it possible to map odors to points in a hyperbolic space. Hyperbolic coordinates have a long but often underappreciated history of relevance to biology. For example, these coordinates approximate distance between species computed along dendrograms, and more generally between points within hierarchical tree-like networks. We find that both natural odors and human perceptual descriptions of smells can be described using a three-dimensional hyperbolic space. This match in geometries can avoid distortions that would otherwise arise when mapping odors to perception. We identify three axes in the perceptual space that are aligned with odor pleasantness, its molecular boiling point and acidity. Because the perceptual space is curved, one can predict odor pleasantness by knowing the coordinates along the molecular boiling point and acidity axes.
3:30 – 4:00pm Tea Break 4:00 – 4:45pm Ed Connor Title: Representation of solid geometry in object vision cortex Abstract: There is a fundamental tension in object vision between the 2D nature of retinal images and the 3D nature of physical reality. Studies of object processing in the ventral pathway of primate visual cortex have focused mainly on 2D image information. Our latest results, however, show that representations of 3D geometry predominate even in V4, the first object-specific stage in the ventral pathway. The majority of V4 neurons exhibit strong responses and clear selectivity for solid, 3D shape fragments. These responses are remarkably invariant across radically different image cues for 3D shape: shading, specularity, reflection, refraction, and binocular disparity (stereopsis). In V4 and in subsequent stages of the ventral pathway, solid shape geometry is represented in terms of surface fragments and medial axis fragments. Whole objects are represented by ensembles of neurons signaling the shapes and relative positions of their constituent parts. The neural tuning dimensionality of these representations includes principal surface curvatures and their orientations, surface normal orientation, medial axis orientation, axial curvature, axial topology, and position relative to object center of mass. Thus, the ventral pathway implements a rapid transformation of 2D image data into explicit representations 3D geometry, providing cognitive access to the detailed structure of physical reality.
4:45 – 5:30pm L. Mahadevan Title: Simple aspects of geometry and probability in perception Abstract: Inspired by problems associated with noisy perception, I will discuss two questions: (i) how might we test people’s perception of probability in a geometric context ? (ii) can one construct invariant descriptions of 2D images using simple notions of probabilistic geometry? Along the way, I will highlight other questions that the intertwining of geometry and probability raises in a broader perceptual context.
Wednesday, April 17Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 9:45am Gily Ginosar Title: The 3D geometry of grid cells in flying bats Abstract: The medial entorhinal cortex (MEC) contains a variety of spatial cells, including grid cells and border cells. In 2D, grid cells fire when the animal passes near the vertices of a 2D spatial lattice (or grid), which is characterized by circular firing-fields separated by fixed distances, and 60 local angles – resulting in a hexagonal structure. Although many animals navigate in 3D space, no studies have examined the 3D volumetric firing of MEC neurons. Here we addressed this by training Egyptian fruit bats to fly in a large room (5.84.62.7m), while we wirelessly recorded single neurons in MEC. We found 3D border cells and 3D head-direction cells, as well as many neurons with multiple spherical firing-fields. 20% of the multi-field neurons were 3D grid cells, exhibiting a narrow distribution of characteristic distances between neighboring fields – but not a perfect 3D global lattice. The 3D grid cells formed a functional continuum with less structured multi-field neurons. Both 3D grid cells and multi-field cells exhibited an anatomical gradient of spatial scale along the dorso-ventral axis of MEC, with inter-field spacing increasing ventrally – similar to 2D grid cells in rodents. We modeled 3D grid cells and multi-field cells as emerging from pairwise-interactions between fields, using an energy potential that induces repulsion at short distances and attraction at long distances. Our analysis shows that the model explains the data significantly better than a random arrangement of fields. Interestingly, simulating the exact same model in 2D yielded a hexagonal-like structure, akin to grid cells in rodents. Together, the experimental data and preliminary modeling suggest that the global property of grid cells is multiple fields that repel each other with a characteristic distance-scale between adjacent fields – which in 2D yields a global hexagonal lattice while in 3D yields only local structure but no global lattice.
Gily Ginosar 1 , Johnatan Aljadeff 2 , Yoram Burak 3 , Haim Sompolinsky 3 , Liora Las 1 , Nachum Ulanovsky 1
(1) Department of Neurobiology, Weizmann Institute of Science, Rehovot 76100, Israel
(2) Department of Bioengineering, Imperial College London, London, SW7 2AZ, UK
(3) The Edmond and Lily Safra Center for Brain Sciences, and Racah Institute of Physics, The Hebrew
University of Jerusalem, Jerusalem, 91904, Israel
9:45 – 10:30am Sandro Romani Title: Neural networks for 3D rotations Abstract: Studies in rodents, bats, and humans have uncovered the existence of neurons that encode the orientation of the head in 3D. Classical theories of the head-direction (HD) system in 2D rely on continuous attractor neural networks, where neurons with similar heading preference excite each other, while inhibiting other HD neurons. Local excitation and long-range inhibition promote the formation of a stable “bump” of activity that maintains a representation of heading. The extension of HD models to 3D is hindered by complications (i) 3D rotations are non-commutative (ii) the space described by all possible rotations of an object has a non-trivial topology. This topology is not captured by standard parametrizations such as Euler angles (e.g. yaw, pitch, roll). For instance, with these parametrizations, a small change of the orientation of the head could result in a dramatic change of neural representation. We used methods from the representation theory of groups to develop neural network models that exhibit patterns of persistent activity of neurons mapped continuously to the group of 3D rotations. I will further discuss how these networks can (i) integrate vestibular inputs to update the representation of heading, and (ii) be used to interpret “mental rotation” experiments in humans.
This is joint work with Hervé Rouault (CENTURI) and Alon Rubin (Weizmann Institute of Science).
10:30 – 11:00am Coffee Break 11:00 – 11:45am Sam Gershman Title: The hippocampus as a predictive map Abstract: A cognitive map has long been the dominant metaphor for hippocampal function, embracing the idea that place cells encode a geometric representation of space. However, evidence for predictive coding, reward sensitivity and policy dependence in place cells suggests that the representation is not purely spatial. I approach this puzzle from a reinforcement learning perspective: what kind of spatial representation is most useful for maximizing future reward? I show that the answer takes the form of a predictive representation. This representation captures many aspects of place cell responses that fall outside the traditional view of a cognitive map. Furthermore, I argue that entorhinal grid cells encode a low-dimensionality basis set for the predictive representation, useful for suppressing noise in predictions and extracting multiscale structure for hierarchical planning.
11:45 – 12:30pm Lucia Jacobs Title: The adaptive geometry of a chemosensor: the origin and function of the vertebrate nose Abstract: A defining feature of a living organism, from prokaryotes to plants and animals, is the ability to orient to chemicals. The distribution of chemicals, whether in water, air or on land, is used by organisms to locate and exploit spatially distributed resources, such as nutrients and reproductive partners. In animals, the evolution of a nervous system coincided with the evolution of paired chemosensors. In contemporary insects, crustaceans, mollusks and vertebrates, including humans, paired chemosensors confer a stereo olfaction advantage on the animal’s ability to orient in space. Among vertebrates, however, this function faced a new challenge with the invasion of land. Locomotion on land created a new conflict between respiration and spatial olfaction in vertebrates. The need to resolve this conflict could explain the current diversity of vertebrate nose geometries, which could have arisen due to species differences in the demand for stereo olfaction. I will examine this idea in more detail in the order Primates, focusing on Old World primates, in particular, the evolution of an external nose in the genus Homo.
12:30 – 1:30pm Lunch 1:30 – 2:15pm Talia Konkle Title: The shape of things and the organization of object-selective cortex Abstract: When we look at the world, we effortlessly recognize the objects around us and can bring to mind a wealth of knowledge about their properties. In part 1, I’ll present evidence that neural responses to objects are organized by high-level dimensions of animacy and size, but with underlying neural tuning to mid-level shape features. In part 2, I’ll present evidence that representational structure across much of the visual system has the requisite structure to predict visual behavior. Together, these projects suggest that there is a ubiquitous “shape space” mapped across all of occipitotemporal cortex that underlies our visual object processing capacities. Based on these findings, I’ll speculate that the large-scale spatial topography of these neural responses is critical for pulling explicit content out of a representational geometry.
2:15 – 3:00pm Vijay Balasubramanian Title: Becoming what you smell: adaptive sensing in the olfactory system Abstract: I will argue that the circuit architecture of the early olfactory system provides an adaptive, efficient mechanism for compressing the vast space of odor mixtures into the responses of a small number of sensors. In this view, the olfactory sensory repertoire employs a disordered code to compress a high dimensional olfactory space into a low dimensional receptor response space while preserving distance relations between odors. The resulting representation is dynamically adapted to efficiently encode the changing environment of volatile molecules. I will show that this adaptive combinatorial code can be efficiently decoded by systematically eliminating candidate odorants that bind to silent receptors. The resulting algorithm for “estimation by elimination” can be implemented by a neural network that is remarkably similar to the early olfactory pathway in the brain. The theory predicts a relation between the diversity of olfactory receptors and the sparsity of their responses that matches animals from flies to humans. It also predicts specific deficits in olfactory behavior that should result from optogenetic manipulation of the olfactory bulb.
3:00 – 3:45pm Ila Feite Title: Invariance, stability, geometry, and flexibility in spatial navigation circuits Abstract: I will describe how the geometric invariances or symmetries of the external world are reflected in the symmetries of neural circuits that represent it, using the example of the brain’s networks for spatial navigation. I will discuss how these symmetries enable spatial memory, evidence integration, and robust representation. At the same time, I will discuss how these seemingly rigid circuits with their inscribed symmetries can be harnessed to represent a range of spatial and non-spatial cognitive variables with high flexibility.
3:45 – 4:00pm L Mahadevan – summary Topology and Dynamics in Quantum Matter Workshop
On September 10-11, 2019, the CMSA will be hosting a second workshop on Topological Aspects of Condensed Matter.
New ideas rooted in topology have recently had a major impact on condensed matter physics, and have led to new connections with high energy physics, mathematics and quantum information theory. The aim of this program will be to deepen these connections and spark new progress by fostering discussion and new collaborations within and across disciplines.
Topics include i) the classification of topological states ii) topological orders in two and three dimensions including quantum spin liquids, quantum Hall states and fracton phases and iii) interplay of symmetry and topology in quantum many body systems, including symmetry protected topological phases, symmetry fractionalization and anomalies iv) topological phenomena in quantum systems driven far from equlibrium v) quantum field theory approaches to topological matter.
This workshop is part of the CMSA’s program on Program on Topological Aspects of Condensed Matter, and is the second of two workshops, in addition to a visitor program and seminars.
The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Click here for a list of restaurants in the area.
Organizers: Michael Hermele (CU Boulder) and Ashvin Vishwanath (Harvard)
Partial list of speakers:
- Nima Arkani-Hamed, IAS
- Jennifer Cano, Stony Brook
- Meng Cheng, Yale
- Lukasz Fidkowski, UW Seattle
- Daniel Freed, Texas
- Jeongwan Haah, Microsoft Research
- Anton Kapustin, Caltech
- Zohar Komargodski, SCGP/Stony Brook
- John McGreevy, UC San Diego
- Prineha Narang, Harvard
- Ying Ran, Boston College
- Shinsei Ryu, Chicago
- Cumrun Vafa, Harvard
- Chong Wang, Perimeter
- Zhenghan Wang, Microsoft Station Q
Videos of the lectures can be found in the Youtube playlist below. Links to talks are also available on the schedule below.
Kickoff Workshop on Topology and Quantum Phases of Matter
On August 27-28, 2018, the CMSA will be hosting a Kickoff workshop on Topology and Quantum Phases of Matter. New ideas rooted in topology have recently had a big impact on condensed matter physics, and have highlighted new connections with high energy physics, mathematics and quantum information theory. Additionally, these ideas have found applications in the design of photonic systems and of materials with novel mechanical properties. The aim of this program will be to deepen these connections by fostering discussion and seeding new collaborations within and across disciplines.
This workshop is a part of the CMSA’s program on Program on Topological Aspects of Condensed Matter, and will be the first of two workshops, in addition to a visitor program and seminars.
The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Please register here
Speakers:
- Zhen Bi, MIT
- Meng Cheng, Yale
- Dima Feldman, Brown
- Dominic Else, UCSB
- Liang Fu, MIT
- Fabian Grusdt, Harvard
- Ying Fei Gu, Harvard
- Bert Halperin, Harvard
- Anton Kapustin, Caltech
- Patrick Lee, MIT
- L. Mahadevan, Harvard
- Brad Marston, Brown
- Max Metlitski, MIT
- Emil V. Prodan, Yeshiva
- Achim Rosch, University of Cologne
- Mathias Scheurer, Harvard
- Marin Soljacic, MIT
- X. G. Wen, MIT
- Cenke Xu, UCSB
- Frank Zhang, Cornell
Static vacuum extensions of Bartnik boundary data near flat domains
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on a joint work with Lan-Hsuan Huang.
Geometry and Physics Seminar
During the summer of 2020, the CMSA will be hosting a new Geometry Seminar. Talks will be scheduled on Mondays at 9:30pm or Tuesdays at 9:30am, depending on the location of the speaker. This seminar is organized by Tsung-Ju Lee, Yoosik Kim, and Du Pei.
To learn how to attend this seminar, please contact Tsung-Ju Lee (tjlee@cmsa.fas.harvard.edu).
Date Speaker Title/Abstract 6/2/2020
9:30am ETSiu-Cheong Lau
Boston UniversityThis meeting will be taking place virtually on Zoom. Speaker: Equivariant Floer theory and SYZ mirror symmetry
Abstract: In this talk, we will first review a symplectic realization of the SYZ program and some of its applications. Then I will explain some recent works on equivariant Lagrangian Floer theory and disc potentials of immersed SYZ fibers. They are joint works with Hansol Hong, Yoosik Kim and Xiao Zheng.
6/8/2020
9:30pm ETYoungjin Bae (KIAS) This meeting will be taking place virtually on Zoom. Title: Legendrian graphs and their invariants
Abstract: Legendrian graphs naturally appear in the study of Weinstein manifolds with a singular Lagrangian skeleton, and a tangle decomposition of Legendrian submanifolds. I will introduce various invariant of Legendrian graphs including DGA type, polynomial type, sheaf theoretic one, and their relationship. This is joint work with Byunghee An, and partially with Tamas Kalman and Tao Su.
6/16/2020
9:30am ETMichael McBreen (CMSA) This meeting will be taking place virtually on Zoom. Title: Loops in hypertoric varieties and symplectic duality
Abstract: Hypertoric varieties are algebraic symplectic varieties associated to graphs, or more generally certain hyperplane arrangements. They make many appearances in modern geometric representation theory. I will discuss certain infinite dimensional or infinite type generalizations of hypertoric varieties which occur in the study of enumerative invariants, focusing on some elementary examples. Joint work with Artan Sheshmani and Shing-Tung Yau.
6/22/2020
9:30pm ETZiming Ma (CUHK) This meeting will be taking place virtually on Zoom. Title: The geometry of Maurer–Cartan equation near degenerate Calabi–Yau varieties
Abstract: In this talk, we construct a \(dgBV algebra PV*(X)\) associated to a possibly degenerate Calabi–Yau variety X equipped with local thickening data. This gives a version of the Kodaira–Spencer dgLa which is applicable to degenerated spaces including both log smooth or maximally degenerated Calabi–Yau. We use this to prove an unobstructedness result about the smoothing of degenerated Log Calabi–Yau varieties X satisfying Hodge–deRham degeneracy property for cohomology of X, in the spirit of Kontsevich–Katzarkov–Pantev. This is a joint work with Kwokwai Chan and Naichung Conan Leung.
6/30/2020
9:30pm ETSunghyuk Park (Caltech) This meeting will be taking place virtually on Zoom. Title: 3-manifolds, q-series, and topological strings
Abstract: \(\hat{Z}\) is an invariant of 3-manifolds valued in q-series (i.e. power series in q with integer coefficients), which has interesting modular properties. While originally from physics, this invariant has been mathematically constructed for a big class of 3-manifolds, and conjecturally it can be extended to all 3-manifolds. In this talk, I will give a gentle introduction to \(\hat{Z}\) and what is known about it, as well as highlighting some recent developments, including the use of R-matrix, generalization to higher rank, large N-limit and interpretation as open topological string partition functions.
7/7/2020
9:30am ETJeremy Lane (McMaster University) This meeting will be taking place virtually on Zoom. Title: Collective integrable systems and global action-angle coordinates
Abstract: A “collective integrable system” on a symplectic manifold is a commutative integrable system constructed from a Hamiltonian action of a non-commutative Lie group. Motivated by the example of Gelfand-Zeitlin systems, we give a construction of collective integrable systems that generate a Hamiltonian torus action on a dense subset of any Hamiltonian K-manifold, where K is any compact connected Lie group. In the case where the Hamiltonian K-manifold is compact and multiplicity free, the resulting Hamiltonian torus action is completely integrable and yields global action angle coordinates. Moreover, the image of the moment map is a (non-simple) convex polytope.
7/13/2020
9:30pm ETPo-Shen Hsin (Caltech) This meeting will be taking place virtually on Zoom. Title: Berry phase in quantum field theory
Abstract: We will discuss Berry phase in family of quantum field theories using effective field theory. The family is labelled by parameters which we promote to be spacetime-dependent sigma model background fields. The Berry phase is equivalent to Wess-Zumino-Witten action for the sigma model. We use Berry phase to study diabolic points in the phase diagram of the quantum field theory and discuss applications to deconfined quantum criticality and new tests for boson/fermion dualities in \((2+1)d\).
7/20/2020
9:30pm ETSangwook Lee (KIAS) This meeting will be taking place virtually on Zoom. Title: A geometric construction of orbifold Jacobian algebras
Abstract: We review the definition of a twisted Jacobian algebra of a Landau-Ginzburg orbifold due to Kaufmann et al. Then we construct an A-infinity algebra of a weakly unobstructed Lagrangian submanifold in a symplectic orbifold. We work on an elliptic orbifold sphere and see that above two algebras are isomorphic, and furthermore their structure constants are related by a modular identity which was used to prove the mirror symmetry of closed string pairings. This is a joint work with Cheol-Hyun Cho.
7/27/2020 9:30pm ET Mao Sheng (USTC) This meeting will be taking place virtually on Zoom. Title: Parabolic de Rham bundles: motivic vs periodic
Abstract: Let \($C$\) be a complex smooth projective curve. We consider the set of parabolic de Rham bundles over \($C$\) (with rational weights in parabolic structure). Many examples arise from geometry: let \($f: X\to U$\) be a smooth projective morphism over some nonempty Zariski open subset \($U\subset C$\). Then the Deligne–Iyer–Simpson canonical parabolic extension of the Gauss–Manin systems associated to \($f$\) provides such examples. We call a parabolic de Rham bundle \emph{motivic}, if it appears as a direct summand of such an example of geometric origin. It is a deep question in the theory of linear ordinary differential equations and in Hodge theory, to get a characterization of motivic parabolic de Rham bundles. In this talk, I introduce another subcategory of parabolic de Rham bundles, the so-called \emph{periodic} parabolic de Rham bundles. It is based on the work of Lan–Sheng–Zuo on Higgs-de Rham flows, with aim towards linking the Simpson correspondence over the field of complex numbers and the Ogus–Vologodsky correspondence over the finite fields. We show that motivic parabolic de Rham bundles are periodic, and conjecture that they are all periodic parabolic de Rham bundles. The conjecture for rank one case follows from the solution of Grothendieck–Katz p-curvature conjecture, and for some versions of rigid cases should follow from Katz’s work on rigid local systems. The conjecture implies that in a spread-out of any complex elliptic curve, there will be infinitely many supersingular primes, a result of N. Elkies for rational elliptic curves. Among other implications of the conjecture, we would like to single out the conjectural arithmetic Simpson correspondence, which asserts that the grading functor is an equivalence of categories from the category of periodic parabolic de Rham bundles to the category of periodic parabolic Higgs bundles. This is a joint work in progress with R. Krishnamoorthy.
8/4/2020
9:30am EtPavel Safronov (University of Zurich) This meeting will be taking place virtually on Zoom. Title: Kapustin–Witten TFT on 3-manifolds and skein modules
Abstract: Kapustin and Witten have studied a one-parameter family of topological twists of \(4d N=4\) super Yang–Mills. They have shown that the categories of boundary conditions on a surface are exactly the categories participating in the geometric Langlands program of Beilinson and Drinfeld. Moreover, S-duality is manifested as a quantum geometric Langlands duality after the topological twist. In this talk I will describe some mathematical formalizations of Hilbert spaces of states on a 3-manifold. I will outline an equivalence between two such possible formalizations: complexified Floer homology of Abouzaid–Manolescu and skein modules. This is a report on work in progress joint with Sam Gunningham.8/11/2020
9:30amXujia Chen (Stonybrook) This meeting will be taking place virtually on Zoom. Title: Lifting cobordisms and Kontsevich-type recursions for counts of real curves
Abstract: Kontsevich’s recursion, proved in the early 90s, is a recursion formula for the counts of rational holomorphic curves in complex manifolds. For complex fourfolds and sixfolds with a real structure (i.e. a conjugation), signed invariant counts of real rational holomorphic curves were defined by Welschinger in 2003. Solomon interpreted Welschinger’s invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline of a potential approach of proving them. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon’s recursions by re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne-Mumford moduli spaces of stable real curves (which is different from Solomon’s approach).
8/18/2020
9:30am ETDongmin Gang (Asia Pacific Center for Theoretical Physics) This meeting will be taking place virtually on Zoom. Title: M-theoretic genesis of topological phases
Abstract: I will talk about a novel way of constructing \((2+1)d\) topological phases using M-theory. They emerge as macroscopic world-volume theories of M5-branes wrapped on non-hyperbolic 3-manifolds. After explaining the algorithm of extracting modular structures of the topological phase from topological data of the 3-manifold, I will discuss the possibility of full classification of topological orders via the geometrical construction.
8/25/2020
9:30pm ETMykola Dedushenko (Caltech) This meeting will be taking place virtually on Zoom. Title: Algebras and traces at the boundary of \(4d N=4\) SYM
Abstract: I will describe how the structure of supersymmetric boundary correlators in \(4d N=4\) SYM can be encoded in a class of associative algebras equipped with twisted traces. In the case of interfaces, this yields a new connection to integrability.
C-P-T Fractionalization, and Quantum Criticality Beyond the Standard Model
Member Seminar
Speaker: Juven Wang
Title: C-P-T Fractionalization, and Quantum Criticality Beyond the Standard Model
Abstract: Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as a Z2-involution on the spacetime coordinates; but together with a charge conjugation C and the fermion parity (−1)^F, these symmetries can be further fractionalized forming nonabelian C-P-R-T-(−1)^F group structures, in various examples such as relativistic Lorentz invariant Dirac spinor quantum field theories (QFT), or nonrelativistic quantum many-body systems (involving Majorana zero modes). This result answers Prof. Shing-Tung Yau’s question on “Can C-P-T symmetries be fractionalized more than involutions?” based on arxiv:2109.15320.
In the second part of my talk, I will sketch to explain how can we modify the so(10) Grand Unified Theory (GUT) by adding a new topological term such that two GUTs of Georgi-Glashow and Pati-Salam can smoother into each other in a quantum phase transition, where the Standard Model and new dark sector physics can occur naturally near the critical region. The new modified so(10) GUT requires a double Spin structure that we name DSpin. This phenomenon is inspired by the “deconfined quantum criticality” in condensed matter. Based on arxiv:2106.16248.
Peeling properties of the spinor fields and the solutions to nonlinear Dirac equations
Abstract: The Dirac equation is a relativistic equation that describes the spin-1/2 particles. We talk about Dirac equations in Minkowski spacetime. In a geometric viewpoint, we can see that the spinor fields satisfying the Dirac equations enjoy the so-called peeling properties. It means the null components of the solution will decay at different rates along the null hypersurface. Based on this decay mechanism, we can obtain a fresh insight to the spinor null forms which is used to prove a small data global existence result especially for some quadratic Dirac models.
Wall-crossing from Higgs bundles to vortices
Speaker: Du Pei
Title: Wall-crossing from Higgs bundles to vortices
Abstract: Quantum field theories can often be used to uncover hidden algebraic structures in geometry and hidden geometric structures in algebra. In this talk, I will demonstrate how such “wall-crossing” can relate the moduli space of Higgs bundles with the moduli space of vortices.
The Large D Limit of Einstein’s Equations
Abstract: Taking the large dimension limit of Einstein’s equations is a useful strategy for solving and understanding the dynamics that these equations encode. I will introduce the underlying ideas and the progress that has resulted in recent years from this line of research. Most of the discussion will be classical in nature and will concern situations where there is a black hole horizon. A main highlight of this approach is the formulation of effective membrane theories of black hole dynamics. These have made possible to efficiently study, with relatively simple techniques, some of the thorniest problems in black hole physics, such as the non-linear evolution of the instabilities of black strings and black branes, and the collisions and mergers of higher-dimensional black holes. Open directions and opportunities will also be discussed. To get a flavor of what this is about, you may read the first few pages of the review (with C.P. Herzog) e-Print: 2003.11394.
Instability of naked singularities in general relativity
Member Seminar
Speaker: Jue Liu
Title: Instability of naked singularities in general relativity
Abstract: One of the fundamental problems in mathematical relativity is the weak cosmic censorship conjecture, proposed by Penrose, which roughly states that for generic physical spacetime, the singularities (if existed) must be hidden behind the black holes. Unfortunately, the singularities visible to faraway observers, which are called by naked singularities, indeed exist. The first example constructed by Christodoulou in 1994 is a family of self-similar spherically symmetric spacetime, in which the naked singularity forms due to a self-gravitating scalar field. Therefore the suitable censorship conjecture should be reduced to prove the instability of the naked singularities. In 1999 Christodoulou succeeded to prove the weak cosmic censorship conjecture in spherically symmetric cases, and recently the co-author and I found that the corresponding results have a big probability to be extended to spacetime without symmetries. In this talk I will discuss how to prove the instability of naked singularities using the energy method, and it is this wild method that helps us to extend some results to the asymmetric cases.
The classical interior of charged black holes with AdS asymptotics
Abstract: The gravitational dual to the grand canonical ensemble of a large N holographic theory is a charged black hole. These spacetimes can have Cauchy horizons that render the classical gravitational dynamics of the black hole interior incomplete. We show that a (spatially uniform) deformation of the CFT by a neutral scalar operator generically leads to a black hole with no inner horizon. There is instead a spacelike Kasner singularity in the interior. For relevant deformations, Cauchy horizons never form. We then consider charged scalars, which are known to condense at low temperatures, thus providing a holographic realization of superconductivity. We look inside the horizon of these holographic superconductors and find intricate dynamical behavior. The spacetime ends at a spacelike Kasner singularity, and there is no Cauchy horizon. Before reaching the singularity, there are several intermediate regimes which we study both analytically and numerically. These include strong Josephson oscillations in the condensate and possible `Kasner inversions’ in which after many e-folds of expansion, the Einstein-Rosen bridge contracts towards the singularity. Due to the Josephson oscillations, the number of Kasner inversions depends very sensitively on temperature, and diverges at a discrete set of temperatures that accumulate at the critical temperature. Near this discrete set of temperatures, the final Kasner exponent exhibits fractal-like behavior.
The complex Monge-Ampere equation in K\”ahler geometry
Speaker: Freid Tong
Title: The complex Monge-Ampere equation in Kahler geometry
Abstract: The complex Monge-Ampere equations occupies an central role in K\”ahler geometry, beginning with Yau’s famous solutions of the Calabi conjecture. Later developments has led to many interesting geometric applications and opening of new fields. In this talk, I will introduce the complex Monge-Ampere equation and discuss the interplay between their analysis and geometry, with a particular focus on the a priori C^0 estimates and their various applications. In the end, I will also try to discuss some recent work with B. Guo and D.H. Phong on a new approach for proving sharp C^0 estimates for complex Monge-Ampere equations, this new approach avoids the machinery of pluripotential theory that was previously necessary and has the advantage of generalizing to a large class of fully nonlinear equations.
Knowledge Graph Embeddings and Inference
Member Seminar
Speaker: Michael Douglas
Title: Knowledge Graph Embeddings and Inference
Abstract: A knowledge graph (KG) is a data structure which represents entities and relations as the vertices and edges of a directed graph. Two examples are Wikidata for general knowledge and SemMedDB for biomedical data.
A popular KG representation method is graph embedding, which facilitates question answering, inferring missing edges, and logical reasoning tasks. In this talk we introduce the topic and explain relevant mathematical results on graph embedding. We then analyze KG inference into several mechanisms: motif learning, network learning, and unstructured statistical inference, and describe experiments to measure the contributions of each mechanism.Joint work with M. Simkin, O. Ben-Eliezer, T. Wu, S. P. Chin, T. V. Dang and A. Wood.
Derived categories of nodal quintic del Pezzo threefolds
Abstract: Conifold transitions are important algebraic geometric constructions that have been of special interests in mirror symmetry, transforming Calabi-Yau 3-folds between A- and B-models. In this talk, I will discuss the change of the quintic del Pezzo 3-fold (Fano 3-fold of index 2 and degree 5) under the conifold transition at the level of the bounded derived category of coherent sheaves. The nodal quintic del Pezzo 3-fold X has at most 3 nodes. I will construct a semiorthogonal decomposition for D^b(X) and in the case of 1-nodal X, detail the change of derived categories from its smoothing to its small resolution.
Causality Comparison and Postive Mass
Abstract: Penrose et al. investigated the physical incoherence of the space-time with negative mass via the bending of light. Precise estimates of the time-delay of null geodesics were needed and played a pivotal role in their proof. In this paper, we construct an intermediate diagonal metric and reduce this problem to a causality comparison in the compactified space-time regarding time-like connectedness near conformal infinities. This different approach allows us to avoid encountering the difficulties and subtle issues that Penrose et al. met. It provides a new, substantially simple, and physically natural non-partial differential equation viewpoint to understand the positive mass theorem. This elementary argument modestly applies to asymptotically flat solutions that are vacuum and stationary near infinity
Growth and zero sets of eigenfunctions and of solutions to elliptic partial differential equations
From February 25 to March 1, the CMSA will be hosting a workshop on Growth and zero sets of eigenfunctions and of solutions to elliptic partial differential equations.
Key participants of this workshop include David Jerison (MIT), Alexander Logunov (IAS), and Eugenia Malinnikova (IAS). This workshop will have morning sessions on Monday-Friday of this week from 9:30-11:30am, and afternoon sessions on Monday, Tuesday, and Thursday from 3:00-5:00pm.
The sessions will be held in \(G02\) (downstairs) at 20 Garden, except for Tuesday afternoon, when the talk will be in \(G10\).9/24/2021 General Relativity Seminar
Title: On the Observable Shape of Black Hole Photon Rings
Abstract: The photon ring is a narrow ring-shaped feature, predicted by General Relativity but not yet observed, that appears on images of sources near a black hole. It is caused by extreme bending of light within a few Schwarzschild radii of the event horizon and provides a direct probe of the unstable bound photon orbits of the Kerr geometry. I will argue that the precise shape of the observable photon ring is remarkably insensitive to the astronomical source profile and can therefore be used as a stringent test of strong-field General Relativity. In practice, near-term interferometric observations may be limited to the visibility amplitude alone, which contains incomplete shape information: for convex curves, the amplitude only encodes the set of projected diameters (or “widths”) of the shape. I will describe the freedom in reconstructing a convex curve from its widths, giving insight into the photon ring shape information probed by technically plausible future astronomical measurements.
9/17/2021 General Relativity Seminar
Title: Stable Big Bang formation for the Einstein equations
Abstract: I will discuss recent work concerning stability of cosmological singularities described by the generalized Kasner solutions. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be stable under perturbations of the Kasner initial data, as long as the Kasner exponents are “sub-critical”. We prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the full regime where stable monotonic-type curvature blowup is expected. We treat the 3+1-dimensional Einstein-scalar field system and the D+1-dimensional Einstein-vacuum equations for D≥10. This is joint work with Speck and Fournodavlos.
9/10/2021 General Relativity Seminar
Title: Asymptotic localization, massive fields, and gravitational singularities
Abstract: I will review three recent developments on Einstein’s field equations under low decay or low regularity conditions. First, the Seed-to-Solution Method for Einstein’s constraint equations, introduced in collaboration with T.-C. Nguyen generates asymptotically Euclidean manifolds with the weakest or strongest possible decay (infinite ADM mass, Schwarzschild decay, etc.). The ‘asymptotic localization problem’ is also proposed an alternative to the ‘optimal localization problem’ by Carlotto and Schoen. We solve this new problem at the harmonic level of decay. Second, the Euclidian-Hyperboloidal Foliation Method, introduced in collaboration with Yue Ma, applies to nonlinear wave systems which need not be asymptotically invariant under Minkowski’s scaling field and to solutions with low decay in space. We established the global nonlinear stability of self-gravitating massive matter field in the regime near Minkowski spacetime. Third, in collaboration with Bruno Le Floch and Gabriele Veneziano, I studied spacetimes in the vicinity of singularity hypersurfaces and constructed bouncing cosmological spacetimes of big bang-big crunch type. The notion of singularity scattering map provides a flexible tool for formulating junction conditions and, by analyzing Einstein’s constraint equations, we established a surprising classification of all gravitational bouncing laws. Blog: philippelefloch.org
Threshold phenomena in random graphs and hypergraphs
Member Seminar
Speaker: Michael Simkin
Title: Threshold phenomena in random graphs and hypergraphs
Abstract: In 1959 Paul Erdos and Alfred Renyi introduced a model of random graphs that is the cornerstone of modern probabilistic combinatorics. Now known as the “Erdos-Renyi” model of random graphs it has far-reaching applications in combinatorics, computer science, and other fields.
The model is defined as follows: Given a natural number $n$ and a parameter $p \in [0,1]$, let $G(n;p)$ be the distribution on graphs with $n$ vertices in which each of the $\binom{n}{2}$ possible edges is present with probability $p$, independent of all others. Despite their apparent simplicity, the study of Erdos-Renyi random graphs has revealed many deep and non-trivial phenomena.
A central feature is the appearance of threshold phenomena: For all monotone properties (e.g., connectivity and Hamiltonicity) there is a critical probability $p_c$ such that if $p >> p_c$ then $G(n;p)$ possesses the property with high probability (i.e., with probability tending to 1 as $n \to \infty$) whereas if $p << p_c$ then with high probability $G(n;p)$ does not possess the property. In this talk we will focus on basic properties such as connectivity and containing a perfect matching. We will see an intriguing connection between these global properties and the local property of having no isolated vertices. We will then generalize the Erdos-Renyi model to higher dimensions where many open problems remain.
Stability and convergence issues in mathematical cosmology
Member Seminar
Speaker: Puskar Mondal
Title: Stability and convergence issues in mathematical cosmology
Abstract: The standard model of cosmology is built on the fact that while viewed on a sufficiently coarse-grained scale the portion of our universe that is accessible to observation appears to be spatially homogeneous and isotropic. Therefore this observed `homogeneity and isotropy’ of our universe is not known to be dynamically derived. In this talk, I will present an interesting dynamical mechanism within the framework of the Einstein flow (including physically reasonable matter sources) which suggests that many closed manifolds that do not support homogeneous and isotropic metrics at all will nevertheless evolve to be asymptotically compatible with the observed approximate homogeneity and isotropy of the physical universe. This asymptotic spacetime is naturally isometric to the standard FLRW models of cosmology. In order to conclude to what extent the asymptotic state is physically realized, one needs to study its stability properties. Therefore, I will briefly discuss the stability issue and its consequences (e.g., structure formation, etc).
Geometry, Entanglement and Quasi Local Data
Member Seminar
Speaker: Itamar Shamir
Title: Geometry, Entanglement and Quasi Local Data
Abstract: I will review some general ideas about gravity as motivation for an approach based on quasi local quantities.
Surfacehedra and the Binary Positive Geometry of Particle and “String” Amplitudes
Speaker: Nima Arkani-Hamed, IAS
Title: Surfacehedra and the Binary Positive Geometry of Particle and “String” Amplitudes
Geometric Analysis Seminar, Tuesdays at 9:50am
The seminar on geometric analysis will be held on Tuesdays from 9:50am to 10:50am with time for questions afterwards in CMSA Building, 20 Garden Street, Room G10. The tentative schedule can be found below. Titles will be added as they are provided.
Evolution Equations Seminar, Thursdays at 9:50am
The seminar for evolution equations, hyperbolic equations, and fluid dynamics will be held on Thursdays from 9:50am to 10:50am with time for questions afterwards in CMSA Building, 20 Garden Street, Room G10. The tentative schedule of speakers is below. Titles for the talks will be added as they are received.
Social Science Applications Forum
During the Summer of 2020, the CMSA will be hosting a periodic Social Science Applications Seminar.
The list of speakers is below and will be updated as details are confirmed.
For a list of past Social Science Applications talks, please click here.
Date Speaker Title/Abstract 7/13/2020 10:00-11:00am ET Ludovic Tangpi (Princeton) Please note, this seminar will take place online using Zoom. Title: Convergence of Large Population Games to Mean Field Games with Interaction Through the Controls
Abstract: This work considers stochastic differential games with a large number of players, whose costs and dynamics interact through the empirical distribution of both their states and their controls. We develop a framework to prove convergence of finite-player games to the asymptotic mean field game. Our approach is based on the concept of propagation of chaos for forward and backward weakly interacting particles which we investigate by fully probabilistic methods, and which appear to be of independent interest. These propagation of chaos arguments allow to derive moment and concentration bounds for the convergence of both Nash equilibria and social optima in non-cooperative and cooperative games, respectively. Incidentally, we also obtain convergence of a system of second order parabolic partial differential equations on finite dimensional spaces to a second order parabolic partial differential equation on the Wasserstein space.
For security reasons, you will have to show your full name to join the meeting.7/27/2020
10:00pmMichael Ewens (Caltech) Please note, this seminar will take place online using Zoom. Title: Measuring Intangible Capital with Market Prices
Abstract: Despite the importance of intangibles in today’s economy, current standards prohibit the capitalization of internally created knowledge and organizational capital, resulting in a downward bias of reported assets. As a result, researchers estimate this value by capitalizing prior flows of R&D and SG&A. In doing so, a set of capitalization parameters, i.e. the R&D depreciation rate and the fraction of SG&A that represents a long-lived asset, must be assumed. Parameters now in use are derived from models with strong assumptions or are ad hoc. We develop a capitalization model that motivates the use of market prices of intangibles to estimate these parameters. Two settings provide intangible asset values: (1) publicly traded equity prices and (2) acquisition prices. We use these parameters to estimate intangible capital stocks and subject them to an extensive set of diagnostic analyses that compare them with stocks estimated using existing parameters. Intangible stocks developed from exit price parameters outperform both stocks developed by publicly traded parameters and those stocks developed with existing estimates. (Joint work with Ryan Peters and Sean Wang.)
Small Cosmological Constants in String Theory
Abstract: We construct supersymmetric AdS4 vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by Kachru, Kallosh, Linde, and Trivedi. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the α 0 expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude < 10^{-123} in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol.
Mathematical supergravity and its applications to differential geometry
Virtual and in 20 Garden Street, Room G10Speaker: Carlos S. Shahbazi (Hamburg University)
Title: Mathematical supergravity and its applications to differential geometry
Abstract: I will discuss the recent developments in the mathematical theory of supergravity that lay the mathematical foundations of the universal bosonic sector of four-dimensional ungauged supergravity and its Killing spinor equations in a differential-geometric framework. I will provide the necessary context and background. explaining the results pedagogically from scratch and highlighting several open mathematical problems which arise in the mathematical theory of supergravity, as well as some of its potential mathematical applications. Work in collaboration with Vicente Cortés and Calin Lazaroiu.
Frontiers in Applied Mathematics and Computation
Together with the School of Engineering and Applied Sciences, the CMSA will be hosting a lecture series on the Frontiers in Applied Mathematics and Computation. Talks in this series will aim to highlight current research trends at the interface of applied math and computation and will explore the application of these trends to challenging scientific, engineering, and societal problems.
Lectures will take place on March 25, April 1, and April 29, 2021.
Speakers:
- George Biros (U.T. Austin)
- Laura Grigori (INRIA Paris)
- Samory K. Kpotufe (Columbia)
- Jonas Martin Peters (University of Copenhagen)
- Joseph M. Teran (UCLA)
The schedule below will be updated as talks are confirmed.
Date/Time Speaker Title/Abstract 3/25/2021
10:00 – 11:00am ETJoseph M. Teran Title: Affine-Particle-In-Cell with Conservative Resampling and Implicit Time Stepping for Surface Tension Forces Abstract: The Particle-In-Cell (PIC) method of Harlow is one of the first and most widely used numerical methods for Partial Differential Equations (PDE) in computational physics. Its relative efficiency, versatility and intuitive implementation have made it particularly popular in computational incompressible flow, plasma physics and large strain elastoplasticity. PIC is characterized by its dual particle/grid (Lagrangian/Eulerian) representation of material where particles are generally used to track material transport in a Lagrangian way and a structured Eulerian grid is used to discretize remaining spatial derivatives in the PDE. I will discuss the importance of conserving linear and angular momentum when switching between these two representations and the recent Affine-Particle-In-Cell (APIC) extension to PIC designed for this conservation. I will also discuss a recent APIC technique for discretizing surface tension forces and their linearizations needed for implicit time stepping. This technique is characterized by a novel surface resampling strategy and I will discuss a generalization of the APIC conservation to this setting.
4/1/2021
9:00 – 10:00am ETGeorge Biros Title: Inverse biophysical modeling and its application to neurooncology Abstract: A predictive, patient-specific, biophysical model of tumor growth would be an invaluable tool for causally connecting diagnostics with predictive medicine. For example, it could be used for tumor grading, characterization of the tumor microenvironment, recurrence prediction, and treatment planning, e.g., chemotherapy protocol or enrollment eligibility for clinical trials. Such a model also would provide an important bridge between molecular drivers of tumor growth and imaging-based phenotypic signatures, and thus, help identify and quantify mechanism-based associations between these two. Unfortunately, such a predictive biophysical model does not exist. Existing models undergoing clinical evaluation are too simple–they do not even capture the MRI phenotype. Although many highly complex models have been proposed, the major hurdle in deploying them clinically is their calibration and validation.
In this talk, I will discuss the challenges related to the calibration and validation of biophysical models, and in particular the mathematical structure of the underlying inverse problems. I will also present a new algorithm that localizes the tumor origin within a few millimeters.
4/1/2021
10:00 – 11:00am ETSamory K. Kpotufe Title: From Theory to Clustering Abstract: Clustering is a basic problem in data analysis, consisting of partitioning data into meaningful groups called clusters. Practical clustering procedures tend to meet two criteria: flexibility in the shapes and number of clusters estimated, and efficient processing. While many practical procedures might meet either of these criteria in different applications, general guarantees often only hold for theoretical procedures that are hard if not impossible to implement. A main aim is to address this gap.
We will discuss two recent approaches that compete with state-of-the-art procedures, while at the same time relying on rigorous analysis of clustering. The first approach fits within the framework of density-based clustering, a family of flexible clustering approaches. It builds primarily on theoretical insights on nearest-neighbor graphs, a geometric data structure shown to encode local information on the data density. The second approach speeds up kernel k-means, a popular Hilbert space embedding and clustering method. This more efficient approach relies on a new interpretation – and alternative use – of kernel-sketching as a geometry-preserving random projection in Hilbert space.
Finally, we will present recent experimental results combining the benefits of both approaches in the IoT application domain.
The talk is based on various works with collaborators Sanjoy Dasgupta, Kamalika Chaudhuri, Ulrike von Luxburg, Heinrich Jiang, Bharath Sriperumbudur, Kun Yang, and Nick Feamster.4/29/2021
12:00 – 1:00pm ETJonas Martin Peters Title: Causality and Distribution Generalization Abstract: Purely predictive methods do not perform well when the test distribution changes too much from the training distribution. Causal models are known to be stable with respect to distributional shifts such as arbitrarily strong interventions on the covariates, but do not perform well when the test distribution differs only mildly from the training distribution. We discuss anchor regression, a framework that provides a trade-off between causal and predictive models. The method poses different (convex and non-convex) optimization problems and relates to methods that are tailored for instrumental variable settings. We show how similar principles can be used for inferring metabolic networks. If time allows, we discuss extensions to nonlinear models and theoretical limitations of such methodology.
4/29/2021
1:00 – 2:00pm ETLaura Grigori Title: Randomization and communication avoiding techniques for large scale linear algebra Abstract: In this talk we will discuss recent developments of randomization and communication avoiding techniques for solving large scale linear algebra operations. We will focus in particular on solving linear systems of equations and we will discuss a randomized process for orthogonalizing a set of vectors and its usage in GMRES, while also exploiting mixed precision. We will also discuss a robust multilevel preconditioner that allows to further accelerate solving large scale linear systems on parallel computers.
Decoding Divergent Distances
Abstract: Motivated by a relationship between the Zamolodchikov and NLSM metrics to the so-called quantum information metric, I will discuss recent work (2106.11313) on understanding infinite distance limits within the context of information theory. I will describe how infinite distance points represent theories that are hyper-distinguishable, in the sense that they can be distinguished from “nearby” theories with certainty in relatively few measurements. I will then discuss necessary and sufficient ingredients for the appearance of these infinite distance points, illustrate these in simple examples, and describe how this perspective can help the swampland program.
2020 Big Data Conference (Virtual)
On August 24-25, 2020 the CMSA hosted our sixth annual Conference on Big Data. The Conference featured many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics. The 2020 Big Data Conference took place virtually.
Organizers:
- Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University
- Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business
- Horng-Tzer Yau, Professor of Mathematics, Harvard University
- Sergiy Verstyuk, CMSA, Harvard University
Speakers:
- Sanjeev Arora, Princeton University
- Juan Camilo Castillo, University of Pennsylvania
- Joseph Dexter, Dartmouth College
- Nicole Immorlica, Microsoft
- Amin Saberi, Stanford University
- Vira Semenova, University of California, Berkeley
- Varda Shalev, Tel Aviv University
Schedule:
Computational Biology Symposium
On May 3, 2021 the CMSA will be hosting a Computational Biology Symposium virtually on Zoom. This symposium will be organized by Vijay Kuchroo.
The symposium will begin at 10:00am ET. There will be a morning and afternoon session, with an hour break for lunch.
Videos of the talks can be found in this Youtube playlist. Links are also available in the schedule below.
Confirmed participants:
- Uri Alon, Weizmann Institute
- Elana Fertig, Johns Hopkins
- Martin Hemberg, Brigham and Women’s Hospital
- Peter Kharchenko, Harvard University
- Smita Krishnaswamy, Yale University
- John Marioni, EMBL-EBI
- Eran Segal, Weizmann Institute
- Meromit Singer, Harvard Medical School
Schedule:
Mathematical Physics Seminar, Mondays
The seminar on mathematical physics will be held on Mondays from 10:00 – 11:00am ET on Zoom. Please email the seminar organizers to learn how toattend. This year’s Seminar will be organized by Yoosik Kim (yoosik@cmsa.fas.harvard.edu), Tsung-Ju Lee (tjlee@cmsa.fas.harvard.edu), and Yang Zhou (yangzhou@cmsa.fas.harvard.edu).
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The list of speakers for the upcoming academic year will be posted below and updated as details are confirmed. Titles and abstracts for the talks will be added as they are received.
Spring 2021:
Date Speaker Title/Abstract 2/1/2021 Choa Dongwook
(KIAS)VideoTitle: Fukaya category of Landau-Ginzburg orbifolds. Abstract: Landau-Ginzburg orbifold is just another name for a holomorphic function W with its abelian symmetry G. Its Fukaya category can be viewed as a categorification of a homology group of its Milnor fiber. In this introductory talk, we will start with some classical results on the topology of isolated singularities and its Fukaya-Seidel category. Then I will explain a new construction for such category to deal with a non-trivial symmetry group G. The main ingredients are classical variation map and the Reeb dynamics at the contact boundary. If time permits, I will show its application to mirror symmetry of LG orbifolds and its Milnor fiber. This is a joint work with C.-H. Cho and W. Jeong
2/8/2021 Jérémy Guéré (Fourier Institute) Title: Congruences on K-theoretic Gromov-Witten invariants Abstract: K-theoretic Gromov-Witten invariants of smooth projective varieties have been introduced by YP Lee, using the Euler characteristic of a virtual structure sheaf. In particular, they are integers. In this talk, I look at these invariants for the quintic threefold and I will explain how to compute them modulo 41, using the virtual localization formula under a finite group action, up to genus 19 and degree 40.
2/15/2021 Zhiwei Zheng (Max Planck Institute) Title: Some new results on automorphisms of hypersurfaces Abstract: It is natural to study automorphisms of hypersurfaces in projective spaces. In this talk, I will discuss a new approach to determine all possible orders of automorphisms of smooth hypersurfaces with fixed degree and dimension. Then we consider the specific case of cubic fourfolds, and discuss the relation with Hodge theory.
2/22/2021 Yu-Shen Lin (Boston University) Title: Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces Abstract: Strominger–Yau–Zaslow conjecture predicts the existence of special Lagrangian fibrations on Calabi–Yau manifolds. The conjecture inspires the development of mirror symmetry while the original conjecture has little progress. In this talk, I will confirm the conjecture for the complement of a smooth anti-canonical divisor in del Pezzo surfaces. Moreover, I will also construct the dual torus fibration on its mirror. As a consequence, the special Lagrangian fibrations detect a non-standard semi-flat metric and some Ricci-flat metrics that don’t obviously appear in the literature. This is based on a joint work with T. Collins and A. Jacob.
3/1/2021 Carlos S. Shahbazi (Hamburg University) Title: Mathematical supergravity and its applications to differential geometry. Abstract: I will discuss the recent developments in the mathematical theory of supergravity that lay the mathematical foundations of the universal bosonic sector of four-dimensional ungauged supergravity and its Killing spinor equations in a differential-geometric framework. I will provide the necessary context and background. explaining the results pedagogically from scratch and highlighting several open mathematical problems which arise in the mathematical theory of supergravity, as well as some of its potential mathematical applications. Work in collaboration with Vicente Cortés and Calin Lazaroiu.
3/8/2021 Miguel Moreira (ETH) Title: Virasoro constraints for stable pairs. Abstract: The theory of stable pairs (PT) with descendents, defined on a 3-fold X, is a sheaf theoretical curve counting theory. Conjecturally, it is equivalent to the Gromov-Witten (GW) theory of X via a universal (but intricate) transformation, so we can expect that the Virasoro conjecture on the GW side should have a parallel in the PT world. In joint work with A. Oblomkov, A. Okounkov, and R. Pandharipande, we formulated such a conjecture and proved it for toric 3-folds in the stationary case. The Hilbert scheme of points on a surface S might be regarded as a component of the moduli space of stable pairs on S x P1, and the Virasoro conjecture predicts a new set of relations satisfied by tautological classes on S[n] which can be proven by reduction to the toric case.
3/15/2021 Spring break 3/22/2021 Ying Xie (Shanghai Center for Mathematical Sciences) Title: Derived categories for Grassmannian flips Abstract: Flip is a fundamental surgery operation for constructing minimal models in higher-dimensional birational geometry. In this talk, I will introduce a series of flips from Lie theory and investigate their derived categories. This is a joint program with Conan Leung.
3/29/2021 Emanuel Scheidegger (Peking University) Title: On the quantum K-theory of the quintic.
Abstract: Quantum cohomology is a deformation of the cohomology of a projective variety governed by counts of stable maps from a curve into this variety. Quantum K-theory is in a similar way a deformation of K-theory but also of quantum cohomology, It has recently attracted attention in physics since a realization in a physical theory has been found. Currently, both the structure and examples in quantum K-theory are far less understood than in quantum cohomology.
We will explain the properties of quantum K-theory in comparison with quantum cohomology, and we will discuss the examples of projective space and the quintic hypersurface in P^4.4/5/2021 Gaëtan Borot (HU Berlin) Title: Topological recursion in 4d N = 2 supersymmetric gauge theories Abstract: According to the Alday-Gaiotto-Tachikawa conjecture (proved in this case by Schiffman and Vasserot), the instanton partition function in 4d N = 2 SU(r) supersymmetric gauge theory on P^2 with equivariant parameters \epsilon_1,\epsilon_2 is the norm of a Whittaker vector for W(gl_r) algebra. I will explain how these Whittaker vectors can be computed (at least perturbatively in the energy scale) by topological recursion for \epsilon_1 +\epsilon_2 = 0, and by a non-commutation version of the topological recursion in the Nekrasov-Shatashvili regime where \epsilon_1/\epsilon_2 is fixed. This is a joint work to appear with Bouchard, Chidambaram and Creutzig.
4/12/2021 Fei Yan (Rutgers) Title: Networks and quantization Abstract: I will describe two quantization scenarios. The first scenario involves the construction of a quantum trace map computing a link “invariant” (with possible wall-crossing behavior) for links L in a 3-manifold M, where M is a Riemann surface C times a real line. This construction unifies the computation of familiar link invariant with the refined counting of framed BPS states for line defects in 4d N=2 theories of class S. Certain networks on C play an important role in the construction. The second scenario concerns the study of Schroedinger equations and their higher order analogues, which could arise in the quantization of Seiberg-Witten curves in 4d N=2 theories. Here similarly certain networks play an important part in the exact WKB analysis for these Schroedinger-like equations. At the end of my talk I will also try to sketch a possibility to bridge these two scenarios.
4/19/2021 Hazel Mak (Brown University) Title: Branching Rules and Young Tableaux Methods: 10D & 11D Supergravity Abstract: In this talk, I will review 4D, N = 1 off-shell supergravity. Then I present explorations to construct 10D and 11D supergravity theories in two steps. The first step is to decompose scalar superfield into Lorentz group representations which involves branching rules and related methods. Interpretations of component fields by Young tableaux methods will be presented. The second step is to implement an analogue of Breitenlohner’s approach for 4D supergravity to 10D and 11D theories.
4/26/2021 Owen Gwilliam (UMass. Amherst) Title: Topological-holomorphic field theories and their BV quantizations Abstract: Topological field theories and holomorphic field theories have each had a substantial impact in both physics and mathematics, so it is natural to consider theories that are hybrids of the two, which we call topological-holomorphic and denote as THFTs. Examples include Kapustin’s twist of N=2, D=4 supersymmetric Yang-Mills theory and Costello’s 4-dimensional Chern-Simons theory. In this talk about joint work with Rabinovich and Williams, I will define THFTs, describe several examples, and then explain how to quantize them rigorously and explicitly, by building on techniques of Si Li. Time permitting, I will indicate how these results offer a novel perspective on the Gaudin model via 3-dimensional field theories.
Fall 2020:
Date Speaker Title/Abstract 9/14/2020 Lino Amorim (Kansas State University) Title: Non-commutative Gromov-Witten invariants Abstract: I will describe an analogue of Saito’s theory of primitive forms for Calabi-Yau A-infinity categories. Under some conditions on the Hochschild cohomology of the category, this construction recovers the (genus zero) Gromov-Witten invariants of a symplectic manifold from its Fukaya category. This includes many compact toric manifolds, in particular projective spaces.
9/21/2020 Yuhan Sun (Rutgers) Title: Displacement energy of Lagrangian 3-spheres Abstract: We study local and global Hamiltonian dynamical behaviors of some Lagrangian submanifolds near a Lagrangian sphere S in a symplectic manifold X. When dim S = 2, we show that there is a one-parameter family of Lagrangian tori near S, which are nondisplaceable in X. When dim S = 3, we obtain a new estimate of the displacement energy of S, by estimating the displacement energy of a one-parameter family of Lagrangian tori near S.
9/28/2020 Shota Komatsu (CERN) Title: Wilson loops as matrix product states Abstract: In this talk, I will discuss a reformulation of the Wilson loop in large N gauge theories in terms of matrix product states. The construction is motivated by the analysis of supersymmetric Wilson loops in the maximally super Yang–Mills theory in four dimensions, but can be applied to any other large N gauge theories and matrix models, although less effective. For the maximally super Yang–Mills theory, one can further perform the computation exactly as a function of ‘t Hooft coupling by combining our formulation with the relation to integrable spin chains.
10/5/2020 Ming Zhang (UBC) Title: Verlinde/Grassmannian correspondence and applications. Abstract: In the 90s’, Witten gave a physical derivation of an isomorphism between the Verlinde algebra of $GL(n)$ of level $l$ and the quantum cohomology ring of the Grassmannian $\text{Gr}(n,n+l)$. In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten’s work by relating the $\text{GL}_{n}$ Verlinde numbers to the level $l$ quantum K-invariants of the Grassmannian $\text{Gr}(n,n+l)$, and refer to it as the Verlinde/Grassmannian correspondence.
The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there. In this talk, I will discuss the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner. At the end of the talk, I will describe some applications of this correspondence.
10/12/2020 Cancelled -Columbus Day 10/19/2020 Ben Gammage (Harvard) Title: 3d mirror symmetry for abelian gauge groups Abstract: 3d mirror symmetry is a proposed duality relating a pair of 3-dimensional supersymmetric gauge theories. Various consequences of this duality have been heavily explored by representation theorists in recent years, under the name of “symplectic duality”. In joint work in progress with Justin Hilburn, for the case of abelian gauge groups, we provide a fully mathematical explanation of this duality in the form of an equivalence of 2-categories of boundary conditions for topological twists of these theories. We will also discuss some applications to homological mirror symmetry and geometric Langlands duality.
10/26/2020 Cancelled 11/2/2020 Haoyu Sun (Berkeley) Title: Double-Janus linear sigma models and generalized quadratic reciprocity
Abstract: We study the supersymmetric partition function of a 2d linear sigma-model whose target space is a torus with a complex structure that varies along one worldsheet direction and a Kähler modulus that varies along the other. This setup is inspired by the dimensional reduction of a Janus configuration of 4d N=4 U(1) Super-Yang-Mills theory compactified on a mapping torus (T^2 fibered over S^1) times a circle with an SL(2,Z) duality wall inserted on S^1, but our setup has minimal supersymmetry. The partition function depends on two independent elements of SL(2,Z), one describing the duality twist, and the other describing the geometry of the mapping torus. It is topological and can be written as a multivariate quadratic Gauss sum. By calculating the partition function in two different ways, we obtain identities relating different quadratic Gauss sums, generalizing the Landsberg-Schaar relation. These identities are a subset of a collection of identities discovered by F. Deloup. Each identity contains a phase which is an eighth root of unity, and we show how it arises as a Berry phase in the supersymmetric Janus-like configuration. Supersymmetry requires the complex structure to vary along a semicircle in the upper half-plane, as shown by Gaiotto and Witten in a related context, and that semicircle plays an important role in reproducing the correct Berry phase.11/9/2020 An Huang (Brandeis) Title: p-adic strings, Einstein equations, Green’s functions, and Tate’s thesis
Abstract: I shall discuss a recent work on how p-adic strings can produce perturbative quantum gravity, and an adelic physics interpretation of Tate’s thesis.11/16/2020
10:00am ETMatt Kerr (WUSTL) Title: Differential equations and mixed Hodge structures Abstract: We report on a new development in asymptotic Hodge theory, arising from work of Golyshev–Zagier and Bloch–Vlasenko, and connected to the Gamma Conjectures in Fano/LG-model mirror symmetry. The talk will focus exclusively on the Hodge/period-theoretic aspects through two main examples.
Given a variation of Hodge structure M on a Zariski open in P^1, the periods of the limiting mixed Hodge structures at the punctures are interesting invariants of M. More generally, one can try to compute these asymptotic invariants for iterated extensions of M by “Tate objects”, which may arise for example from normal functions associated to algebraic cycles. The main point of the talk will be that (with suitable assumptions on M) these invariants are encoded in an entire function called the motivic Gamma function, which is determined by the Picard-Fuchs operator L underlying M. In particular, when L is hypergeometric, this is easy to compute and we get a closed-form answer (and a limiting motive). In the non-hypergeometric setting, it yields predictions for special values of normal functions; this part of the story is joint with V. Golyshev and T. Sasaki.11/23/2020 11:30am ET
Kyoung-Seog Lee (U of Miami) Title: Derived categories and motives of moduli spaces of vector bundles on curves Abstract: Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. On the other hand, moduli spaces of vector bundles on curves provide attractive and important examples of algebraic varieties and there have been intensive works studying them. In this talk, I will discuss derived categories and motives of moduli spaces of vector bundles on curves. This talk is based on joint works with I. Biswas and T. Gomez.
11/30/2020 Zijun Zhou (IPMU) Title: 3d N=2 toric mirror symmetry and quantum K-theory Abstract: In this talk, I will introduce a new construction for the K-theoretic mirror symmetry of toric varieties/stacks, based on the 3d N=2 mirror symmetry introduced by Dorey-Tong. Given the toric datum, i.e. a short exact sequence 0 -> Z^k -> Z^n -> Z^{n-k} -> 0, we consider the toric Artin stack of the form [C^n / (C^*)^k]. Its mirror is constructed by taking the Gale dual of the defining short exact sequence. As an analogue of the 3d N=4 case, we consider the K-theoretic I-function, with a suitable level structure, defined by counting parameterized quasimaps from P^1. Under mirror symmetry, the I-functions of a mirror pair are related to each other under the mirror map, which exchanges the K\”ahler and equivariant parameters, and maps q to q^{-1}. This is joint work with Yongbin Ruan and Yaoxiong Wen.
12/7/2020 Thomas Grimm (Utrecht) Title: Moduli Space Holography and the Finiteness of Flux Vacua Abstract: In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using asymptotic Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The bulk reconstruction from the boundary data is provided by the sl(2)-orbit theorem of Schmid and Cattani, Kaplan, Schmid, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us, in work with C. Schnell, to prove that the number of self-dual flux vacua is finite
For a listing of previous Mathematical Physics Seminars, please click here.
Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces
Speaker: Yu-Shen Lin (Boston University)
Title: Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts the existence of special Lagrangian fibrations on Calabi-Yau manifolds. The conjecture inspires the development of mirror symmetry while the original conjecture has little progress. In this talk, I will confirm the conjecture for the complement of a smooth anti-canonical divisor in del Pezzo surfaces. Moreover, I will also construct the dual torus fibration on its mirror. As a consequence, the special Lagrangian fibrations detect a non-standard semi-flat metric and some Ricci-flat metrics that don’t obviously appear in the literature. This is based on a joint work with T. Collins and A. Jacob.
CMSA Math-Science Literature Lecture: Subfactors–in Memory of Vaughan Jones
Zhengwei Liu (Tsinghua University)
Title: Subfactors–in Memory of Vaughan Jones
Abstract: Jones initiated modern subfactor theory in early 1980s and investigated this area for his whole academic life. Subfactor theory has both deep and broad connections with various areas in mathematics and physics. One well-known peak in the development of subfactor theory is the discovery of the Jones polynomial, for which Jones won the Fields Metal in 1990. Let us travel back to the dark room at the beginning of the story, to appreciate how radically our viewpoint has changed.
Talk chair: Arthur Jaffe
Previous Random Matrix & Probability Theory Seminars
Spring 2020:
Date Speaker Title/Abstract 2/26/2020 Louigi Addario-Berry (McGill University) Title: Hipster random walks and their ilk Abstract: I will describe how certain recursive distributional equations can be solved by importing rigorous results on the convergence of approximation schemes for degenerate PDEs, from numerical analysis. This project is joint work with Luc Devroye, Hannah Cairns, Celine Kerriou, and Rivka Maclaine Mitchell.
4/1/2020 Ian Jauslin (Princeton) This meeting will be taking place virtually on Zoom. Title: A simplified approach to interacting Bose gases
Abstract: I will discuss some new results about an effective theory introduced by Lieb in 1963 to approximate the ground state energy of interacting Bosons at low density. In this regime, it agrees with the predictions of Bogolyubov. At high densities, Hartree theory provides a good approximation. In this talk, I will show that the ’63 effective theory is actually exact at both low and high densities, and numerically accurate to within a few percents in between, thus providing a new approach to the quantum many body problem that bridges the gap between low and high density.4/22/2020 Martin Gebert (UC Davis) This meeting will be taking place virtually on Zoom. Title: Lieb-Robinson bounds for a class of continuum many-body fermion systems
Abstract: We introduce a class of UV-regularized two-body interactions for
fermions in $\R^d$ and prove a Lieb-Robinson estimate for the dynamics
of this class of many-body systems. As a step towards this result, we
also prove a propagation bound of Lieb-Robinson type for continuum
one-particle Schr\“odinger operators. We apply the propagation bound to
prove the existence of a strongly continuous infinite-volume dynamics on
the CAR algebra.4/29/2020 Marcin Napiórkowski (University of Warsaw) This meeting will be taking place virtually on Zoom. Title: Free energy asymptotics of the quantum Heisenberg spin chain
Abstract: Spin wave theory suggests that low temperature properties of the Heisenberg model can be described in terms of noninteracting quasiparticles called magnons. In my talk I will review the basic concepts and predictions of spin wave approximation and report on recent rigorous results in that direction. Based on joint work with Robert Seiringer.
5/6/2020 Antti Knowles (University of Geneva) Title: Field theory as a limit of interacting quantum Bose gases
Abstract: We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d = 1,2,3. For d > 1 the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. The proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials. Joint work with Jürg Fröhlich, Benjamin Schlein, and Vedran Sohinger.5/13/2020 Sven Bachmann (University of British Columbia) Title: Quantized quantum transport and Abelian anyons Abstract: I’ll discuss recent developments in the study of quantized quantum transport, focussing on the quantum Hall effect. Beyond presenting an index taking rational values, and which is the Hall conductance in the adapted setting, I will explain how the index is intimately paired with the existence of quasi-particle excitations having non-trivial braiding properties.
5/20/2020 Kristina Schubert (TU Dortmund) Title: Fluctuation Results for General Ising Models — Block Spin Ising Models and Random Interactions Abstract: Starting from the classical Curie-Weiss model in statistical mechanics, we will consider more general Ising models. On the one hand, we introduce a block structure, i.e. a model of spins in which the vertices are divided into a finite number of blocks and where pair interactions are given according to their blocks. The magnetization is then the vector of magnetizations within each block, and we are interested in its behaviour and in particular in its fluctuations. On the other hand, we consider Ising models on Erdős-Rényi random graphs. Here, I will also present results on the fluctuations of the magnetization.
Fall 2019:
Date Speaker Title/Abstract 9/11/2019 Subhabrata Sen Title: Sampling convergence for random graphs: graphexes and multigraphexes Abstract: We will look at structural properties of large, sparse random graphs through the lens of sampling convergence (Borgs, Chayes, Cohn and Veitch ’17). Sam- pling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a graphex. We will introduce this framework and motivate the components of a graphex. Subsequently, we will discuss the graphex limit for several well-known sparse random (multi)graph models. This is based on joint work with Christian Borgs, Jennifer Chayes, and Souvik Dhara.
9/25/2019 Jeff Schenker (Michigan State) Title: An ergodic theorem for homogeneously distributed quantum channels with applications to matrix product states Abstract: Quantum channels represent the most general physical evolution of a quantum system through unitary evolution and a measurement process. Mathematically, a quantum channel is a completely positive and trace preserving linear map on the space of $D\times D$ matrices. We consider ergodic sequences of channels, obtained by sampling channel valued maps along the trajectories of an ergodic dynamical system. The repeated composition of these maps along such a sequence could represent the result of repeated application of a given quantum channel subject to arbitrary correlated noise. It is physically natural to assume that such repeated compositions are eventually strictly positive, since this is true whenever any amount of decoherence is present in the quantum evolution. Under such an hypothesis, we obtain a general ergodic theorem showing that the composition of maps converges exponentially fast to a rank-one — “entanglement breaking’’ – channel. We apply this result to describe the thermodynamic limit of ergodic matrix product states and prove that correlations of observables in such states decay exponentially in the bulk. (Joint work with Ramis Movassagh)
10/3/2019 Thursday
4:30pm
Jian Ding (UPenn) Title: Distances associated with Liouville quantum gravity Abstract: I will review some recent progresses on distances associated with Liouville quantum gravity, which is a random measure obtained from exponentiating a planar Gaussian free field.
The talk is based on works with Julien Dubédat, Alexander Dunlap, Hugo Falconet, Subhajit Goswami, Ewain Gwynne, Ofer Zeitouni and Fuxi Zhang in various combinations.
10/9/2019 Ruth Williams (UCSD) Title: Stability of a Fluid Model for Fair Bandwidth Sharing with General File Size Distributions Abstract: Massoulie and Roberts introduced a stochastic model for a data communication network where file sizes are generally distributed and the network operates under a fair bandwidth sharing policy. It has been a standing problem to prove stability of this general model when the average load on the system is less than the network’s capacity. A crucial step in an approach to this problem is to prove stability of an associated measure-valued fluid model. We shall describe prior work on this question done under various strong assumptions and indicate how to prove stability of the fluid model under mild conditions.
This talk is based on joint work with Yingjia Fu.
10/11/2019 Cancelled 10/16/2019 Wei-Kuo Chen (University of Minnesota) Title: The generalized TAP free energy Abstract: Spin glasses are disordered spin systems initially invented by theoretical physicists with the aim of understanding some strange magnetic properties of certain alloys. In particular, over the past decades, the study of the Sherrington-Kirkpatrick (SK) mean-field model via the replica method has received great attention. In this talk, I will discuss another approach to studying the SK model proposed by Thouless-Anderson-Palmer (TAP). I will explain how the generalized TAP correction appears naturally and give the corresponding generalized TAP representation for the free energy. Based on a joint work with D. Panchenko and E. Subag.
10/23/2019 Souvik Dhara (MIT) Title: A new universality class for critical percolation on networks with heavy-tailed degrees Abstract: The talk concerns critical behavior of percolation on finite random networks with heavy-tailed degree distribution. In a seminal paper, Aldous (1997) identified the scaling limit for the component sizes in the critical window of phase transition for the Erdős-Rényi random graph. Subsequently, there has been a surge in the literature identifying two universality classes for the critical behavior depending on whether the asymptotic degree distribution has a finite or infinite third moment.
In this talk, we will present a completely new universality class that arises in the context of degrees having infinite second moment. Specifically, the scaling limit of the rescaled component sizes is different from the general description of multiplicative coalescent given by Aldous and Limic (1998). Moreover, the study of critical behavior in this regime exhibits several surprising features that have never been observed in any other universality classes so far.
This is based on joint works with Shankar Bhamidi, Remco van der Hofstad, Johan van Leeuwaarden.
10/30/2019 Aram Harrow (MIT) Title: Random quantum circuits, phase transitions and complexity Abstract: Random unitary dynamics are a toy model for chaotic quantum dynamics and also have applications to quantum information theory and computing. Recently, random quantum circuits were the basis of Google’s announcement of “quantum computational supremacy,” meaning performing a task on a programmable quantum computer that would difficult or infeasible for any classical computer. Google’s approach is based on the conjecture that random circuits are as hard to classical computers to simulate as a worst-case quantum computation would be. I will describe evidence in favor of this conjecture for deep random circuits and against this conjecture for shallow random circuits. (Deep/shallow refers to the number of time steps of the quantum circuit.) For deep random circuits in Euclidean geometries, we show that quantum dynamics match the first few moments of the Haar measure after roughly the amount of time needed for a signal to propagate from one side of the system to the other. In non-Euclidean geometries, such as the Schwarzschild metric in the vicinity of a black hole, this turns out not to be always true. I will also explain how shallow quantum circuits are easier to simulate when the gates are randomly chosen than in the worst case. This uses a simulation algorithm based on tensor contraction which is analyzed in terms of an associated stat mech model.
This is based on joint work with Saeed Mehraban (1809.06957) and with John Napp, Rolando La Placa, Alex Dalzell and Fernando Brandao (to appear).
11/6/2019 Bruno Nachtergaele (UC Davis) Title: The transmission time and local integrals of motion for disordered spin chains Abstract: We investigate the relationship between zero-velocity Lieb-Robinson bounds and the existence of local integrals of motion (LIOMs) for disordered quantum spin chains. We also study the effect of dilute random perturbations on the dynamics of many-body localized spin chains. Using a notion of transmission time for propagation in quantum lattice systems we demonstrate slow propagation by proving a lower bound for the transmission time. This result can be interpreted as a robustness property of slow transport in one dimension. (Joint work with Jake Reschke)
11/13/2019 Gourab Ray (University of Victoria) Title: Logarithmic variance of height function of square-iceAbstract: A homomorphism height function on a finite graph is a integer-valued function on the set of vertices constrained to have adjacent vertices take adjacent integer values. We consider the uniform distribution over all such functions defined on a finite subgraph of Z^2 with predetermined values at some fixed boundary vertices. This model is equivalent to the height function of the six-vertex model with a = b = c = 1, i.e. to the height function of square-ice. Our main result is that in a subgraph of Z^2 with zero boundary conditions, the variance grows logarithmically in the distance to the boundary. This establishes a strong form of roughness of the planar uniform homomorphisms. Joint work with: Hugo Duminil Copin, Matan Harel, Benoit Laslier and Aran Raoufi.
11/20/2019 Vishesh Jain (MIT) Title: A combinatorial approach to the quantitative invertibility of random matrices. Abstract: Abstract: Let $s_n(M_n)$ denote the smallest singular value of an $n\times n$ random matrix $M_n$. We will discuss a novel combinatorial approach (in particular, not using either inverse Littlewood–Offord theory or net arguments) for obtaining upper bounds on the probability that $s_n(M_n)$ is smaller than $\eta \geq 0$ for quite general random matrix models. Such estimates are a fundamental part of the non-asymptotic theory of random matrices and have applications to the strong circular law, numerical linear algebra etc. In several cases of interest, our approach provides stronger bounds than those obtained by Tao and Vu using inverse Littlewood–Offord theory.
2018-2019
Date Speaker Title/Abstract 9/28/2018 *Friday, 10:00am*
Yash Deshpande (MIT) Title: Estimating low-rank matrices in noise: phase transitions from spin glass theory Abstract: Estimating low-rank matrices from noisy observations is a common task in statistical and engineering applications. Following the seminal work of Johnstone, Baik, Ben-Arous and Peche, versions of this problem have been extensively studied using random matrix theory. In this talk, we will consider an alternative viewpoint based on tools from mean field spin glasses. We will present two examples that illustrate how these tools yield information beyond those from classical random matrix theory. The first example is the two-groups stochastic block model (SBM), where we will obtain a full information-theoretic understanding of the estimation phase transition. In the second example, we will augment the SBM with covariate information at nodes, and obtain results on the altered phase transition.
This is based on joint works with Emmanuel Abbe, Andrea Montanari, Elchanan Mossel and Subhabrata Sen.
10/3/2018 Ian Jauslin (IAS) Title: Liquid Crystals and the Heilmann-Lieb model Abstract: In 1979, O.Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of a nematic liquid crystal phase in it. In such a phase, dimers spontaneously align, but there is no long range translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. I will discuss a recent proof of this conjecture. This is joint work with Elliott H. Lieb.
10/10/2018 Afonso Bandeira (NYU Title: Statistical estimation under group actions: The Sample Complexity of Multi-Reference Alignment Abstract: Many problems in signal/image processing, and computer vision amount to estimating a signal, image, or tri-dimensional structure/scene from corrupted measurements. A particularly challenging form of measurement corruption are latent transformations of the underlying signal to be recovered. Many such transformations can be described as a group acting on the object to be recovered. Examples include the Simulatenous Localization and Mapping (SLaM) problem in Robotics and Computer Vision, where pictures of a scene are obtained from different positions and orientations; Cryo-Electron Microscopy (Cryo-EM) imaging where projections of a molecule density are taken from unknown rotations, and several others.
One fundamental example of this type of problems is Multi-Reference Alignment: Given a group acting in a space, the goal is to estimate an orbit of the group action from noisy samples. For example, in one of its simplest forms, one is tasked with estimating a signal from noisy cyclically shifted copies. We will show that the number of observations needed by any method has a surprising dependency on the signal-to-noise ratio (SNR), and algebraic properties of the underlying group action. Remarkably, in some important cases, this sample complexity is achieved with computationally efficient methods based on computing invariants under the group of transformations.
10/17/2018 3:30pm
Thomas Chen (UT Austin) Title: Dynamics of a heavy quantum tracer particle in a Bose gas Abstract: We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in R^3. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to 1/N where N is the expected particle number. Assuming that the mass of the tracer particle is proportional to N, we derive generalized Hartree equations in the limit where N tends to infinity. Moreover, we prove the global well-posedness of the associated Cauchy problem for sufficiently weak interaction potentials. This is joint work with Avy Soffer (Rutgers University).
10/24/2018 *Room G02*
Tselil Schramm (Harvard/MIT) Title: (Nearly) Efficient Algorithms for the Graph Matching Problem in Correlated Random Graphs Abstract: The Graph Matching problem is a robust version of the Graph Isomorphism problem: given two not-necessarily-isomorphic graphs, the goal is to find a permutation of the vertices which maximizes the number of common edges. We study a popular average-case variant; we deviate from the common heuristic strategy and give the first quasi-polynomial time algorithm, where previously only sub-exponential time algorithms were known.
Based on joint work with Boaz Barak, Chi-Ning Chou, Zhixian Lei, and Yueqi Sheng.
10/30/2018 *Tuesday
10:30am
SC 507*
Lauren Williams (Harvard) Title: Introduction to the asymmetric simple exclusion process (from a combinatorialist’s point of view) Abstract: The asymmetric simple exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice, subject to the condition that there is at most one particle per site. This model was introduced in 1970 by biologists (as a model for translation in protein synthesis) but has since been shown to display a rich mathematical structure. There are many variants of the model — e.g. the lattice could be a ring, or a line with open boundaries. One can also allow multiple species of particles with different “weights.” I will explain how one can give combinatorial formulas for the stationary distribution using various kinds of tableaux. I will also explain how the ASEP is related to interesting families of orthogonal polynomials, including Askey-Wilson polynomials, Koornwinder polynomials, and Macdonald polynomials.
11/7/2018 Willhelm Schlag (Yale) Title: on the Bourgain-Dyatlov fractal uncertainty principle Abstract: We will present the Bourgain-Dyatlov theorem on the line, it’s connection with other uncertainty principles in harmonic analysis, and my recent partial progress with Rui Han on the problem of higher dimensions.
11/14/2018 David Gamarnik (MIT) Title: Two Algorithmic Hardness Results in Spin Glasses and Compressive Sensing. Abstract: I will discuss two computational problems in the area of random combinatorial structures. The first one is the problem of computing the partition function of a Sherrington-Kirkpatrick spin glass model. While the the problem of computing the partition functions associated with arbitrary instances is known to belong to the #P complexity class, the complexity of the problem for random instances is open. We show that the problem of computing the partition function exactly (in an appropriate sense) for the case of instances involving Gaussian couplings is #P-hard on average. The proof uses Lipton’s trick of computation modulo large prime number, reduction of the average case to the worst case instances, and the near uniformity of the ”stretched” log-normal distribution.
In the second part we will discuss the problem of explicit construction of matrices satisfying the Restricted Isometry Property (RIP). This challenge arises in the field of compressive sensing. While random matrices are known to satisfy the RIP with high probability, the problem of explicit (deterministic) construction of RIP matrices eluded efforts and hits the so-called ”square root” barrier which I will discuss in the talk. Overcoming this barrier is an open problem explored widely in the literature. We essentially resolve this problem by showing that an explicit construction of RIP matrices implies an explicit construction of graphs satisfying a very strong form of Ramsey property, which has been open since the seminal work of Erdos in 1947.
11/28/2018 Sean O’ Rourke (UC Boulder) Title: Universality and least singular values of random matrix products Abstract: We consider the product of m independent iid random matrices as m is fixed and the sizes of the matrices tend to infinity. In the case when the factor matrices are drawn from the complex Ginibre ensemble, Akemann and Burda computed the limiting microscopic correlation functions. In particular, away from the origin, they showed that the limiting correlation functions do not depend on m, the number of factor matrices. We show that this behavior is universal for products of iid random matrices under a moment matching hypothesis. In addition, we establish universality results for the linear statistics for these product models, which show that the limiting variance does not depend on the number of factor matrices either. The proofs of these universality results require a near-optimal lower bound on the least singular value for these product ensembles.
12/5/2018 *Room G02*
Omer Angel (UBC) Title: balanced excited random walks Abstract: I will present results on the scaling limit and asymptotics of the balanced excited random walk and related processes. This is a walk the that moves vertically on the first visit to a vertex, and horizontally on every subsequent visit. We also analyze certain versions of “clairvoyant scheduling” of random walks.
Joint work with Mark Holmes and Alejandro Ramirez.
2/7/2019 Science Center 530
Ramis Movassagh (IMB Research) Title: Generic Gaplessness, and Hamiltonian density of states from free probability theory Abstract: Quantum many-body systems usually reside in their lowest energy states. This among other things, motives understanding the gap, which is generally an undecidable problem. Nevertheless, we prove that generically local quantum Hamiltonians are gapless in any dimension and on any graph with bounded maximum degree.
We then provide an applied and approximate answer to an old problem in pure mathematics. Suppose the eigenvalue distributions of two matrices M_1 and M_2 are known. What is the eigenvalue distribution of the sum M_1+M_2? This problem has a rich pure mathematics history dating back to H. Weyl (1912) with many applications in various fields. Free probability theory (FPT) answers this question under certain conditions. We will describe FPT and show examples of its powers for approximating physical quantities such as the density of states of the Anderson model, quantum spin chains, and gapped vs. gapless phases of some Floquet systems. These physical quantities are often hard to compute exactly (provably NP-hard). Nevertheless, using FPT and other ideas from random matrix theory excellent approximations can be obtained. Besides the applications presented, we believe the techniques will find new applications in fresh new contexts.
2/14/2019 Nike Sun (MIT) Title: Capacity lower bound for the Ising perceptron Abstract: The perceptron is a toy model of a simple neural network that stores a collection of given patterns. Its analysis reduces to a simple problem in high-dimensional geometry, namely, understanding the intersection of the cube (or sphere) with a collection of random half-spaces. Despite the simplicity of this model, its high-dimensional asymptotics are not well understood. I will describe what is known and present recent results.
2/21/2019 Michael Loss (Georgia Tech) Title: Some results for functionals of Aharonov-Bohm type Abstract: In this talk I present some variational problems of Aharonov-Bohm type, i.e., they include a magnetic flux that is entirely concentrated at a point. This is maybe the simplest example of a variational problems for systems, the wave function being necessarily complex. The functional is rotationally invariant and the issue to be discussed is whether the optimizer have this symmetry or whether it is broken.
3/6/2019 4:15pm
Science Center 411
Ilya Kachkovskiy (Michigan State University) Title: Localization and delocalization for interacting 1D quasiperiodic particles. Abstract: We consider a system of two interacting one-dimensional quasiperiodic particles as an operator on $\ell^2(\mathbb Z^2)$. The fact that particle frequencies are identical, implies a new effect compared to generic 2D potentials: the presence of large coupling localization depends on symmetries of the single-particle potential. If the potential has no cosine-type symmetries, then we are able to show large coupling localization at all energies, even if the interaction is not small (with some assumptions on its complexity). If symmetries are present, we can show localization away from finitely many energies, thus removing a fraction of spectrum from consideration. We also demonstrate that, in the symmetric case, delocalization can indeed happen if the interaction is strong, at the energies away from the bulk spectrum. The result is based on joint works with Jean Bourgain and Svetlana Jitomirskaya.
3/14/2019 5:45pm
Science Center 232
Anna Vershynina (University of Houston) Title: How fast can entanglement be generated in quantum systems? Abstract: We investigate the maximal rate at which entanglement can be generated in bipartite quantum systems. The goal is to upper bound this rate. All previous results in closed systems considered entanglement entropy as a measure of entanglement. I will present recent results, where entanglement measure can be chosen from a large class of measures. The result is derived from a general bound on the trace-norm of a commutator, and can, for example, be applied to bound the entanglement rate for Renyi and Tsallis entanglement entropies.
3/28/2019 Room G02
Xuwen Chen (University of Rochester) Title: The Derivation of the Energy-critical NLS from Quantum Many-body Dynamics Abstract: We derive the 3D energy-critical quintic NLS from quantum many-body dynamics with 3-body interaction in the T^3 (periodic) setting. Due to the known complexity of the energy critical setting, previous progress was limited in comparison to the 2-body interaction case yielding energy subcritical cubic NLS. We develop methods to prove the convergence of the BBGKY hierarchy to the infinite Gross-Pitaevskii (GP) hierarchy, and separately, the uniqueness of large GP solutions. Since the trace estimate used in the previous proofs of convergence is the false sharp trace estimate in our setting, we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de Finetti theorem. For the large solution uniqueness argument, we discover the new HUFL (hierarchical uniform frequency localization) property for the GP hierarchy and use it to prove a new type of uniqueness theorem.
4/4/2019 Paul Bourgade (NYU) Title: Log-correlations and branching structures in analytic number theory Abstract: Fyodorov, Hiary and Keating have predicted the size of local maxima of L-function along the critical axis, based on analogous random matrix statistics. I will explain this prediction in the context of the log-correlated universality class and branching structures. In particular I will explain why the Riemann zeta function exhibits log-correlations, and outline the proof for the leading order of the maximum in the Fyodorov, Hiary and Keating prediction. Joint work with Arguin, Belius, Radziwill and Soundararajan.
4/9/2019 Tuesday
12:00pm
Room G02
Giulio Biroli (ENS Paris) Title: Large deviations for the largest eigenvalues and eigenvectors of spiked random matrices Abstract: I consider matrices formed by a random $N\times N$ matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength $\theta$, and focus on the largest eigenvalue, $x$, and the component, $u$, of the corresponding eigenvector in the direction associated to the rank-one perturbation. I will show how to obtain the large deviation principle governing the atypical joint fluctuations of $x$ and $u$. Interestingly, for $\theta>1$, in large deviations characterized by a small value of $u$, i.e. $u<1-1/\theta$, the second-largest eigenvalue pops out from the Wigner semi-circle and the associated eigenvector orients in the direction corresponding to the rank-one perturbation. These results can be generalized to the Wishart Ensemble, and extended to the first $n$ eigenvalues and the associated eigenvectors.
Finally, I will discuss motivations and applications of these results to the study of the geometric properties of random high-dimensional functions—a topic that is currently attracting a lot of attention in physics and computer science.
4/11/2019 Rui Han (Georgia Tech) Title: Spectral gaps in graphene structures Abstract: We present a full analysis of the spectrum of graphene in magnetic fields with constant flux through every hexagonal comb. In particular, we provide a rigorous foundation for self-similarity by showing that for irrational flux, the spectrum of graphene is a zero measure Cantor set. We also show that for vanishing flux, the spectral bands have nontrivial overlap, which proves the discrete Bethe-Sommerfeld conjecture for the graphene structure. This is based on joint works with S. Becker, J. Fillman and S. Jitomirskaya.
4/25/2019 Benjamin Fehrman (Oxford) Title: Pathwise well-posedness of nonlinear diffusion equations with nonlinear, conservative noise Abstract: We present a pathwise well-posedness theory for stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, approximate the Dean-Kawasaki equation in fluctuating fluid dynamics, describe the fluctuating hydrodynamics of the zero range process, and model the evolution of a thin film in the regime of negligible surface tension. Motivated by the theory of stochastic viscosity solutions, we pass to the equation’s kinetic formulation, where the noise enters linearly and can be inverted using the theory of rough paths. The talk is based on joint work with Benjamin Gess.
4/30/2019 TBA TBA 5/2/2019 Jian Ding (UPenn) TBA 2017-2018
Date………… Name……………. Title/Abstract 2-16-20183:30pm G02
Reza Gheissari (NYU) Dynamics of Critical 2D Potts ModelsAbstract: The Potts model is a generalization of the Ising model to $q\geq 3$ states with inverse temperature $\beta$. The Gibbs measure on $\mathbb Z^2$ has a sharp transition between a disordered regime when $\beta<\beta_c(q)$ and an ordered regime when $\beta>\beta_c(q)$. At $\beta=\beta_c(q)$, when $q\leq 4$, the phase transition is continuous while when $q>4$, the phase transition is discontinuous and the disordered and ordered phases coexist. We will discuss recent progress, joint with E. Lubetzky, in analyzing the time to equilibrium (mixing time) of natural Markov chains (e.g., heat bath/Metropolis) for the 2D Potts model, where the mixing time on an $n \times n$ torus should transition from $O(\log n)$ at high temperatures to $\exp(c_\beta n)$ at low temperatures, via a critical slowdown at $\beta_c(q)$ that is polynomial in $n$ when $q \leq 4$ and exponential in $n$ when $q>4$.
2-23-20183:30pm G02
Mustazee Rahman (MIT) On shocks in the TASEPAbstract: The TASEP particle system runs into traffic jams when the particle density to the left is smaller than the density to the right. Macroscopically, the particle density solves Burgers’ equation and traffic jams correspond to its shocks. I will describe work with Jeremy Quastel on a specialization of the TASEP shock whereby we identify the microscopic fluctuations around the shock by using exact formulas for the correlation functions of TASEP and its KPZ scaling limit. The resulting laws are related to universal laws of random matrix theory. For the curious, here is a video of the shock forming in Burgers’ equation:
4-20-20182:00-3:00pm Carlo Lucibello(Microsoft Research NE) The Random Perceptron Problem: thresholds, phase transitions, and geometryAbstract: The perceptron is the simplest feedforward neural network model, the building block of the deep architectures used in modern machine learning practice. In this talk, I will review some old and new results, mostly focusing on the case of binary weights and random examples. Despite its simplicity, this model provides an extremely rich phenomenology: as the number of examples per synapses is increased, the system undergoes different phase transitions, which can be directly linked to solvers’ performances and to information theoretic bounds. A geometrical analysis of the solution space shows how two different types of solutions, akin to wide and sharp minima, have different generalization capabilities when presented with new examples. Solutions in dense clusters generalize remarkably better, partially closing the gap with Bayesian optimal estimators. Most of the results I will present were first obtained using non rigorous techniques from spin glass theory and many of them haven’t been rigorously established yet, although some big steps forward have been taken in recent years. 4-20-20183:00-4:00pm Yash Despande(MIT) Phase transitions in estimating low-rank matricesAbstract: Low-rank perturbations of Wigner matrices have been extensively studied in random matrix theory; much information about the corresponding spectral phase transition can be gleaned using these tools. In this talk, I will consider an alternative viewpoint based on tools from spin glass theory, and two examples that illustrate how these they yield information beyond traditional spectral tools. The first example is the stochastic block model,where we obtain a full information-theoretic picture of estimation. The second example demonstrates how side information alters the spectral threshold. It involves a new phase transition that interpolates between the Wigner and Wishart ensembles. Date Name Title/Abstract 9-27-17 Herbert Spohn, Technische Universität München Hydrodynamics of integrable classical and quantum systems Abstract: In the cold atoms community there is great interest in developing Euler-type hydrodynamics for one-dimensional integrable quantum systems, in particular with application to domain wall initial states. I provide some mathematical physics background and also compare with integrable classical systems.
10-23-17 *12:00-1:00pm, Science Center 232*
Madhu Sudan, Harvard SEAS General Strong Polarization A recent discovery (circa 2008) in information theory called Polar Coding has led to a remarkable construction of error-correcting codes and decoding algorithms, resolving one of the fundamental algorithmic challenges in the field. The underlying phenomenon studies the “polarization” of a “bounded” martingale. A bounded martingale, X_0,…,X_t,… is one where X_t in [0,1]. This martingale is said to polarize if Pr[lim_{t\to infty} X_t \in {0,1}] = 1. The questions of interest to the results in coding are the rate of convergence and proximity: Specifically, given epsilon and tau > 0 what is the smallest t after which it is the case that Pr[X_t in (tau,1-tau)] < epsilon? For the main theorem, it was crucial that t <= min{O(log(1/epsilon)), o(log(1/tau))}. We say that a martingale polarizes strongly if it satisfies this requirement. We give a simple local criterion on the evolution of the martingale that suffices for strong polarization. A consequence to coding theory is that a broad class of constructions of polar codes can be used to resolve the afore-mentioned algorithmic challenge.
In this talk I will introduce the concepts of polarization and strong polarization. Depending on the audience interest I can explain why this concept is useful to construct codes and decoding algorithms, or explain the local criteria that help establish strong polarization (and the proof of why it does so).
Based on joint work with Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, and Atri Rudra.
10-25-17 *2:00-4:00pm*
Subhabrata Sen (Microsoft and MIT) Noga Alon,(Tel Aviv University)
Subhabrata Sen, “Partitioning sparse random graphs: connections with mean-field spin glasses” Abstract: The study of graph-partition problems such as Maxcut, max-bisection and min-bisection have a long and rich history in combinatorics and theoretical computer science. A recent line of work studies these problems on sparse random graphs, via a connection with mean field spin glasses. In this talk, we will look at this general direction, and derive sharp comparison inequalities between cut-sizes on sparse Erd\ ̋{o}s-R\'{e}nyi and random regular graphs.
Based on joint work with Aukosh Jagannath.
Noga Alon, “Random Cayley Graphs”
Abstract: The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators. Several intriguing questions that remain open will be mentioned as well.
11-1-17 *2:00-4:00pm*
Kay Kirkpatrick (Illinois) and
Wei-Ming Wang (CNRS)
Kay Kirkpatrick, Quantum groups, Free Araki-Woods Factors, and a Calculus for Moments Abstract: We will discuss a central limit theorem for quantum groups: that the joint distributions with respect to the Haar state of the generators of free orthogonal quantum groups converge to free families of generalized circular elements in the large (quantum) dimension limit. We also discuss a connection to free Araki-Woods factors, and cases where we have surprisingly good rates of convergence. This is joint work with Michael Brannan. Time permitting, we’ll mention another quantum central limit theorem for Bose-Einstein condensation and work in progress.
Wei-Min Wang, Quasi-periodic solutions to nonlinear PDE’s
Abstract: We present a new approach to the existence of time quasi-periodic solutions to nonlinear PDE’s. It is based on the method of Anderson localization, harmonic analysis and algebraic analysis. This can be viewed as an infinite dimensional analogue of a Lagrangian approach to KAM theory, as suggested by J. Moser.
11-8-17 Elchanan Mossel Optimal Gaussian Partitions. Abstract: How should we partition the Gaussian space into k parts in a way that minimizes Gaussian surface area, maximize correlation or simulate a specific distribution.
The problem of Gaussian partitions was studied since the 70s first as a generalization of the isoperimetric problem in the context of the heat equation. It found a renewed interest in context of the double bubble theorem proven in geometric measure theory and due to connection to problems in theoretical computer science and social choice theory.
I will survey the little we know about this problem and the major open problems in the area.
11-10-17 *12pm SC 232*
Zhe Wang (NYU) A Driven Tagged Particle in One-dimensional Simple Exclusion Process Abstract: We study the long-time behavior of a driven tagged particle in a one-dimensional non-nearest- neighbor simple exclusion process. We will discuss two scenarios when the tagged particle has a speed. Particularly, for the ASEP, the tagged particle can have a positive speed even when it has a drift with negative mean; for the SSEP with removals, we can compute the speed explicitly. We will characterize some nontrivial invariant measures of the environment process by using coupling arguments and color schemes.
11-15-17 *4:00-5:00pm*
*G02*
Daniel Sussman (BU) Multiple Network Inference: From Joint Embeddings to Graph Matching Abstract: Statistical theory, computational methods, and empirical evidence abound for the study of individual networks. However, extending these ideas to the multiple-network framework remains a relatively under-explored area. Individuals today interact with each other through numerous modalities including online social networks, telecommunications, face-to-face interactions, financial transactions, and the sharing and distribution of goods and services. Individually these networks may hide important activities that are only revealed when the networks are studied jointly. In this talk, we’ll explore statistical and computational methods to study multiple networks, including a tool to borrow strength across networks via joint embeddings and a tool to confront the challenges of entity resolution across networks via graph matching.
11-20-17 *Monday
12:00-1:00pm*
Yue M. Lu (Harvard)
Asymptotic Methods for High-Dimensional Inference: Precise Analysis, Fundamental Limits, and Optimal DesignsAbstract: Extracting meaningful information from the large datasets being compiled by our society presents challenges and opportunities to signal and information processing research. On the one hand, many classical methods, and the assumptions they are based on, are simply not designed to handle the explosive growth of the dimensionality of the modern datasets. On the other hand, the increasing dimensionality offers many benefits: in particular, the very high-dimensional settings allow one to apply powerful asymptotic methods from probability theory and statistical physics to obtain precise characterizations that would otherwise be too complicated in moderate dimensions. I will mention recent work on exploiting such blessings of dimensionality via sharp asymptotic methods. In particular, I will show (1) the exact characterization of a widely-used spectral method for nonconvex signal recoveries; (2) the fundamental limits of solving the phase retrieval problem via linear programming; and (3) how to use scaling and mean-field limits to analyze nonconvex optimization algorithms for high-dimensional inference and learning. In these problems, asymptotic methods not only clarify some of the fascinating phenomena that emerge with high-dimensional data, they also lead to optimal designs that significantly outperform commonly used heuristic choices.11-29-17 David Gamarink (MIT) (Arguably) Hard on Average Constraint Satisfaction Problems Abstract: Many combinatorial optimization problems defined on random instances such as random graphs, exhibit an apparent gap between the optimal value, which can be estimated by non-constructive means, and the best values achievable by fast (polynomial time) algorithms. Through a combined effort of mathematicians, computer scientists and statistical physicists, it became apparent that a potential and in some cases a provable obstruction for designing algorithms bridging this gap is an intricate geometry of nearly optimal solutions, in particular the presence of chaos and a certain Overlap Gap Property (OGP), which we will introduce in this talk. We will demonstrate how for many such problems, the onset of the OGP phase transition indeed nearly coincides with algorithmically hard regimes. Our examples will include the problem of finding a largest independent set of a graph, finding a largest cut in a random hypergrah, random NAE-K-SAT problem, the problem of finding a largest submatrix of a random matrix, and a high-dimensional sparse linear regression problem in statistics.
Joint work with Wei-Kuo Chen, Quan Li, Dmitry Panchenko, Mustazee Rahman, Madhu Sudan and Ilias Zadik.
12-6-17 *2:00-4:00pm*
Philippe Rigollet (MIT) 2-3 pm
&
Ankur Moitra (MIT)
3-4 pm
Philippe Rigollet (MIT), Exact Recovery in the Ising Block Model Abstract: Over the past fifteen years, the problem of learning Ising models from independent samples has been of significant interest in the statistics, machine learning, and statistical physics communities. Much of the effort has been directed towards finding algorithms with low computational cost for various restricted classes of models, primarily in the case where the interaction graph is sparse. In parallel, stochastic blockmodels have played a more and more preponderant role in community detection and clustering as an average case model for the minimum bisection model. In this talk, we introduce a new model, called Ising blockmodel for the community structure in an Ising model. It imposes a block structure on the interactions of a dense Ising model and can be viewed as a structured perturbation of the celebrated Curie-Weiss model. We show that interesting phase transitions arise in this model and leverage this probabilistic analysis to develop an algorithm based on semidefinite programming that recovers exactly the community structure when the sample size is large enough. We also prove that exact recovery of the block structure is actually impossible with fewer samples.
This is joint work with Quentin Berthet (University of Cambridge) and Piyush Srivastava (Tata Institute).
Ankur Moitra (MIT), A New Approach to Approximate Counting and Sampling
Abstract: Over the past sixty years, many remarkable connections have been made between statistical physics, probability, analysis and theoretical computer science through the study of approximate counting. While tight phase transitions are known for many problems with pairwise constraints, much less is known about problems with higher-order constraints.
Here we introduce a new approach for approximately counting and sampling in bounded degree systems. Our main result is an algorithm to approximately count the number of solutions to a CNF formula where the degree is exponential in the number of variables per clause. Our algorithm extends straightforwardly to approximate sampling, which shows that under Lovasz Local Lemma-like conditions, it is possible to generate a satisfying assignment approximately uniformly at random. In our setting, the solution space is not even connected and we introduce alternatives to the usual Markov chain paradigm.12-14-17 TBD Strongly Correlated Quantum Materials and High-Temperature Superconductors Series
In the 2020-2021 academic year, the CMSA will be hosting a lecture series on Strongly Correlated Materials and High Tc Superconductor. All talks will take place from 10:30-12:00pm ET virtually on Zoom.
Cuprate high-temperature superconductors are a classic quantum material system to demonstrate the beauty of “Emergence and Entanglement” in the quantum phases of matter. Merely by adding more holes into an antiferromagnetic insulator, several fascinating phases emerge, including a d-wave superconductor, a pseudo-gap metal, and strange metal. After intensive studies from experimental, theoretical, and numerical communities for more than three decades, remarkable progress has been made, but basic questions remain:
- What is the origin of the superconductivity? What are the relative contributions of electron-phonon coupling, spin fluctuations, or resonating-valence-bonds?
- How do we explain the pseudo-gap and the Fermi arc in the underdoped region above the critical temperature? Are they from some symmetry breaking order parameters, or do we need an unconventional picture involving fractionalization?
- Is the strange metal at optimal doping associated with a quantum critical point? And if so, what is the driving force of this phase transition?
The cuprate quantum materials have been a major source for many new concepts in modern condensed matter physics, such as quantum spin liquids, topological order, and non-Fermi liquids. In the coming years, it is clear that the study of the cuprates will continually motivate new concepts and development of new techniques. In this seminar series, we hope to accelerate this process by bringing together deeper conversations between experimental, theoretical, and numerical experts with different backgrounds and perspectives.
The Strongly Correlated Quantum Materials and High-Temperature Superconductors series is a part of the Quantum Matter in Mathematics and Physics seminar.
Seminar organizers: Juven Wang (Harvard CMSA) and Yahui Zhang (Harvard).
Scientific program advisors: Professor Subir Sachdev (Harvard), Professor Patrick Lee (MIT).
In order to learn how to attend this series, please fill out this form.
For more information, please contact Juven Wang (jw@cmsa.fas.harvard.edu) and Yahui Zhang (yahui_zhang@g.harvard.edu)
Spring 2022
April 20, 2022 | 11:30 – 1:00 pm ET
Harold Y. Hwang (Stanford University & SLAC National Accelerator Laboratory)
Title: Superconductivity in infinite-layer nickelates
Abstract: Since its discovery, unconventional superconductivity in cuprates has motivated the search for materials with analogous electronic or atomic structure. We have used soft chemistry approaches to synthesize superconducting infinite layer nickelates from their perovskite precursor phase. We will present the synthesis and transport properties of the nickelates, observation of a doping-dependent superconducting dome, and our current understanding of their electronic and magnetic structure.
February 3, 2022 | 11:30 – 1:00 pm ET
Lu Li (U Michigan)
Title: Quantum Oscillations of Electrical Resistivity in an Insulator
Abstract: In metals, orbital motions of conduction electrons are quantized in magnetic fields, which is manifested by quantum oscillations in electrical resistivity. This Landau quantization is generally absent in insulators, in which all the electrons are localized. Here we report a notable exception in an insulator — ytterbium dodecaboride (YbB12). The resistivity of YbB12, despite much larger than that of usual metals, exhibits profound quantum oscillations under intense magnetic fields. This unconventional oscillation is shown to arise from the insulating bulk instead of conducting surface states. The large effective masses indicate strong correlation effects between electrons. Our result is the first discovery of quantum oscillations in the electrical resistivity of a strongly correlated insulator and will bring crucial insight into understanding the ground state in gapped Kondo systems.
2020 – 2021
September 2, 2020 | 10:30am ET
Subir Sachdev (Harvard) Title: Metal-to-metal quantum phase transitions not described by symmetry-breaking orders
Abstract: Numerous experiments have explored the phases of the cuprates with increasing doping density p from the antiferromagnetic insulator. There is now strong evidence that the small p region is a novel phase of matter, often called the pseudogap metal, separated from conventional Fermi liquid at larger p by a quantum phase transition. Symmetry-breaking orders play a spectator role, at best, at this quantum phase transition. I will describe trial wavefunctions across this metal-metal transition employing hidden layers of ancilla qubits (proposed by Ya-Hui Zhang). Quantum fluctuations are described by a gauge theory of ghost fermions that carry neither spin nor charge. I will also
describe a separate approach to this transition in a t-J model with random exchange interactions in the limit of large dimensions. This approach leads to a partly solvable SYK-like critical theory of holons and spinons, and a linear in temperature resistivity from time reparameterization fluctuations. Near criticality, both approaches have in common emergent fractionalized excitations, and a significantly larger entropy than naively expected.September 23, 2020 | 10:30am ET
Subir Sachdev (Harvard) Title: Metal-to-metal quantum phase transitions not described by symmetry-breaking orders II
Abstract: In this second talk, I will focus on (nearly) solvable models of metal-metal transition in random systems. The t-J model with random and all-to-all hopping and exchange can be mapped onto a quantum impurity model coupled self-consistently to an environment (the mapping also applies to a t-J model in a large dimension lattice, with random nearest-neighbor exchange). Such models will be argued to exhibit metal-metal quantum phase transitions in the universality class of the SYK model, accompanied by a linear-in-T resistivity from time reparameterization fluctuations. I will also present the results of exact diagonalization of random t-J clusters, obtained recently with Henry Shackleton, Alexander Wietek, and Antoine Georges.
September 24, 2020 | 12:00pm ET
Inna Vishik (University of California, Davis)
Title: Universality vs materials-dependence in cuprates: ARPES studies of the model cuprate Hg1201Abstract: The cuprate superconductors exhibit the highest ambient-pressure superconducting transition temperatures (T c ), and after more than three decades of extraordinary research activity, continue to pose formidable scientific challenges. A major experimental obstacle has been to distinguish universal phenomena from materials- or technique-dependent ones. Angle-resolved photoemission spectroscopy (ARPES) measures momentum-dependent single-particle electronic excitations and has been invaluable in the endeavor to determine the anisotropic momentum-space properties of the cuprates. HgBa 2 CuO 4+d (Hg1201) is a single-layer cuprate with a particularly high optimal T c and a simple crystal structure; yet there exists little information from ARPES about the electronic properties of this model system. I will present recent ARPES studies of doping-, temperature-, and momentum-dependent systematics of near-nodal dispersion anomalies in Hg1201. The data reveal a hierarchy of three distinct energy scales which establish several universal phenomena, both in terms of connecting multiple experimental techniques for a single material, and in terms of connecting comparable spectral features in multiple structurally similar cuprates.VideoOctober 15, 2020 | 10:30am ET
Louis Taillefer (Université de Sherbrooke) Title: New signatures of the pseudogap phase of cuprate superconductors
Abstract: The pseudogap phase of cuprate superconductors is arguably the most enigmatic phase of quantum matter. We aim to shed new light on this phase by investigating the non- superconducting ground state of several cuprate materials at low temperature across a wide doping range, suppressing superconductivity with a magnetic field. Hall effect measurements across the pseudogap critical doping p* reveal a sharp drop in carrier density n from n = 1 + p above p* to n = p below p, signaling a major transformation of the Fermi surface. Angle-dependent magneto-resistance (ADMR) directly reveals a change in Fermi surface topology across p. From specific heat measurements, we observe the classic thermodynamic signatures of quantum criticality: the electronic specific heat C el shows a sharp peak at p, where it varies in temperature as C el ~ – T logT. At p and just above, the electrical resistivity is linear in T at low T, with an inelastic scattering rate that obeys the Planckian limit. Finally, the pseudogap phase is found to have a large negative thermal Hall conductivity, which extends to zero doping. We show that the pseudogap phase makes phonons become chiral. Understanding the mechanisms responsible for these various new signatures will help elucidate the nature of the pseudogap phase.
October 28, 2020 | 10:30am ET
Patrick Lee (MIT) Title: The not-so-normal normal state of underdoped Cuprate
Abstract: The underdoped Cuprate exhibits a rich variety of unusual properties that have been exposed after years of experimental investigations. They include a pseudo-gap near the anti-nodal points and “Fermi arcs” of gapless excitations, together with a variety of order such as charge order, nematicity and possibly loop currents and time reversal and inversion breaking. I shall argue that by making a single assumption of strong pair fluctuations at finite momentum (Pair density wave), a unified description of this phenomenology is possible. As an example, I will focus on a description of the ground state that emerges when superconductivity is suppressed by a magnetic field which supports small electron pockets. [Dai, Senthil, Lee, Phys Rev B101, 064502 (2020)] There is some support for the pair density wave hypothesis from STM data that found charge order at double the usual wave-vector in the vicinity of vortices, as well as evidence for a fragile form of superconductivity persisting to fields much above Hc2. I shall suggest a more direct experimental probe of the proposed fluctuating pair density wave.
November 6, 2020 |12:30pm ET
Zhi-Xun Shen (Stanford University) Title: Essential Ingredients for Superconductivity in Cupper Oxide Superconductors
Abstract: High‐temperature superconductivity in cupper oxides, with critical temperature well above what wasanticipated by the BCS theory, remains a major unsolved physics problem. The problem is fascinating because it is simultaneously simple ‐ being a single band and 1⁄2 spin system, yet extremely rich ‐ boasting d‐wave superconductivity, pseudogap, spin and charge orders, and strange metal phenomenology. For this reason, cuprates emerge as the most important model system for correlated electrons – stimulating conversations on the physics of Hubbard model, quantum critical point, Planckian metal and beyond.
Central to this debate is whether the Hubbard model, which is the natural starting point for the undoped
magnetic insulator, contains the essential ingredients for key physics in cuprates. In this talk, I will discuss our photoemission evidence for a multifaceted answer to this question [1‐3]. First, we show results that naturally points to the importance of Coulomb and magnetic interactions, including d‐wave superconducting gap structure [4], exchange energy (J) control of bandwidth in single‐hole dynamics [5]. Second, we evidence effects beyond the Hubbard model, including band dispersion anomalies at known phonon frequencies [6, 7], polaronic spectral lineshape and the emergence of quasiparticle with doping [8]. Third, we show properties likely of hybrid electronic and phononic origin, including the pseudogap [9‐11], and the almost vertical phase boundary near the critical 19% doping [12]. Fourth, we show examples of small q phononic coupling that cooperates with d‐wave superconductivity [13‐15]. Finally, we discuss recent experimental advance in synthesizing and investigating doped one‐dimensional (1D) cuprates [16]. As theoretical calculations of the 1D Hubbard model are reliable, a robust comparison can be carried out. The experiment reveals a near‐neighbor attractive interaction that is an order of magnitude larger than the attraction generated by spin‐superexchange in the Hubbard model. Addition of such an attractive term, likely of phononic origin, into the Hubbard model with canonical parameters provides a quantitative explanation for all important experimental observable: spinon and holon dispersions, and holon‐ holon attraction. Given the structural similarity of the materials, It is likely that an extended two‐dimensional
(2D) Hubbard model with such an attractive term, will connect the dots of the above four classes of
experimental observables and provide a holistic understanding of cuprates, including the elusive d‐wave superconductivity in 2D Hubbard model.[1] A. Damascelli, Z. Hussain, and Z.‐X. Shen, Review of Modern Physics, 75, 473 (2003)
[2] M. Hashimoto et al., Nature Physics 10, 483 (2014)
[3] JA Sobota, Y He, ZX Shen ‐ arXiv preprint arXiv:2008.02378, 2020; submitted to Rev. of Mod. Phys.
[4] Z.‐X. Shen et al., Phys. Rev. Lett. 70, 1553 (1993)
[5] B.O. Wells et al., Phys. Rev. Lett. 74, 964 (1995)
[6] A. Lanzara et al., Nature 412, 510 (2001)
[7] T. Cuk et al., Phys. Rev. Lett., 93, 117003 (2004)
[8] K.M. Shen et al., Phys. Rev. Lett., 93, 267002 (2004)
[9] D.M. King et al., J. of Phys. & Chem of Solids 56, 1865 (1995)
[10] D.S. Marshall et al., Phy. Rev. Lett. 76, 484 (1996)
[11] A.G. Loeser et al., Science 273, 325 (1996)
[12] S. Chen et al., Science, 366, 6469 (2019)
[13] T.P. Devereaux, T. Cuk, Z.X. Shen, N. Nagaosa, Phys. Rev. Lett., 93, 117004 (2004)
[14] S. Johnston et al., Phys. Rev. Lett. 108, 166404 (2012)
[15] Yu He et al., Science, 362, 62 (Oct. 2018)
[16] Z. Chen, Y. Wang et al., preprint, 2020November 12, 2020 |10:30am ET
Chandra Varma (Visting Professor, University of California, Berkeley.
Emeritus Distinguished Professor, University of California, Riverside.)Title: Loop-Current Order and Quantum-Criticality in CupratesThis talk is organized as follows:
1. Physical Principles leading to Loop-current order and quantum criticality as the central feature in the physics of Cuprates.
2. Summary of the essentially exact solution of the dissipative xy model for Loop-current fluctuations.
3. Quantitative comparison of theory for the quantum-criticality with a variety of experiments.
4. Topological decoration of loop-current order to understand ”Fermi-arcs” and small Fermi-surface magneto-oscillations.Time permitting,
(i) Quantitative theory and experiment for fluctuations leading to d-wave superconductivity.
(ii) Extensions to understand AFM quantum-criticality in heavy-fermions and Fe-based superconductors.
(iii) Problems.VideoNovember 18, 2020 |10:30am ET
Antoine Georges (Collège de France, Paris and Flatiron Institute, New York) Abstract: Simplified as it is, the Hubbard model embodies much of the complexity of the `strong correlation problem’ and has established itself as a paradigmatic model in the field. In this talk, I will argue that several key aspects of its physics in two dimensions can now be established beyond doubt, thanks to the development of controlled and accurate computational methods. These methods implement different and complementary points of view on the quantum many-body problem. Along with pushing forward each method, the community has recently embarked into a major effort to combine and critically compare these approaches, and in several instances a consistent picture of the physics has emerged as a result. I will review in this perspective our current understanding of the emergence of a pseudogap in both the weak and strong coupling regimes. I will present recent progress in understanding how the pseudogap phase may evolve into a stripe-dominated regime at low temperature, and briefly address the delicate question of the competition between stripes and superconductivity. I will also emphasize outstanding questions which are still open, such as the possibility of a Fermi surface reconstruction without symmetry breaking. Whenever possible, connections to the physics of cuprate superconductors will be made. If time permits, I may also address the question of Planckian transport and bad metallic transport at high temperature.
November 19, 2020 |10:30am ET
Eduardo Fradkin (University of Illinois at Urbana-Champaign) Title: Pair Density Waves and Intertwined Orders in High Tc Superconductors
Abstract: I will argue that the orders that are present in high temperature superconductors naturally arise with the same strength and are better regarded as intertwined rather than competing. I illustrate this concept in the context of the orders that are present in the pair-density-wave state and the phase diagrams that result from this analysis.November 25, 2020 |10:30am ET
Qimiao Si (Rice University) Title: Bad Metals and Electronic Orders – Nematicity from Iron Pnictides to Graphene Moiré Systems
Abstract: Strongly correlated electron systems often show bad-metal behavior, as operationally specified in terms of a resistivity at room temperature that reaches or exceeds the Mott-Ioffe-Regel limit. They display a rich landscape of electronic orders, which provide clues to the underlying microscopic physics. Iron-based superconductors present a striking case study, and have been the subject of extensive efforts during the past decade or so. They are well established to be bad metals, and their phase diagrams prominently feature various types of electronic orders that are essentially always accompanied by nematicity. In this talk, I will summarize these characteristic features and discuss our own efforts towards understanding the normal state through the lens of the electronic orders and their fluctuations. Implications for superconductivity will be briefly discussed. In the second part of the talk, I will consider the nematic correlations that have been observed in the graphene-based moiré narrow-band systems. I will present a theoretical study which demonstrates nematicity in a “fragile insulator”, predicts its persistence in the bad metal regime and provides an overall perspective on the phase diagram of these correlated systems.
December 2, 2020 |10:30am ET
Andrey Chubukov (University of Minnesota) Title: Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal
Abstract: I discuss the interplay between non-Fermi liquid behaviour and pairing near a quantum-critical point (QCP) in a metal. These tendencies are intertwined in the sense that both originate from the same interaction mediated by gapless fluctuations of a critical order parameter. The two tendencies compete because fermionic incoherence destroys the Cooper logarithm, while the pairing eliminates scattering at low energies and restores fermionic coherence. I discuss this physics for a class of models with an effective dynamical interaction V (Ω) ~1/|Ω|^γ (the γ-model). This model describes, in particular, the pairing at a 2D Ising-nematic critical point in (γ=1/3), a 2D antiferromagnetic critical point (γ=1/2) and the pairing by an Einstein phonon with vanishing dressed Debye frequency (γ=2). I argue the pairing wins, unless the pairing component of the interaction is artificially reduced, but because of fermionic incoherence in the normal state, the system develops a pseudogap, preformed pairs behaviour in the temperature range between the onset of the pairing at Tp and the onset of phase coherence at the actual superconducting Tc. The ratio Tc/Tp decreases with γ and vanishes at γ =2. I present two complementary arguments of why this happens. One is the softening of longitudinal gap fluctuations, which become gapless at γ =2. Another is the emergence of a 1D array of dynamical vortices, whose number diverges at γ =2. I argue that once the number of vortices becomes infinite, quasiparticle energies effectively get quantized and do not get re-arranged in the presence of a small phase variation. I show that a new non-superconducting ground state emerges at γ >2.December 9, 2020 |10:30am ET
David Hsieh (Caltech) Title: Signatures of anomalous symmetry breaking in the cuprates
Abstract: The temperature versus doping phase diagram of the cuprate high-Tc superconductors features an enigmatic pseudogap region whose microscopic origin remains a subject of intensive study. Experimentally resolving its symmetry properties is imperative for narrowing down the list of possible explanations. In this talk I will give an overview of how optical second harmonic generation (SHG) can be used as a sensitive probe of symmetry breaking, and recap the ways it has been used to solve outstanding problems in condensed matter physics. I will then describe how we have been applying SHG polarimetry and spectroscopy to interrogate the cuprate pseudogap. In particular, I will discuss our data on YBa2Cu3Oy [1], which show an order parameter-like increase in SHG intensity below the pseudogap temperature T* across a broad range of doping levels. I will then focus on our more recent results on a model parent cuprate Sr2CuO2Cl2 [2], where evidence of anomalous broken symmetries surprisingly also exists. Possible connections between these observations will be speculated upon.
[1] L. Zhao, C. A. Belvin, R. Liang, D. A. Bonn, W. N. Hardy, N. P. Armitage and D. Hsieh, “A global inversion-symmetry-broken phase inside the pseudogap region of YBa2Cu3Oy,” Nature Phys. 13, 250 (2017).[2] A. de la Torre, K. L. Seyler, L. Zhao, S. Di Matteo, M. S. Scheurer, Y. Li, B. Yu, M. Greven, S. Sachdev, M. R. Norman and D. Hsieh. “Anomalous mirror symmetry breaking in a model insulating cuprate Sr2CuO2Cl2,” Preprint at https://arxiv.org/abs/2008.06516
December 16, 2020 |10:30am ET
Zheng-Yu Weng (Tsinghua University) Title: Organizing Principle of Mottness and Complex Phenomenon in High Temperature Superconductors
Abstract: The complex phenomenon in the high-Tc cuprate calls for a microscopic understanding based on general principles. In this Lecture, an exact organizing principle for a typical doped Mott insulator will be presented, in which the fermion sign structure is drastically reduced to a mutual statistics. Its nature as a long-range spin-charge entanglement of many-body quantum mechanics will be exemplified by exact numerical calculations. The phase diagram of the cuprate may be unified in a “bottom-up” fashion by a “parent” ground state ansatz with hidden orders constructed based on the organizing principle. Here the pairing mechanism will go beyond the “RVB” picture and the superconducting state is of non-BCS nature with modified London equation and novel elementary excitations. In particular, the Bogoliubov/Landau quasiparticle excitation are emerging with a two-gap structure in the superconducting state and the Fermi arc in a pseudogap regime. A mathematic framework of fractionalization and duality transformation guided by the organizing principle will be introduced to describe the above emergent phenomenon.December 17, 2020 |10:30am ET
Steven Kivelson (Stanford University) Abstract: Despite the fact that papers submitted to glossy journals universally start by bemoaning the absence of theoretical understanding, I will argue that the answer to the title question is “quite a lot.” To focus the discussion, I will take the late P.W. Anderson’s “Last Words on the Cuprates” (arXiv:1612.03919) as a point of departure, although from a perspective that differs from his in many key points.
January 20, 2021 |10:30am ET
Thomas Peter Devereaux (Stanford University) Title: Numerical investigations of models of the cuprates
Abstract: Richard Feynman once said “Anyone who wants to analyze the properties of matter in a real problem might want to start by writing down the fundamental equations and then try to solve them mathematically. Although there are people who try to use such an approach, these people are the failures in this field. . . ”I will summarize efforts to solve microscopic models of the cuprates using quantum Monte Carlo and density matrix renormalization group computational methods, with emphasis on how far one can get before failing to describe the real materials. I will start with an overview of the quantum chemistry of the cuprates that guides our choices of models, and then I will discuss “phases” of these models, both realized and not. I will lastly discuss the transport properties of the models in the “not-so-normal” regions of the phase diagram.
February 3, 2021 |10:30am ET
Philip Phillips (University of Illinois Urbana-Champaign) Title: Beyond BCS: An Exact Model for Superconductivity and Mottness
Abstract: High-temperature superconductivity in the cuprates remains an unsolved problem because the cuprates start off their lives as Mott insulators in which no organizing principle such a Fermi surface can be invoked to treat the electron interactions. Consequently, it would be advantageous to solve even a toy model that exhibits both Mottness and superconductivity. Part of the problem is that the basic model for a Mott insulator, namely the Hubbard model is unsolvable in any dimension we really care about. To address this problem, I will start by focusing on the overlooked Z_2 emergent symmetry of a Fermi surface first noted by Anderson and Haldane. Mott insulators break this emergent symmetry. The simplest model of this type is due to Hatsugai/Kohmoto. I will argue that this model can be thought of a fixed point for Mottness. I will then show exactly[1] that this model when appended with a weak pairing interaction exhibits not only the analogue of Cooper’s instability but also a superconducting ground state, thereby demonstrating that a model for a doped Mott insulator can exhibit superconductivity. The properties of the superconducting state differ drastically from that of the standard BCS theory. The elementary excitations of this superconductor are not linear combinations of particle and hole states but rather are superpositions of doublons and holons, composite excitations signaling that the superconducting ground state of the doped Mott insulator inherits the non-Fermi liquid character of the normal state. Additional unexpected features of this model are that it exhibits a superconductivity-induced transfer of spectral weight from high to low energies and a suppression of the superfluid density as seen in the cuprates.
[1] PWP, L. Yeo, E. Huang, Nature Physics, 16, 1175-1180 (2020).February 10, 2021 |10:30am ET
Senthil Todadri (MIT) Abstract: The strange metal regime is one of the most prominent features of the cuprate phase diagram but yet has remained amongst the most mysterious. Seemingly similar metallic behavior is seen in a few other metals. In this talk, I will discuss, in great generality, some properties of `strange metals’ in an ideal clean system. I will discuss general constraints[1] on the emergent low energy symmetries of any such strange metal. These constraints may be viewed as a generalization of the Luttinger theorem of ordinary Fermi liquids. Many, if not all, non-Fermi liquids will have the same realization of emergent symmetry as a Fermi liquid (even though they could have very different dynamics). Such phases – dubbed ersatz Fermi liquids – share some (but not all) universal properties with Fermi liquids. I will discuss the implications for understanding the strange metal physics observed in experiments . Combined with a few experimental observations, I will show that these general model-independent considerations lead to concrete predictions[2] about a class of strange metals. The most striking of these is a divergent susceptibility of an observable that has the same symmetries as the loop current order parameter.
[1]. Dominic Else, Ryan Thorngren, T. Senthil, https://arxiv.org/abs/2007.07896
[2]. Dominic Else, T. Senthil, https://arxiv.org/abs/2010.10523April 1, 2021 |9:00am ET
Naoto Nagaosa (University of Tokyo) Title: Applied physics of high-Tc theories
Abstract: Since the discovery of high temperature superconductors in cuprates in 1986, many theoretical ideas have been proposed which have enriched condensed matter theory. Especially, the resonating valence bond (RVB) state for (doped) spin liquids is one of the most fruitful idea. In this talk, I would like to describe the development of RVB idea to broader class of materials, especially more conventional magnets. It is related to the noncollinear spin structures with spin chirality and associated quantal Berry phase applied to many phenomena and spintronics applications. It includes the (quantum) anomalous Hall effect, spin Hall effect, topological insulator, multiferroics, various topological spin textures, e.g., skyrmions, and nonlinear optics. I will show that even though the phenomena are extensive, the basic idea is rather simple and common in all of these topics.
April 22, 2021 |10:30am ET
Dung-Hai Lee (UC Berkeley) Title: “Non-abelian bosonization in two and three spatial dimensions and some applications”
Abstract: In this talk, we generalize Witten’s non-abelian bosonization in $(1+1)$-D to two and three spatial dimensions. Our theory applies to fermions with relativistic dispersion. The bosonized theories are non-linear sigma models with level-1 Wess-Zumino-Witten terms. As applications, we apply the bosonization results to the $SU(2)$ gauge theory of the $\pi$ flux mean-field theory of half-filled Hubbard model, critical spin liquids of “bipartite-Mott insulators” in 1,2,3 spatial dimensions, and twisted bilayer graphene.
May 12, 2021 |10:30am ET
André-Marie Tremblay (Université de Sherbrooke) Title: A unified theoretical perspective on the cuprate phase diagram
Abstract: Many features of the cuprate phase diagram are a challenge for the usual tools of solid state physics. I will show how a perspective that takes into account both the localized and delocalized aspects of conduction electrons can explain, at least qualitatively, many of these features. More specifically, I will show that the work of several groups using cluster extensions of dynamical mean-field theory sheds light on the pseudogap, on the quantum-critical point and on d-wave superconductivity. I will argue that the charge transfer gap and oxygen hole content are the best indicators of strong superconductivity and that many observations are a signature of the influence of Mott physics away from half-filling. I will also briefly comment on what information theoretic measures tell us about this problem.
August 11, 2021 |10:30am ET
Piers Coleman (Rutgers) Title: Order Fractionalization*
Abstract: I will discuss the interplay of spin fractionalization with broken
symmetry. When a spin fractionalizes into a fermion, the resulting particle
can hybridize or pair with the mobile electrons to develop a new kind of
fractional order parameter. The concept of “order fractionalization” enables
us to extend the concept of off-diagonal order to encompass the formation of
such order parameters with fractional quantum numbers, such as spinorial
order[1].
A beautiful illustration of this phenomenon is provided by a model
which incorporates the Yao-Lee-Kitaev model into a Kondo lattice[2]. This
model explicitly exhibits order fractionalization and is expected to undergo a
discrete Ising phase transition at finite temperature into an
order-fractionalized phase with gapless Majorana excitations.
The broader implications of these considerations for Quantum
Materials and Quantum Field Theory will be discussed.
Work done in collaboration with Yashar Komijani, Anna Toth and Alexei
Tsvelik.
[1] Order Fractionalization, Yashar Komijani, Anna Toth, Premala Chandra, Piers Coleman, (2018).
[2] Order Fractionalization in a Kitaev Kondo model, Alexei Tsvelik and Piers Coleman, (2021).September 15, 2021 |10:30am ET
Liang Fu (MIT) Title: Three-particle mechanism for pairing and superconductivity
Abstract: I will present a new mechanism and an exact theory of electron pairing due to repulsive interaction in doped insulators. When the kinetic energy is small, the dynamics of adjacent electrons on the lattice is strongly correlated. By developing a controlled kinetic energy expansion, I will show that two doped charges can attract and form a bound state, despite and because of the underlying repulsion. This attraction by repulsion is enabled by the virtual excitation of a third electron in the filled band. This three-particle pairing mechanism leads to a variety of novel phenomena at finite doping, including spin-triplet superconductivity, pair density wave, BCS-BEC crossover and Feshbach resonance involving “trimers”. Possible realizations in moire materials, ZrNCl and WTe2 will be discussed.
[1] V. Crepel and L. Fu, Science Advances 7, eabh2233 (2021)
[2] V. Crepel and L. Fu, arXiv:2103.12060
[3] K. Slagle and L. Fu, Phys. Rev. B 102, 235423 (2020)September 29, 2021 |11:30am ET (special time)
Nai Phuan Ong (Princeton University)
Title:.Abstract: The layered honeycomb magnet alpha-RuCl3 orders below 7 K in a zigzag phase in zero field. An in-plane magnetic field H||a suppresses the zigzag order at 7 Tesla, leaving a spin-disordered phase widely believed to be a quantum spin liquid (QSL) that extends to ~12 T. We have observed oscillations in the longitudinal thermal conductivity Kxx vs. H from 0.4 to 4 K. The oscillations are periodic in 1/H (with a break-in-slope at 7 T). The amplitude function is maximal in the QSL phase (7 –11.5 T). I will describe a benchmark for crystalline disorder, the reproducibility and intrinsic nature of the oscillations, and discuss implications for the QSL state. I will also show detailed data on the thermal Hall conductivity Kxy measured from 0.4 K to 10 K and comment on recent half-quantization results.*Czajka et al., Nature Physics 17, 915 (2021).Collaborators: Czajka, Gao, Hirschberger, Lampen Kelley, Banerjee, Yan, Mandrus and Nagler.Date TBA |10:30am ET
Suchitra Sebastian (University of Cambridge) Title: TBA
Date TBA |10:30am ET
Jenny Hoffman (Harvard University) Title: TBA
Exact symmetries and threshold states in two-dimensional models for QCD
Speaker: Silviu Pufu (Princeton University)
Title: Exact symmetries and threshold states in two-dimensional models for QCD
Abstract: Two-dimensional QCD models form an interesting playground for studying phenomena such as confinement and screening. In this talk I will describe one such model, namely a 2d SU(N) gauge theory with an adjoint and a fundamental fermion, and explain how to compute the spectrum of bound states using discretized light-cone quantization at large N. Surprisingly, the spectrum of the discretized theory exhibits a large number of exact degeneracies, for which I will provide two different explanations. I will also discuss how these degeneracies provide a physical picture of confinement in 2d QCD with just a massless adjoint fermion. This talk is based on joint work with R. Dempsey and I. Klebanov.
Swampland Seminar Series
During the 2021-22 academic year, the CMSA will be co-hosting a seminar on Swampland, with the Harvard Physics Department, organized by Miguel Montero, Cumrun Vafa, Irene Valenzuela. This seminar is a part of the Swampland Program. This seminar will take place on Mondays at 10:00 am – 11:30 am (Boston time). To learn how to attend, please subscribe here.
Talks will be posted on the Swampland Seminars YouTube channel. The schedule below will be updated as talks are confirmed.
Spring 2022
Date Speaker Title/Abstract 1/31/2022 Rafael Álvarez-García (DESY Hamburg) Title: Membrane Limits in Quantum Gravity 2/7/2022 Du Pei (Harvard CMSA) Title: Holomorphic CFTs and topological modular forms Abstract: The theory of topological modular forms leads to many interesting constraints and predictions for two-dimensional quantum field theories, and some of them might have interesting implications for the swampland program. In this talk, I will show that a conjecture by Segal, Stolz and Teichner requires the constant term of the partition function of a bosonic holomorphic CFTs to be divisible by specific integers determined by the central charge. We verify this constraint in large classes of physical examples, and rule out the existence of an infinite set of “extremal CFTs”, including those with central charges c = 48, 72, 96 and 120.
2/28/2022 Tom Rudelius (UC, Berkeley) Title: Generalized Global Symmetries and the Weak Gravity Conjecture 3/7/2022 Fernando Marchesano (UAM-CSIC, Madrid) and Max Wiesner (Harvard CMSA) Title: 4d strings at strong coupling 3/21/2022 Patrick Draper (Univ. of Illinois) and Alvaro Herraez (IPhT Saclay). Open Mic Discussion
Topic: Entropy bounds (species bound, Bekenstein bound, CKN bound, and the like)3/28/2022 Fernando Quevedo (Cambridge) Title: On renormalisation group induced moduli stabilisation and brane-antibrane inflation Abstract: A proposal to use the renormalisation group to address moduli stabilisation in IIB string perturbation theory will be described. We revisit brane-antibrane inflation combining this proposal with non-linearly realised supersymmetry.
4/5/2022 Simon Caron-Huot (McGill University) and Julio Parra (Caltech) Title: Causality constraints on corrections to Einstein gravity Abstract: We study constraints from causality and unitarity on 2→2 graviton scattering in four-dimensional weakly-coupled effective field theories. Together, causality and unitarity imply dispersion relations that connect low-energy observables to high-energy data. Using such dispersion relations, we derive two-sided bounds on gravitational Wilson coefficients in terms of the mass M of new higher-spin states. Our bounds imply that gravitational interactions must shut off uniformly in the limit G→0, and prove the scaling with M expected from dimensional analysis (up to an infrared logarithm). We speculate that causality, together with the non-observation of gravitationally-coupled higher-spin states at colliders, severely restricts modifications to Einstein gravity that could be probed by experiments in the near future.
4/11/2022 Timm Wrase and Eduardo Gonzalo (Lehigh) Title: Type IIB flux compactifications with $h^{1,1}=0$ Abstract: We revisit type IIB flux compactification that are mirror dual to type IIA on rigid Calabi-Yau manifolds. We find a variety of interesting new solutions, like fully stabilized Minkowski vacua and infinite families of AdS$_4$ solutions with arbitrarily large numbers of spacetime filling D3 branes. We discuss how these solutions fit into the web of swampland conjectures.
4/18/2022 José Calderón (IFT Madrid) Open mic Swampland Discussion Topic: Cobordism
5/9/2022 Georges Obie (Harvard) Title: Inflation and light Dark Matter constraints from the Swampland Abstract: I will explore the interplay between Swampland conjectures and models of inflation and light Dark Matter. To that end, I will briefly review the weak gravity conjecture (WGC) and the related Festina Lente (FL) bound. These have implications for light darkly and milli-charged particles and can disfavor a large portion of parameter space. The FL bound also implies strong restrictions on the field content of our universe during inflation and presents an opportunity for inflationary model building. At the same time, it rules out some popular models like chromo-natural inflation and gauge-flation. Finally, I will review another Swampland conjecture related to Stückelberg photon masses and discuss its implications for astro-particle physics.
Fall 2021
Date Speaker Title/Abstract 9/13/2021 John Stout (Harvard) Title: Decoding Divergent Distances Abstract: Motivated by a relationship between the Zamolodchikov and NLSM metrics to the so-called quantum information metric, I will discuss recent work (2106.11313) on understanding infinite distance limits within the context of information theory. I will describe how infinite distance points represent theories that are hyper-distinguishable, in the sense that they can be distinguished from “nearby” theories with certainty in relatively few measurements. I will then discuss necessary and sufficient ingredients for the appearance of these infinite distance points, illustrate these in simple examples, and describe how this perspective can help the swampland program.
9/20/2021 Manki Kim (MIT) Title: Small Cosmological Constants in String Theory Abstract: We construct supersymmetric AdS4 vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by Kachru, Kallosh, Linde, and Trivedi. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the α 0 expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude < 10^{-123} in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol.
9/27/2021 Eran Palti (Ben Gurion) Title: Convexity of Charged Operators in CFTs and the Weak Gravity Conjecture Abstract: In this talk I will introduce a particular formulation of the Weak Gravity Conjecture in AdS space in terms of the self-binding energy of a particle. The holographic CFT dual of this formulation corresponds to a certain convex-like structure for operators charged under continuous global symmetries. Motivated by this, we propose a conjecture that this convexity is a general property of all CFTs, not just those with weakly-curved gravitational duals. It is possible to test this in simple CFTs, the conjecture passes all the tests performed so far.
10/18/2021 Thomas Van Riet (KU Leuven) Title: The Festina Lente Bound Abstract: I will explain what the Festina Lente bound means and where it comes from. Then I discuss its possible implications for phenomenology, both top-down and bottom-up.
10/25/2021 Joe Conlon (Oxford) Title: Exploring the Holographic Swampland Abstract: I describe our work looking at `traditional’ scenarios of moduli stabilisation from a holographic perspective. This reveals some interesting structure that is not apparent from the top-down perspective. For vacua in the extreme regions of moduli space, such as LVS in type IIB or the DGKT flux vacua in type IIA, the dual moduli conformal dimensions reduce to fixed values – in a certain sense, the low-conformal dimension part of the CFT is unique and independent of the large number of flux choices. For the DGKT flux vacua these conformal dimensions are also integer, for reasons we do not understand.
11/01/2021 Pieter Bomans (Princeton) Title: Bubble instability of mIIA on AdS_4 x S^6 Abstract: Recently, a set of non-supersymmetric AdS_4 vacua of massive type IIA string theory has been constructed. These vacua are perturbatively stable with respect to the full KK spectrum of type mIIA supergravity and furthermore, they are stable against a variety of non-perturbative decay channels. Hence, at this point, they represent a serious challenge to the AdS swampland conjecture. In my talk, I will review in detail the construction of these vacua as well as introduce a new decay channel, ultimately sealing their fate as being unstable.
11/15/2021 Nima Arkani-Hamed (IAS), and Gary Shiu (UW-Madison) This week’s seminar will be an open mic discussion which will be led by Nima Arkani-Hamed (IAS), and by Gary Shiu (UW-Madison), and the topic will be “Swampland constraints, Unitarity and Causality”. They will start with a brief introduction sharing their thoughts about the topic and moderate a discussion afterwards. 11/22/2021 Thomas Grimm (Utrecht University) Title: Taming the Landscape Abstract: In this talk I will introduce a generalized notion of finiteness that provides a structural principle for the set of effective theories that can be consistently coupled to quantum gravity. More concretely, I will propose a ‘tameness conjecture’ that states that all scalar field spaces and coupling functions that appear in such an effective theory must be definable in an o-minimal structure. The fascinating field of tame geometry has seen much recent progress and I will argue that the results can be used to support the above swampland conjecture. The strongest evidence arises from a new finiteness theorem for the flux landscape which is shown using the tameness of the period map.
11/29/2021 Timm Wrase (Lehigh University) Title: Scale separated AdS vacua? Abstract: In this talk I will review massive type IIA flux compactifications that seem to give rise to infinite families of supersymmetric 4d AdS vacua. These vacua provide an interesting testing ground for the swampland program. After reviewing potential shortcomings of this setup, I will discuss recent progress on overcoming them and getting a better understanding of these solutions.
12/6/2021 Lars Aalsma (University of Wisconsin-Madison) Title: Extremal Black Hole Corrections from Iyer-Wald Abstract: Extremal black holes play a key role in our understanding of various swampland conjectures and in particular the WGC. The mild form of the WGC states that higher-derivative corrections should decrease the mass of extremal black holes at fixed charge. Whether or not this conjecture is satisfied depends on the sign of the combination of Wilson coefficients that control corrections to extremality. Typically, corrections to extremality need to be computed on a case-by-case basis, but in this talk I will present a universal derivation of extremal black hole corrections using the Iyer-Wald formalism. This leads to a formula that expresses general corrections to the extremality bound in terms of the stress tensor of the perturbations under consideration, clarifying the relation between the WGC and energy conditions. This shows that a necessary condition for the mild form of the WGC to be satisfied is a violation of the Dominant Energy Condition. This talk is based on 2111.04201.
Global Anomalies on the Hilbert Space
Speaker: Jaume Gomis (Perimeter PI)
Title: Global Anomalies on the Hilbert Space
Abstract: We will discuss an elementary way of detecting some global anomalies from the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory, give a physical description of the imprint of the “layers” that enter in the cobordism classification of anomalies and discuss applications, including how anomalies can imply a supersymmetric spectrum in strongly coupled (nonsupersymmetric) gauge theories.
2/18/2021 Quantum Matter Seminar
Speaker: Xiao-Gang Wen (MIT)
Title: A solution to the chiral fermion problem
Abstract: Motivated by the relation between anomaly and topological/SPT order in one higher dimension, we propose a solution to the chiral fermion problem. In particular, we find several sufficient conditions, such that a chiral fermion field theory can be regularized by an interacting lattice model in the same dimension. We also discuss some related issues, such as mass without mass term, and why ‘topological’ phase transitions are usually not “topological” phase transitions.
Hybrid Fracton Orders
Nathanan Tantivasadakarn (Harvard) Title: Hybrid Fracton Orders Abstract: I will introduce a family of gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders called “Hybrid Fracton Orders”. First, I will present the simplest example of such an order: the “Hybrid X-cube” model, where excitations can be labeled identically to those of the Z2 toric code tensored with the Z2 X-cube model, but exhibit fusion and braiding properties between the two sets of excitations. Next, I will provide a general construction of hybrid fracton orders which inputs a finite group G and an abelian normal subgroup N and produces an exactly solvable model. Such order can host non-abelian fracton excitations when G is non-abelian. Furthermore, the mobilities of a general excitation is dictated by the choice of N, from which by varying, one can view as “interpolating” between a pure 3D topological order and a pure fracton order.
Based on 2102.09555 and 2106.03842
The nu=5/2 enigma: Recent insights from theory and experiment
peaker: Ady Stern & David Mross (Weizmann)
Speaker: Ady Stern & David Mross (Weizmann
Title: The nu=5/2 enigma: Recent insights from theory and experiment
Abstract: Non-Abelian phases of matter have long inspired quantum physicists across various disciplines. The strongest experimental evidence of such a phase arises in quantum Hall systems at the filling factor 5/2 but conflicts with decades of numerical works. We will briefly introduce the 5/2 plateau and explain some of the key obstacles to identifying its topological order. We will then describe recent experimental and theoretical progress, including a proposal for resolving the 5/2 enigma based on electrical conductance measurements.
General Relativity 2021-22
During the 2021–22 academic year, the CMSA will be hosting a seminar on General Relativity, organized by Aghil Alaee, Jue Liu, Daniel Kapec, and Puskar Mondal. This seminar will take place on Thursdays at 9:30am – 10:30am (Eastern time). The meetings will take place virtually on Zoom. To learn how to attend, please fill out this form.
The schedule below will be updated as talks are confirmed.
Spring 2022
Date Speaker Title/Abstract 2/10/2022 Tin Yau Tsang (UC Irvine) Title: Dihedral ridigity and mass Abstract: To characterise scalar curvature, Gromov proposed the dihedral rigidity conjecture which states that a positively curved polyhedron having dihedral angles less than those of a corresponding flat polyhedron should be isometric to a flat one. In this talk, we will discuss some recent progress on this conjecture and its connection with general relativity (ADM mass and quasilocal mass).
2/17/2022 Shiraz Minwalla
(Tata Institute of Fundamental Research, Mumbai)Title: Black Hole dynamics at Large D Abstract: I demonstrate that black hole dynamics simplifies – without trivializing – in the limit in which the number of spacetime dimensions D in which the black holes live is taken to infinity. In the strict large D limit and under certain conditions I show the equations that govern black hole dynamics reduce to the equations describing the dynamics of a non gravitational membrane propagating in an unperturbed spacetime (e.g. flat space). In the stationary limit black hole thermodynamics maps to membrane thermodynamics, which we formulate in a precise manner. We also demonstrate that the large D black hole membrane agrees with the fluid gravity map in the appropriate regime.
2/24/2022 Achilleas Porfyriadis
(Harvard Black Hole Initiative)Title: Extreme Black Holes: Anabasis and Accidental Symmetry Abstract: The near-horizon region of black holes near extremality is universally AdS_2-like. In this talk I will concentrate on the simplest example of AdS_2 x S^2 as the near-horizon of (near-)extreme Reissner-Nordstrom. I will first explain the SL(2) transformation properties of the spherically symmetric linear perturbations of
AdS_2 x S^2 and show how their backreaction leads to the Reissner-Nordstrom black hole. This backreaction with boundary condition change is called an anabasis. I will then show that the linear Einstein equation near AdS_2 x S^2, with or without additional matter, enjoys an accidental symmetry that may be thought of as an on-shell large diffeomorphism of AdS_2.3/10/2022 David Fajman (University of Vienna) Title: The Einstein-flow on manifolds of negative curvature
Abstract: We consider the Cauchy problem for the Einstein equations for cosmological spacetimes, i.e. spacetimes with compact spatial hypersurfaces. Various classes of those dynamical spacetimes have been constructed and analyzed using CMC foliations or equivalently the CMC-Einstein flow. We will briefly review the Andersson-Moncrief stability result of negative Einstein metrics under the vacuum Einstein flow and then present various recent generalizations to the nonvacuum case. We will emphasize what difficulties arise in those generalizations, how they can be handled depending on the matter model at hand, and what implications we can draw from these results for cosmology. We then turn to a scenario where the CMC Einstein flow leads to a large data result in 2+1-dimensions.3/21/2022 Prof. Arick Shao (Queen Mary University of London) Title: Bulk-boundary correspondence for vacuum asymptotically Anti-de Sitter spacetimes Abstract: The AdS/CFT conjecture in physics posits the existence of a correspondence between gravitational theories in asymptotically Anti-de Sitter (aAdS) spacetimes and field theories on their conformal boundary. In this presentation, we prove rigorous mathematical statements toward this conjecture.
In particular, we show there is a one-to-one correspondence between aAdS solutions of the Einstein-vacuum equations and a suitable space of data on the conformal boundary (consisting of the boundary metric and the boundary stress-energy tensor). We also discuss consequences of this result, as well as the main ingredient behind its proof: a unique continuation property for wave equations on aAdS spacetimes.
This is joint work with Gustav Holzegel (and makes use of joint works with Alex McGill and Athanasios Chatzikaleas).
3/24/2022 Qian Wang, University of Oxford Title: Rough solutions of the $3$-D compressible Euler equations Abstract: I will talk about my work on the compressible Euler equations. We prove the local-in-time existence the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, mega) \in H^s\times H^s\times H^{s’}$, $2<s'<s$. The result extends the sharp result of Smith-Tataru and Wang, established in the irrotational case, i.e $mega=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e. with a constant density, the result is known to hold for $mega\in H^s$, $s>3/2$ and fails for $s\le 3/2$, see the work of Bourgain-Li. It is thus natural to conjecture that the optimal result should be $(v,\varrho, mega) \in H^s\times H^s\times H^{s’}$, $s>2, \, s’>\frac{3}{2}$. We view our work as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of a sound wave from the transported part and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.
3/28/2022 Emanuele Berti, Johns Hopkins University Title: Black Hole Spectroscopy Abstract: According to general relativity, the remnant of a binary black hole merger should be a perturbed Kerr black hole. Perturbed Kerr black holes emit “ringdown” radiation which is well described by a superposition of quasinormal modes, with frequencies and damping times that depend only on the mass and spin of the remnant. Therefore the observation of gravitational radiation emitted by black hole mergers might finally provide direct evidence of black holes with the same certainty as, say, the 21 cm line identifies interstellar hydrogen. I will review the current status of this “black hole spectroscopy” program. I will focus on two important open issues: (1) When is the waveform well described by linear black hole perturbation theory? (2) What is the current observational status of black hole spectroscopy?
4/7/2022 CMSA General Relativity Conference 4/14/2022 Chao Liu, Huazhong University of Science and Technology Title: Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n≥4 Abstract: In this talk, we briefly introduce our recent work on establishing the global existence and stability to the future of non-linear perturbation of de Sitter-like solutions to the Einstein-Yang-Mills system in n≥4 spacetime dimension. This generalizes Friedrich’s (1991) Einstein-Yang-Mills stability results in dimension n=4 to all higher dimensions. This is a joint work with Todd A. Oliynyk and Jinhua Wang.
4/21/2022 Jinhua Wang,
Xiamen UniversityTitle: Future stability of the $1+3$ Milne model for the Einstein-Klein-Gordon system Abstract: We study the small perturbations of the $1+3$-dimensional Milne model for the Einstein-Klein-Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on the $1+3$ splitting of the Bianchi-Klein-Gordon equations. Moreover, we treat the Bianchi-Klein-Gordon equations as evolution equations and establish the energy scheme in the sense that we only commute the Bianchi-Klein-Gordon equations with spatially covariant derivatives while normal derivative is not allowed. We propose some refined estimates for lapse and the hierarchies of energy estimates to close the energy argument.
4/28/2022 Allen Fang, Sorbonne University Title: A new proof for the nonlinear stability of slowly-rotating Kerr-de Sitter Abstract: The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vectorfield techniques to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.
Fall 2021
Date Speaker Title/Abstract 9/10/2021 (10:30am – 11:30am (Boston time)
Philippe G. LeFloch, Sorbonne University and CNRS Title: Asymptotic localization, massive fields, and gravitational singularities Abstract: I will review three recent developments on Einstein’s field equations under low decay or low regularity conditions. First, the Seed-to-Solution Method for Einstein’s constraint equations, introduced in collaboration with T.-C. Nguyen generates asymptotically Euclidean manifolds with the weakest or strongest possible decay (infinite ADM mass, Schwarzschild decay, etc.). The ‘asymptotic localization problem’ is also proposed an alternative to the ‘optimal localization problem’ by Carlotto and Schoen. We solve this new problem at the harmonic level of decay. Second, the Euclidian-Hyperboloidal Foliation Method, introduced in collaboration with Yue Ma, applies to nonlinear wave systems which need not be asymptotically invariant under Minkowski’s scaling field and to solutions with low decay in space. We established the global nonlinear stability of self-gravitating massive matter field in the regime near Minkowski spacetime. Third, in collaboration with Bruno Le Floch and Gabriele Veneziano, I studied spacetimes in the vicinity of singularity hypersurfaces and constructed bouncing cosmological spacetimes of big bang-big crunch type. The notion of singularity scattering map provides a flexible tool for formulating junction conditions and, by analyzing Einstein’s constraint equations, we established a surprising classification of all gravitational bouncing laws. Blog: philippelefloch.org
9/17/2021 (10:30am – 11:30am (Boston time)
Igor Rodnianski, Princeton University Title: Stable Big Bang formation for the Einstein equations Abstract: I will discuss recent work concerning stability of cosmological singularities described by the generalized Kasner solutions. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be stable under perturbations of the Kasner initial data, as long as the Kasner exponents are “sub-critical”. We prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the full regime where stable monotonic-type curvature blowup is expected. We treat the 3+1-dimensional Einstein-scalar field system and the D+1-dimensional Einstein-vacuum equations for D≥10. This is joint work with Speck and Fournodavlos.
9/24/2021 (10:30am – 11:30am (Boston time)
Alex Lupsasca Title: On the Observable Shape of Black Hole Photon Rings Abstract: The photon ring is a narrow ring-shaped feature, predicted by General Relativity but not yet observed, that appears on images of sources near a black hole. It is caused by extreme bending of light within a few Schwarzschild radii of the event horizon and provides a direct probe of the unstable bound photon orbits of the Kerr geometry. I will argue that the precise shape of the observable photon ring is remarkably insensitive to the astronomical source profile and can therefore be used as a stringent test of strong-field General Relativity. In practice, near-term interferometric observations may be limited to the visibility amplitude alone, which contains incomplete shape information: for convex curves, the amplitude only encodes the set of projected diameters (or “widths”) of the shape. I will describe the freedom in reconstructing a convex curve from its widths, giving insight into the photon ring shape information probed by technically plausible future astronomical measurements.
10/1/2021 (10:30am – 11:30am (Boston time)
Zhongshan An, University of Connecticut Title: Static vacuum extensions of Bartnik boundary data near flat domains Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on a joint work with Lan-Hsuan Huang.
10/8/2021 (10:30am – 11:30am (Boston time)
Xiaoning Wu, Chinese Academy of Sciences Title: Causality Comparison and Postive Mass Abstract: Penrose et al. investigated the physical incoherence of the space-time with negative mass via the bending of light. Precise estimates of the time-delay of null geodesics were needed and played a pivotal role in their proof. In this paper, we construct an intermediate diagonal metric and reduce this problem to a causality comparison in the compactified space-time regarding time-like connectedness near conformal infinities. This different approach allows us to avoid encountering the difficulties and subtle issues that Penrose et al. met. It provides a new, substantially simple, and physically natural non-partial differential equation viewpoint to understand the positive mass theorem. This elementary argument modestly applies to asymptotically flat solutions that are vacuum and stationary near infinity
10/15/2021 (10:30am – 11:30am (Boston time)
Jiong-Yue Li, Sun Yat-Sen University Title: Peeling properties of the spinor fields and the solutions to nonlinear Dirac equations Abstract: The Dirac equation is a relativistic equation that describes the spin-1/2 particles. We talk about Dirac equations in Minkowski spacetime. In a geometric viewpoint, we can see that the spinor fields satisfying the Dirac equations enjoy the so-called peeling properties. It means the null components of the solution will decay at different rates along the null hypersurface. Based on this decay mechanism, we can obtain a fresh insight to the spinor null forms which is used to prove a small data global existence result especially for some quadratic Dirac models.
10/22/2021 (11:00am – 12:30pm (Boston time)
Roberto Emparan, University of Barcelona Title: The Large D Limit of Einstein’s Equations Abstract: Taking the large dimension limit of Einstein’s equations is a useful strategy for solving and understanding the dynamics that these equations encode. I will introduce the underlying ideas and the progress that has resulted in recent years from this line of research. Most of the discussion will be classical in nature and will concern situations where there is a black hole horizon. A main highlight of this approach is the formulation of effective membrane theories of black hole dynamics. These have made possible to efficiently study, with relatively simple techniques, some of the thorniest problems in black hole physics, such as the non-linear evolution of the instabilities of black strings and black branes, and the collisions and mergers of higher-dimensional black holes. Open directions and opportunities will also be discussed. To get a flavor of what this is about, you may read the first few pages of the review (with C.P. Herzog) e-Print: 2003.11394.
10/28/2021 Jorge Santos, University of Cambridge Title: The classical interior of charged black holes with AdS asymptotics Abstract: The gravitational dual to the grand canonical ensemble of a large N holographic theory is a charged black hole. These spacetimes can have Cauchy horizons that render the classical gravitational dynamics of the black hole interior incomplete. We show that a (spatially uniform) deformation of the CFT by a neutral scalar operator generically leads to a black hole with no inner horizon. There is instead a spacelike Kasner singularity in the interior. For relevant deformations, Cauchy horizons never form. We then consider charged scalars, which are known to condense at low temperatures, thus providing a holographic realization of superconductivity. We look inside the horizon of these holographic superconductors and find intricate dynamical behavior. The spacetime ends at a spacelike Kasner singularity, and there is no Cauchy horizon. Before reaching the singularity, there are several intermediate regimes which we study both analytically and numerically. These include strong Josephson oscillations in the condensate and possible `Kasner inversions’ in which after many e-folds of expansion, the Einstein-Rosen bridge contracts towards the singularity. Due to the Josephson oscillations, the number of Kasner inversions depends very sensitively on temperature, and diverges at a discrete set of temperatures that accumulate at the critical temperature. Near this discrete set of temperatures, the final Kasner exponent exhibits fractal-like behavior.
11/4/2021
at 10 am ETElena Giorgi, Columbia University Title: The stability of charged black holes Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
11/11/2021
*9:30 am ET*Siyuan Ma, Sorbonne University Title: Sharp decay for Teukolsky equation in Kerr spacetimes Abstract: Teukolsky equation in Kerr spacetimes governs the dynamics of the spin $s$ components, $s=0, \pm 1, \pm 2$ corresponding to the scalar field, the Maxwell field, and the linearized gravity, respectively. I will discuss recent joint work with L. Zhang on proving the precise asymptotic profiles for these spin $s$ components in Schwarzschild and Kerr spacetimes.
11/19/2021 (10:30–11:30 am ET)
Nishanth Gudapati, Clark University Title: On Curvature Propagation and ‘Breakdown’ of the Einstein Equations on U(1) Symmetric Spacetimes Abstract: The analysis of global structure of the Einstein equations for general relativity, in the context of the initial value problem, is a difficult and intricate mathematical subject. Any additional structure in their formulation is welcome, in order to alleviate the problem. It is expected that the initial value problem of the Einstein equations on spacetimes admitting a translational, fixed-point free, spatial U(1) isometry group are globally well-posed. In our previous works, we discussed the special structure provided by the dimensional reduction of 3+1 dimensional U(1) symmetric Einstein equations to 2+1 Einstein-wave map system and demonstrated global existence in the equivariant case for large data. In this talk, after discussing some preliminaries and background, we shall discuss about yet another structure of the U(1) symmetric Einstein equations, namely the analogy with Yang-Mills theory via the Cartan formalism and reconcile with the dimensionally reduced field equations. We shall also discuss implications for ‘breakdown’ criteria of U(1) symmetric Einstein equations.
12/2/2021 Professor Geoffrey Comp
ére, Université Libre de BruxellesTitle: Kerr Geodesics and Self-consistent match between Inspiral and Transition-to-merger Abstract: The two-body motion in General Relativity can be solved perturbatively in the small mass ratio expansion. Kerr geodesics describe the leading order motion. After a short summary of the classification of polar and radial Kerr geodesic motion, I will consider the inspiral motion of a point particle around the Kerr black hole subjected to the self-force. I will describe its quasi-circular inspiral motion in the radiation timescale expansion. I will describe in parallel the transition-to-merger motion around the last stable circular orbit and prove that it is controlled by the Painlevé transcendental equation of the first kind. I will then prove that one can consistently match the two motions using the method of asymptotically matched expansions.
12/16/2021 Xinliang An, University of Singapore Title: Low regularity ill-posedness for 3D elastic waves and for 3D ideal compressible MHD driven by shock formation Abstract: We construct counterexamples to the local existence of low-regularity solutions to elastic wave equations and to the ideal compressible magnetohydrodynamics (MHD) system in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad’s classic results on the scalar wave equation by showing that the Cauchy problems for 3D elastic waves and for 3D MHD system are ill-posed in $H^3(R^3)$ and $H^2(R^3)$, respectively. Both elastic waves and MHD are physical systems with multiple wave speeds. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. In particular, when the magnetic field is absent in MHD, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proofs for elastic waves and for MHD are based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D elastic waves and the 3D ideal MHD equations into $6\times 6$ and $7\times 7$ non-strictly hyperbolic systems. Via detailed calculations, we reveal their hidden subtle structures. With them, we give a complete description of solutions’ dynamics up to the earliest singular event, when a shock forms. This talk is based on joint works with Haoyang Chen and Silu Yin.
2021 Summer Introduction to Mathematical Research
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithm, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited to 10 people, so don’t wait too long to apply.
CMSA Math-Science Literature Lecture: Birational geometry
Vyacheslav V. Shokurov (Johns Hopkins University)
Title: Birational geometry
Abstract: About main achievements in birational geometry during the last fifty years.
Talk chair: Caucher Birkar
CMSA Math-Science Literature Lecture: Nonlinear stability of Kerr black holes for small angular momentum
Sergiu Klainerman (Princeton University)
Title: Nonlinear stability of Kerr black holes for small angular momentum
Abstract: According to a well-known conjecture, initial data sets, for the Einstein vacuum equations, sufficiently close to a Kerr solution with parameters $a, m$, $|a|/m <1$, have maximal developments with complete future null infinity and with domain of outer communication (i.e complement of a future event horizon) which approaches (globally) a nearby Kerr solution. I will describe the main ideas in my recent joint work with Jeremie Szeftel concerning the resolution of the conjecture for small angular momentum, i.e. $, $|a|/m $ sufficiently small. The work, ArXiv:2104.11857v1, also depends on forthcoming work on solutions of nonlinear wave equations in realistic perturbations of Kerr, with Szeftel and Elena Giorgi, which I will also describe.
Talk chair: Lydia Bieri
CMSA Math-Science Literature Lecture: Black Hole Formation
Lydia Bieri (University of Michigan)
Title: Black Hole Formation
Abstract: Can black holes form through the focusing of gravitational waves? This was an outstanding question since the early days of general relativity. In his breakthrough result of 2008, Demetrios Chrstodoulou answered this question with “Yes!” In order to investigate this result, we will delve deeper into the dynamical mathematical structures of the Einstein equations. Black holes are related to the presence of trapped surfaces in the spacetime manifold. Christodoulou proved that in the regime of pure general relativity and for arbitrarily dispersed initial data, trapped surfaces form through the focusing of gravitational waves provided the incoming energy is large enough in a precisely defined way. The proof combines new ideas from geometric analysis and nonlinear partial differential equations as well as it introduces new methods to solve large data problems. These methods have many applications beyond general relativity. D. Christodoulou’s result was generalized in various directions by many authors. It launched mathematical activities going into multiple fields in mathematics and physics. In this talk, we will discuss the mathematical framework of the above question. Then we will outline the main ideas of Christodoulou’s result and its generalizations, show relations to other questions and give an overview of implications in other fields.
FRG Workshop on Geometric Methods for Analyzing Discrete Shapes
This workshop will take place May 7-9 (Friday-Sunday), 2021 virtually on Zoom
The aim of the workshop is to bring together a community of researchers in mathematics, computer science, and data science who develop theoretical and computational models to characterize shapes and analysis of image data.
This workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes.
The first half of the workshop will feature talks aimed at graduate students, newcomers, and a broad spectrum of audiences. Christopher Bishop (Stony Brook) and Keenan Crane (Carnegie Mellon) will each give two featured talks. The remaining part will have both background and research talks. There will also be organized discussions of open problems and potential applications.
For the discussions, we are soliciting open problems in mathematical theory and applications of shape analysis. You are encouraged to post problems by sending an email to geometricproblemsfrg@gmail.com.
We invite junior researchers to present a short talk in the workshop. The session will be held on Friday, May 7th or Saturday, May 8th at 4pm and are expected to be 15-20 minutes in length. It is a great opportunity to share your work and get to know others at the workshop. Depending on the number of contributed talks, the organizers will review the submissions and let you know if you have been selected. If you are interested please send your title and abstract to tianqi@cmsa.fas.harvard.edu by the end of May 2nd.
Organizers:
- David Glickenstein, University of Arizona
- Joel Hass, University of California, Davis
- Patrice Koehl, University of California, Davis
- Feng Luo, Rutgers University, New Brunswick
- Tianqi Wu, Harvard University
- Shing-Tung Yau, Harvard University
Featured lectures:
- Christopher Bishop, Stony Brook
- Keenan Crane, Carnegie Mellon
Speakers include:
- Miri Ben-Chen, Technion – Israel Institute of Technology
- Alexander Bobenko, Technische Universität Berlin, Germany
- Ulrike Buecking, Free University, Germany
- Nadav Dym, Duke University
- Ivan Izmestiev, Vienna University of Technology
- Yanwen Luo, Rutgers
- Stephan Tillmann, The University of Sydney
- Max Wardetzky, University of Goettingen
- Xu Xu, Wuhan University
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Simplices in the Calabi–Yau web
Abstract: Calabi–Yau manifolds of a given dimension are connected by an intricate web of birational maps. This web has deep consequences for the derived categories of coherent sheaves on such manifolds, and for the associated string theories. In particular, for 4-folds and beyond, I will highlight certain simplices appearing in the web, and identify corresponding derived category structures.
Previous Colloquia
The CMSA Colloquium will take place every Wednesday from 4:30-5:30pm in CMSA Building, 20 Garden Street, G10.
Spring 2020
Date Speaker Title/Abstract 1/29/2020 David Yang (Harvard) Abstract: Data-intensive technologies such as AI may reshape the modern world. We propose that two features of data interact to shape innovation in data-intensive economies: first, states are key collectors and repositories of data; second, data is a non-rival input in innovation. We document the importance of state-collected data for innovation using comprehensive data on Chinese facial recognition AI firms and government contracts. Firms produce more commercial software and patents, particularly data-intensive ones, after receiving government public security contracts. Moreover, effects are largest when contracts provide more data. We then build a directed technical change model to study the state’s role in three applications: autocracies demanding AI for surveillance purposes, data-driven industrial policy, and data regulation due to privacy concerns. When the degree of non-rivalry is as strong as our empirical evidence suggests, the state’s collection and processing of data can shape the direction of innovation and growth of data-intensive economies.
2/5/2020 Scott Aaronson (UT Austin) Title: Gentle Measurement of Quantum States and Differential Privacy Abstract: I’ll discuss a recent connection between two seemingly unrelated problems: how to measure a collection of quantum states without damaging them too much (“gentle measurement”), and how to provide statistical data without leaking too much about individuals (“differential privacy,” an area of classical CS). This connection leads, among other things, to a new protocol for “shadow tomography”
of quantum states (that is, answering a large number of questions about a quantum state given few copies of it).Based on joint work with Guy Rothblum (arXiv:1904.08747)
2/12/2020 Scott Kominers (Harvard) Title: A Compact, Logical Approach to Large-Market Analysis Abstract: In game theory, we often use infinite models to represent “limit” settings, such as markets with a large number of agents or games with a long time horizon. Yet many game-theoretic models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. Here, we show how to extend key results from (finite) models of matching, games on graphs, and trading networks to infinite models by way of Logical Compactness, a core result from Propositional Logic. Using Compactness, we prove the existence of man-optimal stable matchings in infinite economies, as well as strategy-proofness of the man-optimal stable matching mechanism. We then use Compactness to eliminate the need for a finite start time in a dynamic matching model. Finally, we use Compactness to prove the existence of both Nash equilibria in infinite games on graphs and Walrasian equilibria in infinite trading networks.
2/19/2020 Peter Shor (MIT) Title: Quantum Money from Lattices Abstract: Quantum money is a cryptographic protocol for quantum computers. A quantum money protocol consists of a quantum state which can be created (by the mint) and verified (by anybody with a quantum computer who knows what the “serial number” of the money is), but which cannot be duplicated, even by somebody with a copy of the quantum state who knows the verification protocol. Several previous proposals have been made for quantum money protocols. We will discuss the history of quantum money and give a protocol which cannot be broken unless lattice cryptosystems are insecure.
2/26/2020 Daneil Wise (McGill) Title: The Cubical Route to Understanding Groups Abstract: Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that culminated in the resolution of the virtual Haken conjecture for 3-manifolds and simultaneously dramatically extended our understanding of many infinite groups.3/4/2020 4:45 – 5:45pm
Salil Vadhan (Harvard) Title: Derandomizing Algorithms via Spectral Graph Theory Abstract: Randomization is a powerful tool for algorithms; it is often easier to design efficient algorithms if we allow the algorithms to “toss coins” and output a correct answer with high probability. However, a longstanding conjecture in theoretical computer science is that every randomized algorithm can be efficiently “derandomized” — converted into a deterministic algorithm (which always outputs the correct answer) with only a polynomial increase in running time and only a constant-factor increase in space (i.e. memory usage).
In this talk, I will describe an approach to proving the space (as opposed to time) version of this conjecture via spectral graph theory. Specifically, I will explain how randomized space-bounded algorithms are described by random walks on directed graphs, and techniques in algorithmic spectral graph theory (e.g. solving Laplacian systems) have yielded deterministic space-efficient algorithms for approximating the behavior of such random walks on undirected graphs and Eulerian directed graphs (where every vertex has the same in-degree as out-degree). If these algorithms can be extended to general directed graphs, then the aforementioned conjecture about derandomizing space-efficient algorithms will be resolved.
3/11/2020 Postponed
Jose Scheinkman (Columbia)
This colloquium will be rescheduled at a later date. Title: Menu Costs and the Volatility of Inflation
Abstract: We present a state-dependent equilibrium pricing model that generates inflation rate fluctuations from idiosyncratic shocks to the cost of price changes of individual firms. A firm’s nominal price increase lowers other firms’ relative prices, thereby inducing further nominal price increases. We first study a mean-field limit where the equilibrium is characterized by a variational inequality and exhibits a constant rate of inflation. We use the limit model to show that in the presence of a large but finite number n of firms the snowball effect of repricing causes fluctuations to the aggregate price level and these fluctuations converge to zero slowly as n grows. The fluctuations caused by this mechanism are larger when the density of firms at the repricing threshold is high, and the density at the threshold is high when the trend inflation level is high. However a calibration to US data shows that this mechanism is quantitatively important even at modest levels of trend inflation and can account for the positive relationship between inflation level and volatility that has been observed empirically.
3/12/2020 4:00 – 5:00pm
Daniel Forger (University of Michigan) This meeting will be taking place virtually on Zoom. Title: Math, Music and the Mind; Mathematical analysis of the performed Trio Sonatas of J. S. Bach
Abstract: I will describe a collaborative project with the University of Michigan Organ Department to perfectly digitize many performances of difficult organ works (the Trio Sonatas by J.S. Bach) by students and faculty at many skill levels. We use these digitizations, and direct representations of the score to ask how music should encoded in the mind. Our results challenge the modern mathematical theory of music encoding, e.g., based on orbifolds, and reveal surprising new mathematical patterns in Bach’s music. We also discover ways in which biophysical limits of neuronal computation may limit performance.
Daniel Forger is the Robert W. and Lynn H. Browne Professor of Science, Professor of Mathematics and Research Professor of Computational Medicine and Bioinformatics at the University of Michigan. He is also a visiting scholar at Harvard’s NSF-Simons Center and an Associate of the American Guild of Organists.
3/25/2020 Cancelled 4/1/2020 Mauricio Santillana (Harvard) This meeting will be taking place virtually on Zoom. Title: Data-driven machine learning approaches to monitor and predict events in healthcare. From population-level disease outbreaks to patient-level monitoring
Abstract: I will describe data-driven machine learning methodologies that leverage Internet-based information from search engines, Twitter microblogs, crowd-sourced disease surveillance systems, electronic medical records, and weather information to successfully monitor and forecast disease outbreaks in multiple locations around the globe in near real-time. I will also present data-driven machine learning methodologies that leverage continuous-in-time information coming from bedside monitors in Intensive Care Units (ICU) to help improve patients’ health outcomes and reduce hospital costs.
4/8/2020 Juven Wang (CMSA) This meeting will be taking place virtually on Zoom. Title: Quantum Matter Adventure to Fundamental Physics and Mathematics (Continued)
Abstract: In 1956, Parity violation in Weak Interactions is confirmed in particle physics. The maximal parity violation now is a Standard Model physics textbook statement, but it goes without any down-to-earth explanation for long. Why? We will see how the recent physics development in Quantum Matter may guide us to give an adventurous story and possibly a new elementary
explanation. We will see how the topology and cobordism in mathematics may come into play of anomalies and non-perturbative interactions in
fundamental physics. Perhaps some of you (geometers, string theorists, etc.) can team up with me to understand the “boundary conditions” of the Standard Model and Beyond4/15/2020 Lars Andersson (Max-Planck Institute for Gravitational Physics)This meeting will be taking place virtually on Zoom. Title: Stability of spacetimes with supersymmetric compactifications
Abstract: Spacetimes with compact directions, which have special holonomy such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss the global, non-linear stability for the vacuum Einstein equations on a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. I will start by giving a brief overview of related stability problems which have received a lot of attention recently, including the black hole stability problem. This is based on joint work with Pieter Blue, Zoe Wyatt and Shing-Tung Yau.
4/22/2020 William Minicozzi (MIT) This meeting will be taking place virtually on Zoom. Title: Mean curvature flow in high codimension
Abstract: I will talk about joint work with Toby Colding on higher codimension mean curvature flow. Some of the ideas come from function theory on manifolds with Ricci curvature bounds.
4/29/2020 Gerhard Huisken (Tübingen University / MFO) This meeting will be taking place virtually on Zoom. Title: Mean curvature flow of mean-convex embedded 2-surfaces in 3-manifolds
Abstract: The lecture describes joint work with Simon Brendle on the deformation of embedded surfaces with positive mean curvature in Riemannian 3-manifolds in direction of their mean curvature vector. It is described how to find long-time solutions of this flow, possibly including singularities that are overcome by surgery, leading to a comprehensive description of embedded mean-convex surfaces and the regions they bound in a 3-manifold. The flow can be used to sweep out the region between space-like infinity and the outermost horizon in asymptotically flat 3-manifolds arising in General Relativity. (Joint with Simon Brendle.)
5/6/2020 Lydia Bieri (UMich) This meeting will be taking place virtually on Zoom. Title: Energy, Mass and Radiation in General Spacetimes
Abstract: In Mathematical General Relativity (GR) the Einstein equations describe the laws of the universe. Isolated gravitating systems such as binary stars, black holes or galaxies can be described in GR by asymptotically flat (AF) solutions of these equations. These are solutions that look like flat Minkowski space outside of spatially compact regions. There are well-defined notions for energy and mass for such systems. The energy-matter content as well as the dynamics of such a system dictate the decay rates at which the solution tends to the flat one at infinity. Interesting questions occur for very general AF systems of slow decay. We are also interested in spacetimes with pure radiation. In this talk, I will review what is known for these systems. Then we will concentrate on spacetimes with pure radiation. In particular, we will compare the situations of incoming radiation and outgoing radiation under various circumstances and what we can read off from future null infinity.
5/13/2020 Mikhail Lukin (Harvard) This meeting will be taking place virtually on Zoom. Title: Exploring New Frontiers of Quantum Science with Programmable Atom Arrays
Abstract: We will discuss recent work at a new scientific interface between many-body physics and quantum information science. Specifically, we will describe the advances involving programmable, coherent manipulation of quantum many-body systems using atom arrays excited into Rydberg states. Within this system we performed quantum simulations of one dimensional spin models, discovered a new type of non-equilibrium quantum dynamics associated with the so-called many body scars and created large-scale entangled states. We will also describe the most recent developments that now allow the control over 200 atoms in two-dimensional arrays. Ongoing efforts to study exotic many-body phenomena and to realize and test quantum optimization algorithms within such systems will be discussed.
5/20/2020 This meeting will be taking place virtually on Zoom. Fall 2019
Date Speaker Title/Abstract 9/18/2019 Bill Helton (UC San Diego) Title: A taste of noncommutative convex algebraic geometry Abstract: The last decade has seen the development of a substantial noncommutative (in a free algebra) real and complex algebraic geometry. The aim of the subject is to develop a systematic theory of equations and inequalities for (noncommutative) polynomials or rational functions of matrix variables. Such issues occur in linear systems engineering problems, in free probability (random matrices), and in quantum information theory. In many ways the noncommutative (NC) theory is much cleaner than classical (real) algebraic geometry. For example,
◦ A NC polynomial, whose value is positive semidefinite whenever you plug matrices into it, is a sum of squares of NC polynomials.
◦ A convex NC semialgebraic set has a linear matrix inequality representation.
◦ The natural Nullstellensatz are falling into place.
The goal of the talk is to give a taste of a few basic results and some idea of how these noncommutative problems occur in engineering. The subject is just beginning and so is accessible without much background. Much of the work is joint with Igor Klep who is also visiting CMSA for the Fall of 2019.
9/25/2019 Pavel Etingof (MIT) Title: Double affine Hecke algebras Abstract: Double affine Hecke algebras (DAHAs) were introduced by I. Cherednik in the early 1990s to prove Macdonald’s conjectures. A DAHA is the quotient of the group algebra of the elliptic braid group attached to a root system by Hecke relations. DAHAs and their degenerations are now central objects of representation theory. They also have numerous connections to many other fields — integrable systems, quantum groups, knot theory, algebraic geometry, combinatorics, and others. In my talk, I will discuss the basic properties of double affine Hecke algebras and touch upon some applications.
10/2/2019 Spiro Karigiannis (University of Waterloo) Title: Cohomologies on almost complex manifolds and their applications Abstract: We define three cohomologies on an almost complex manifold (M, J), defined using the Nijenhuis-Lie derivations induced from the almost complex structure J and its Nijenhuis tensor N, regarded as vector-valued forms on M. One of these can be applied to distinguish non-isomorphic non-integrable almost complex structures on M. Another one, the J-cohomology, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex structures. The J-cohomology encodes whether a complex manifold satisfies the “del-delbar-lemma”, and more generally in the non-integrable case the J-cohomology encodes whether (M, J) satisfies a generalization of this lemma. We also mention some other potential cohomologies on almost complex manifolds, related to an interesting question involving the Nijenhuis tensor. This is joint work with Ki Fung Chan and Chi Cheuk Tsang.
10/9/2019 Hans Lindblad (Johns Hopkins University) Title: Global Existence and Scattering for Einstein’s equations and related equations satisfying the weak null condition Abstract: Einstein’s equations in harmonic or wave coordinates are a system of nonlinear wave equations for a Lorentzian metric, that in addition satisfy the preserved wave coordinate condition.
Christodoulou-Klainerman proved global existence for Einstein vacuum equations for small asymptotically flat initial data. Their proof avoids using coordinates since it was believed the metric in harmonic coordinates would blow up for large times.
John had noticed that solutions to some nonlinear wave equations blow up for small data, whereas lainerman came up with the ‘null condition’, that guaranteed global existence for small data. However Einstein’s equations do not satisfy the null condition.
Hormander introduced a simplified asymptotic system by neglecting angular derivatives which we expect decay faster due to the rotational invariance, and used it to study blowup. I showed that the asymptotic system corresponding to the quasilinear part of Einstein’s equations does not blow up and gave an example of a nonlinear equation of this form that has global solutions even though it does not satisfy the null condition.
Together with Rodnianski we introduced the ‘weak null condition’ requiring that the corresponding asymptotic system have global solutions and we showed that Einstein’s equations in wave coordinates satisfy the weak null condition and we proved global existence for this system. Our method reduced the proof to afraction and has now been used to prove global existence also with matter fields.
Recently I derived precise asymptotics for the metric which involves logarithmic corrections to the radiation field of solutions of linear wave equations. We are further imposing these asymptotics at infinity and solve the equationsbackwards to obtain global solutions with given data at infinity.
10/16/2019 Aram Harrow (MIT) Title: Monogamy of entanglement and convex geometry Abstract: The SoS (sum of squares) hierarchy is a flexible algorithm that can be used to optimize polynomials and to test whether a quantum state is entangled or separable. (Remarkably, these two problems are nearly isomorphic.) These questions lie at the boundary of P, NP and the unique games conjecture, but it is in general open how well the SoS algorithm performs. I will discuss how ideas from quantum information (the “monogamy” property of entanglement) can be used to understand this algorithm. Then I will describe an alternate algorithm that relies on apparently different tools from convex geometry that achieves similar performance. This is an example of a series of remarkable parallels between SoS algorithms and simpler algorithms that exhaustively search over carefully chosen sets. Finally, I will describe known limitations on SoS algorithms for these problems.
10/23/2019 No talk 10/30/2019 Nima Arkani-Hamed (IAS) Title: Spacetime, Quantum Mechanics and Positive Geometry at Infinity 11/6/2019 Kevin Costello (Perimeter Institute) Title: A unified perspective on integrability Abstract: Two dimensional integrable field theories, and the integrable PDEs which are their classical limits, play an important role in mathematics and physics. I will describe a geometric construction of integrable field theories which yields (essentially) all known integrable theories as well as many new ones. Billiard dynamical systems will play a surprising role. Based on work (partly in progress) with Gaiotto, Lee, Yamazaki, Witten, and Wu.
11/13/2019 Heather Harrington (University of Oxford) Title: Algebra, Geometry and Topology of ERK Enzyme Kinetics Abstract: In this talk I will analyse ERK time course data by developing mathematical models of enzyme kinetics. I will present how we can use differential algebra and geometry for model identifiability and topological data analysis to study these the wild type dynamics of ERK and ERK mutants. This work is joint with Lewis Marsh, Emilie Dufresne, Helen Byrne and Stanislav Shvartsman.
11/20/2019 Xi Yin (Harvard) Title: An Introduction to the Non-Perturbative Bootstrap Abstract: I will discuss non-perturbative definitions of quantum field theories, some properties of correlation functions of local operators, and give a brief overview of some results and open questions concerning the conformal bootstrap
11/25/2019 Monday
Madhu Sudan (Harvard) Abstract: The task of manipulating randomness has been a subject of intense investigation in the theory of computer science. The classical definition of this task consider a single processor massaging random samples from an unknown source and trying to convert it into a sequence of uniform independent bits.In this talk I will talk about a less studied setting where randomness is distributed among different players who would like to convert this randomness to others forms with relatively little communication. For instance players may be given access to a source of biased correlated bits, and their goal may be to get a common random bit out of this source. Even in the setting where the source is known this can lead to some interesting questions that have been explored since the 70s with striking constructions and some surprisingly hard questions. After giving some background, I will describe a recent work which explores the task of extracting common randomness from correlated sources with bounds on the number of rounds of interaction.
Based on joint works with Mitali Bafna (Harvard), Badih Ghazi (Google) and Noah Golowich (Harvard).
12/4/2019 Xiao-Gang Wen (MIT)
VideoTitle: Emergence of graviton-like excitations from a lattice model Abstract: I will review some construction of lattice rotor model which give rise to emergent photons and graviton-like excitations. The appearance of vector-like charge and symmetric tensor field may be related to gapless fracton phases.
2018-2019
Date Speaker Title/Abstract 9/26/2018 Xiao-Gang Wen (MIT) Title: A classification of low dimensional topological orders and fully extended TQFTs Abstract: In this talk, I will review the recent progress on classification of gapped phases of quantum matter (ie topological orders) in 1,2, and 3 spatial dimensions for boson systems. In 1-dimension, there is no non-trivial topological orders. In 2-dimensions, the topological orders are classified by modular tensor category theory. In 3-dimensions, the topological orders are classified by a simple class of braided fusion 2-categories. The classification of topological orders may correspond to a classification of fully extended unitary TQFTs.
10/03/2018 Richard Schoen (Stanford) Title: Perspectives on the scalar curvature Abstract: This will be a general talk concerning the role that the scalar curvature plays in Riemannian geometry and general relativity. We will describe recent work on extending the known results to all dimensions, and other issues which are being actively studied.
10/10/2018 Justin Solomon (MIT) Title: Correspondence and Optimal Transport for Geometric Data Processing Abstract: Correspondence problems involving matching of two or more geometric domains find application across disciplines, from machine learning to computer vision. A basic theoretical framework involving correspondence along geometric domains is optimal transport (OT). Dating back to early economic applications, the OT problem has received renewed interest thanks to its applicability to problems in machine learning, computer graphics, geometry, and other disciplines. The main barrier to wide adoption of OT as a modeling tool is the expense of optimization in OT problems. In this talk, I will summarize efforts in my group to make large-scale transport tractable over a variety of domains and in a variety of application scenarios, helping transition OT from theory to practice. In addition, I will show how OT can be used as a unit in algorithms for solving a variety of problems involving the processing of geometrically-structured data.
10/17/2018 Jeremy England (MIT) Title: Wisdom of the Jumble Abstract: There are certain, specific behaviors that are particularly distinctive of life. For example, living things self-replicate, harvest energy from challenging environmental sources, and translate experiences of past and present into actions that accurately anticipate the predictable parts of their future. What all of these activities have in common from a physics standpoint is that they generally take place under conditions where the pronounced flow of heat sharpens the arrow of time. We have therefore sought to use thermodynamics to understand the emergence and persistence of life-like phenomena in a wide range of messy systems made of many interacting components.
In this talk I will discuss some of the recent insights we have gleaned from studying emergent fine-tuning in disordered collections of matter exposed to complexly patterned environments. I will also point towards future possible applications in the design of new, more life-like ways of computing that have the potential to either be cheaper or more powerful than existing means.
10/31/2018 Moon Duchin (Tufts) Title: Exploring the (massive) space of graph partitions Abstract: The problem of electoral redistricting can be set up as a search of the space of partitions of a graph (representing the units of a state or other jurisdiction) subject to constraints (state and federal rules about the properties of districts). I’ll survey the problem and some approaches to studying it, with an emphasis on the deep mathematical questions it raises, from combinatorial enumeration to discrete differential geometry to dynamics.
11/14/2018 Dusa McDuff (Columbia) Title: The virtual fundamental class in symplectic geometry Abstract: Essential to many constructions and applications of symplectic geometry is the ability to count J-holomorphic curves. The moduli spaces of such curves have well understood compactifications, and if cut out transversally are oriented manifolds of dimension equal to the index of the problem, so that they a fundamental class that can be used to count curves. In the general case, when the defining equation is not transverse, there are various different approaches to constructing a representative for this class, We will discuss and compare different approaches to such a construction e.g. using polyfolds or various kinds of finite dimensional reduction. Most of this is joint work with Katrin Wehrheim.
11/19/2018 Xiaoqin Wang (Johns Hopkins) Title: Computational Principles of Auditory Cortex Abstract: Auditory cortex is located at the top of a hierarchical processing pathway in the brain that encodes acoustic information. This brain region is crucial for speech and music perception and vocal production. Auditory cortex has long been considered a difficult brain region to study and remained one of less understood sensory cortices. Studies have shown that neural computation in auditory cortex is highly nonlinear. In contrast to other sensory systems, the auditory system has a longer pathway between sensory receptors and the cerebral cortex. This unique organization reflects the needs of the auditory system to process time-varying and spectrally overlapping acoustic signals entering the ears from all spatial directions at any given time. Unlike visual or somatosensory cortices, auditory cortex must also process and differentiate sounds that are externally generated or self-produced (during speaking). Neural representations of acoustic information in auditory cortex are shaped by auditory feedback and vocal control signals during speaking. Our laboratory has developed a unique and highly vocal non-human primate model (the common marmoset) and quantitative tools to study neural mechanisms underlying audition and vocal communication.
11/28/2018 Robert Haslhofer (University of Toronto) Title: Recent progress on mean curvature flow Abstract: A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution in extrinsic geometry and shares many features with Hamilton’s Ricci flow from intrinsic geometry. In the first half of the talk, I will give an overview of the well developed theory in the mean convex case, i.e. when the mean curvature vector everywhere on the surface points inwards. Mean convex mean curvature flow can be continued through all singularities either via surgery or as level set solution, with a precise structure theory for the singular set. In the second half of the talk, I will report on recent progress in the general case without any curvature assumptions. Namely, I will describe our solution of the mean convex neighborhood conjecture and the nonfattening conjecture, as well as a general classification result for all possible blowup limits near spherical or cylindrical singularities. In particular, assuming Ilmanen’s multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed. This is joint work with Kyeongsu Choi and Or Hershkovits.
12/5/2018 Robert McCann (University of Toronto) Title: Displacement convexity of Boltzmann’s entropy characterizes positive energy in general relativity Abstract: Einstein’s theory of gravity is based on assuming that the fluxes of a energy and momentum in a physical system are proportional to a certain variant of the Ricci curvature tensor on a smooth 3+1 dimensional spacetime. The fact that gravity is attractive rather than repulsive is encoded in the positivity properties which this tensor is assumed to satisfy. Hawking and Penrose (1971) used this positivity of energy to give conditions under which smooth spacetimes must develop singularities. By lifting fractional powers of the Lorentz distance between points on a globally hyperbolic spacetime to probability measures on spacetime events, we show that the strong energy condition of Hawking and Penrose is equivalent to convexity of the Boltzmann-Shannon entropy along the resulting geodesics of probability measures. This new characterization of the strong energy condition on globally hyperbolic manifolds also makes sense in (non-smooth) metric measure settings, where it has the potential to provide a framework for developing a theory of gravity which admits certain singularities and can be continued beyond them. It provides a Lorentzian analog of Lott, Villani and Sturm’s metric-measure theory of lower Ricci bounds, and hints at new connections linking gravity to the second law of thermodynamics.
Preprint available at http://www.math.toronto.edu/mccann/papers/GRO.pdf
12/12/2018 Zhiwei Yun (MIT) Title: Shtukas: what and why Abstract: This talk is of expository nature. Drinfeld introduced the notion of Shtukas and the moduli space of them. I will review how Shtukas compare to more familiar objects in geometry, how they are used in the Langlands program, and what remains to be done about them.
1/30/2019 Richard Freeman (Harvard) Title: Innovation in Cell Phones in the US and China: Who Improves Technology Faster? Abstract: Cell phones are the archetypical modern consumer innovation, spreading around the world at an incredible pace, extensively used for connecting people with the Internet and diverse apps. Consumers report spending from 2-5 hours a day at their cell phones, with 44% of Americans saying “couldn’t go a day without their mobile devices.” Cell phone manufacturers introduce new models regularly, embodying additional features while other firms produce new applications that increase demand for the phones. Using newly developed data on the prices, attributes, and sales of different models in the US and China, this paper estimates the magnitude of technological change in the phones in the 2000s. It explores the problems of analyzing a product with many interactive attributes in the standard hedonic price regression model and uses Principal Components Regression to reduce dimensionality. The main finding is that technology improved the value of cell phones at comparable rates in the US and China, despite different market structures and different evaluations of some attributes and brands. The study concludes with a discussion of ways to evaluate the economic surplus created by the cell phones and their contribution to economic well-being.
2/7/2019 *Thursday*
Ulrich Mueller (Princeton) Title: Inference for the Mean Abstract: Consider inference about the mean of a population with finite variance, based on an i.i.d. sample. The usual t-statistic yields correct inference in large samples, but heavy tails induce poor small sample behavior. This paper combines extreme value theory for the smallest and largest observations with a normal approximation for the t-statistic of a truncated sample to obtain more accurate inference. This alternative approximation is shown to provide a refinement over the standard normal approximation to the full sample t-statistic under more than two but less than three moments, while the bootstrap does not. Small sample simulations suggest substantial size improvements over the bootstrap.
2/13/2019 Christian Santangelo (UMass Amherst) Title: 4D printing with folding forms Abstract: 4D printing is the name given to a set of advanced manufacturing techniques for designing flat materials that, upon application of a stimulus, fold and deform into a target three-dimensional shapes. The successful design of such structures requires an understanding of geometry as it applies to the mechanics of thin, elastic sheets. Thus, 4D printing provides a playground for both the development of new theoretical tools as well as old tools applied to new problems and experimental challenges in soft materials. I will describe our group’s efforts to understand and design structures that can fold from an initially flat sheet to target three-dimensional shapes. After reviewing the state-of-the-art in the theory of 4D printing, I will describe recent results on the folding and misfolding of flat structures and highlight the challenges remaining to be overcome.
2/20/2019 Michael Woodford (Columbia) Title: Optimally Imprecise Memory and Biased Forecasts Abstract: We propose a model of optimal decision making subject to a memory constraint. The constraint is a limit on the complexity of memory measured using Shannon’s mutual information, as in models of rational inattention; the structure of the imprecise memory is optimized (for a given decision problem and noisy environment) subject to this constraint. We characterize the form of the optimally imprecise memory, and show that the model implies that both forecasts and actions will exhibit idiosyncratic random variation; that beliefs will fluctuate forever around the rational-expectations (perfect-memory) beliefs with a variance that does not fall to zero; and that more recent news will be given disproportionate weight. The model provides a simple explanation for a number of features of observed forecast bias in laboratory and field settings.
[authors: Rava Azeredo da Silveira (ENS) and Michael Woodford (Columbia)]
2/27/2019 2:30pm
Ian Martin (LSE) Title: Sentiment and Speculation in a Market with Heterogeneous Beliefs Abstract: We present a dynamic model featuring risk-averse investors with heterogeneous beliefs. Individual investors have stable beliefs and risk aversion, but agents who were correct in hindsight become relatively wealthy; their beliefs are overrepresented in market sentiment, so “the market” is bullish following good news and bearish following bad news. Extreme states are far more important than in a homogeneous economy. Investors understand that sentiment drives volatility up, and demand high risk premia in compensation. Moderate investors supply liquidity: they trade against market sentiment in the hope of capturing a variance risk premium created by the presence of extremists. [with Dimitris Papadimitriou]
3/6/2019 2:30pm
Philippe Sosoe (Cornell) Title: A sharp transition for Gibbs measures associated to the nonlinear Schrödinger equation Abstract: In 1987, Lebowitz, Rose and Speer (LRS) showed how to construct formally invariant measures for the nonlinear Schrödinger equation on the torus. This seminal contribution spurred a large amount of activity in the area of partial differential equations with random initial data. In this talk, I will explain LRS’s result, and discuss a sharp transition in the construction of the Gibbs-type invariant measures considered by these authors. (Joint work with Tadahiro Oh and Leonardo Tolomeo)
3/13/2019 5:15pm
Greg Galloway (University of Miami) Title: On the geometry and topology of initial data sets in General Relativity Abstract: A theme of long standing interest (to the speaker!) concerns the relationship between the topology of spacetime and the occurrence of singularities (causal geodesic incompleteness). Many results concerning this center around the notion of topological censorship, which has to do with the idea that the region outside all black holes (and white holes) should be simple. The aim of the results to be presented is to provide support for topological censorship at the pure initial data level, thereby circumventing difficult issues of global evolution. The proofs rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry. The talk will begin with a brief overview of general relativity and topological censorship. The talk is based primarily on joint work with various collaborators: Lars Andersson, Mattias Dahl, Michael Eichmair and Dan Pollack.
3/20/2019 Sonia Jaffe (Microsoft) Title: Quality Externalities on Platforms: The Case of Airbnb Abstract: We explore quality externalities on platforms: when buyers have limited information, a seller’s quality affects whether her buyers return to the platform, thereby impacting other sellers’ future business. We propose an intuitive measure of this externality, applicable across a range of platforms. Guest Return Propensity (GRP) is the aggregate propensity of a seller’s customers to return to the platform. We validate this metric using Airbnb data: matching customers to listings with a one standard deviation higher GRP causes them to take 17% more subsequent trips. By directing buyers to higher-GRP sellers, platforms may be able to increase overall seller surplus. (Joint work with Peter Coles, Steven Levitt, and Igor Popov.)
3/27/2019 5:15pm
Tatyana Sharpee (Salk Institute for Biological Studies) Title: Hyperbolic geometry of the olfactory space. Abstract: The sense of smell can be used to avoid poisons or estimate a food’s nutrition content because biochemical reactions create many by-products. Thus, the presence of certain bacteria in the food becomes associated with the emission of certain volatile compounds. This perspective suggests that it would be convenient for the nervous system encode odors based on statistics of their co-occurrence within natural mixtures rather than based on the chemical structure per se. I will discuss how this statistical perspective makes it possible to map odors to points in a hyperbolic space. Hyperbolic coordinates have a long but often underappreciated history of relevance to biology. For example, these coordinates approximate distance between species computed along dendograms, and more generally between points within hierarchical tree-like networks. We find that these coordinates, which were generated purely based on the statistics of odors in the natural environment, provide a contiguous map of human odor pleasantness. Further, a separate analysis of human perceptual descriptions of smells indicates that these also generate a three dimensional hyperbolic representation of odors. This match in geometries between natural odor statistics and human perception can help to minimize distortions that would otherwise arise when mapping odors to perception. We identify three axes in the perceptual space that are aligned with odor pleasantness, its molecular boiling point and acidity. Because the perceptual space is curved, one can predict odor pleasantness by knowing the coordinates along the molecular boiling point and acidity axes.
4/3/2019 2:30pm
Sarah Moshary (Chicago Booth) Title: Deregulation through Direct Democracy: Lessons from Liquor Abstract: This paper examines the merits of state control versus private provision of spirits retail, using the 2012 deregulation of liquor sales in Washington state as an event study. We document effects along a number of dimensions: prices, product variety, convenience, substitution to other goods, state revenue, and consumption externalities. We estimate a demand system to evaluate the net effect of privatization on consumer welfare. Our findings suggest that deregulation harmed the median Washingtonian, even though residents voted in favor of deregulation by a 16% margin. Further, we find that vote shares for the deregulation initiative do not reflect welfare gains at the ZIP code level. We discuss implications of our findings for the efficacy of direct democracy as a policy tool.
4/10/2019 2:30pm
Pietro Veronesi (Chicago Booth) Title: Inequality Aversion, Populism, and the Backlash Against Globalization Abstract: Motivated by the recent rise of populism in western democracies, we develop a model in which a populist backlash emerges endogenously in a growing economy. In the model, voters dislike inequality, especially the high consumption of “elites.” Economic growth exacerbates inequality due to heterogeneity in risk aversion. In response to rising inequality, rich-country voters optimally elect a populist promising to end globalization. Countries with more inequality, higher financial development, and current account deficits are more vulnerable to populism, both in the model and in the data. Evidence on who voted for Brexit and Trump in 2016 also supports the model.
4/17/2019 Yi-Zhuang You (UCSD) Title: Machine Learning Physics: From Quantum Mechanics to Holographic Geometry Abstract: Inspired by the “third wave” of artificial intelligence (AI), machine learning has found rapid applications in various topics of physics research. Perhaps one of the most ambitious goals of machine learning physics is to develop novel approaches that ultimately allows AI to discover new concepts and governing equations of physics from experimental observations. In this talk, I will present our progress in applying machine learning technique to reveal the quantum wave function of Bose-Einstein condensate (BEC) and the holographic geometry of conformal field theories. In the first part, we apply machine translation to learn the mapping between potential and density profiles of BEC and show how the concept of quantum wave function can emerge in the latent space of the translator and how the Schrodinger equation is formulated as a recurrent neural network. In the second part, we design a generative model to learn the field theory configuration of the XY model and show how the machine can identify the holographic bulk degrees of freedom and use them to probe the emergent holographic geometry.
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[1] C. Wang, H. Zhai, Y.-Z. You. Uncover the Black Box of Machine Learning Applied to Quantum Problem by an Introspective Learning Architecture https://arxiv.org/abs/1901.11103
[2] H.-Y. Hu, S.-H. Li, L. Wang, Y.-Z. You. Machine Learning Holographic Mapping by Neural Network Renormalization Group https://arxiv.org/abs/1903.00804
[3] Y.-Z. You, Z. Yang, X.-L. Qi. Machine Learning Spatial Geometry from Entanglement Features https://arxiv.org/abs/1709.01223
4/24/2019 Shengwu Li (Harvard) Title: Credible MechanismsAbstract: Consider an extensive-form mechanism, run by an auctioneer who communicates sequentially and privately with agents. Suppose the auctioneer can deviate from the rules provided that no single agent detects the deviation. A mechanism is credible if it is incentive-compatible for the auctioneer to follow the rules. We study the optimal auctions in which only winners pay, under symmetric independent private values. The first-price auction is the unique credible static mechanism. The ascending auction is the unique credible strategy-proof mechanism.Date………… Speaker Title 02-09-2018 *Friday Fan Chung (UCSD)
Sequences: random, structured or something in between There are many fundamental problems concerning sequences that arise in many areas of mathematics and computation. Typical problems include finding or avoiding patterns;
testing or validating various `random-like’ behavior; analyzing or comparing different statistics, etc. In this talk, we will examine various notions of regularity or irregularity for sequences and mention numerous open problems.
02-14-2018 Zhengwei Liu (Harvard Physics)
A new program on quantum subgroups Abstract: Quantum subgroups have been studied since the 1980s. The A, D, E classification of subgroups of quantum SU(2) is a quantum analogue of the McKay correspondence. It turns out to be related to various areas in mathematics and physics. Inspired by the quantum McKay correspondence, we introduce a new program that our group at Harvard is developing.
02-21-2018 Don Rubin (Harvard)
Essential concepts of causal inference — a remarkable history Abstract: I believe that a deep understanding of cause and effect, and how to estimate causal effects from data, complete with the associated mathematical notation and expressions, only evolved in the twentieth century. The crucial idea of randomized experiments was apparently first proposed in 1925 in the context of agricultural field trails but quickly moved to be applied also in studies of animal breeding and then in industrial manufacturing. The conceptual understanding seemed to be tied to ideas that were developing in quantum mechanics. The key ideas of randomized experiments evidently were not applied to studies of human beings until the 1950s, when such experiments began to be used in controlled medical trials, and then in social science — in education and economics. Humans are more complex than plants and animals, however, and with such trials came the attendant complexities of non-compliance with assigned treatment and the occurrence of “Hawthorne” and placebo effects. The formal application of the insights from earlier simpler experimental settings to more complex ones dealing with people, started in the 1970s and continue to this day, and include the bridging of classical mathematical ideas of experimentation, including fractional replication and geometrical formulations from the early twentieth century, with modern ideas that rely on powerful computing to implement aspects of design and analysis.
02-26-2018 *Monday Tom Hou (Caltech)
Computer-assisted analysis of singularity formation of a regularized 3D Euler equation Abstract: Whether the 3D incompressible Euler equation can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. In a recent joint work with Dr. Guo Luo, we provided convincing numerical evidence that the 3D Euler equation develops finite time singularities. Inspired by this finding, we have recently developed an integrated analysis and computation strategy to analyze the finite time singularity of a regularized 3D Euler equation. We first transform the regularized 3D Euler equation into an equivalent dynamic rescaling formulation. We then study the stability of an approximate self-similar solution. By designing an appropriate functional space and decomposing the solution into a low frequency part and a high frequency part, we prove nonlinear stability of the dynamic rescaling equation around the approximate self-similar solution, which implies the existence of the finite time blow-up of the regularized 3D Euler equation. This is a joint work with Jiajie Chen, De Huang, and Dr. Pengfei Liu.
03-07-2018 Richard Kenyon (Brown)
Harmonic functions and the chromatic polynomial Abstract: When we solve the Dirichlet problem on a graph, we look for a harmonic function with fixed boundary values. Associated to such a harmonic function is the Dirichlet energy on each edge. One can reverse the problem, and ask if, for some choice of conductances on the edges, one can find a harmonic function attaining any given tuple of edge energies. We show how the number of solutions to this problem is related to the chromatic polynomial, and also discuss some geometric applications. This talk is based on joint work with Aaron Abrams and Wayne Lam.
03-14-2018 03-21-2018 03-28-2018 Andrea Montanari (Stanford) A Mean Field View of the Landscape of Two-Layers Neural Networks Abstract: Multi-layer neural networks are among the most powerful models in machine learning and yet, the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a highly non-convex and high-dimensional objective (risk function), a problem which is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the first case, does this happen because local minima are absent, or because SGD somehow avoids them? In the second, why do local minima reached by SGD have good generalization properties?
We consider a simple case, namely two-layers neural networks, and prove that –in a suitable scaling limit– the SGD dynamics is captured by a certain non-linear partial differential equation. We then consider several specific examples, and show how the asymptotic description can be used to prove convergence of SGD to network with nearly-ideal generalization error. This description allows to `average-out’ some of the complexities of the landscape of neural networks, and can be used to capture some important variants of SGD as well.
[Based on joint work with Song Mei and Phan-Minh Nguyen]03-30-2018 04-04-2018 Ramesh Narayan (Harvard)
Black Holes and Naked Singularities Abstract: Black Hole solutions in General Relativity contain Event Horizons and
Singularities. Astrophysicists have discovered two populations of
black hole candidates in the Universe: stellar-mass objects with
masses in the range 5 to 30 solar masses, and supermassive objects
with masses in the range million to several billion solar
masses. There is considerable evidence that these objects have Event
Horizons. It thus appears that astronomical black hole candidates are
true Black Holes. Direct evidence for Singularities is much harder to
obtain since, at least in the case of Black Holes, the Singularities
are hidden inside the Event Horizon. However, General Relativity also
permits Naked Singularities which are visible to external
observers. Toy Naked Singularity models have been constructed, and
some observational features of accretion flows in these spacetimes
have been worked out.04-11-2018 Pablo Parrilo (MIT)
Graph Structure in Polynomial Systems: Chordal Networks Abstract: The sparsity structure of a system of polynomial equations or an optimization problem can be naturally described by a graph summarizing the interactions among the decision variables. It is natural to wonder whether the structure of this graph might help in computational algebraic geometry tasks (e.g., in solving the system). In this lecture we will provide a gentle introduction to this area, focused on the key notions of chordality and treewidth, which are of great importance in related areas such as numerical linear algebra, database theory, constraint satisfaction, and graphical models. In particular, we will discuss “chordal networks”, a novel representation of structured polynomial systems that provides a computationally convenient decomposition of a polynomial ideal into simpler (triangular) polynomial sets, while maintaining its underlying graphical structure. As we will illustrate through examples from different application domains, algorithms based on chordal networks can significantly outperform existing techniques. Based on joint work with Diego Cifuentes (MIT).
04-18-2018 Washington Taylor (MIT)
On the fibration structure of known Calabi-Yau threefolds Abstract: In recent years, there is increasing evidence from a variety of directions, including the physics of F-theory and new generalized CICY constructions, that a large fraction of known Calabi-Yau manifolds have a genus one or elliptic fibration. In this talk I will describe recent work with Yu-Chien Huang on a systematic analysis of the fibration structure of known toric hypersurface Calabi-Yau threefolds. Among other results, this analysis shows that every known Calabi-Yau threefold with either Hodge number exceeding 150 is genus one or elliptically fibered, and suggests that the fraction of Calabi-Yau threefolds that are not genus one or elliptically fibered decreases roughly exponentially with h_{11}. I will also make some comments on the connection with the structure of triple intersection numbers in Calabi-Yau threefolds.
04-25-2018 Xi Yin (Harvard)
How we can learn what we need to know about M-theory Abstract: M-theory is a quantum theory of gravity that admits an eleven dimensional Minkowskian vacuum with super-Poincare symmetry and no dimensionless coupling constant. I will review what was known about M-theory based on its relation to superstring theories, then comment on a number of open questions, and discuss how they can be addressed from holographic dualities. I will outline a strategy for extracting the S-matrix of M-theory from correlation functions of dual superconformal field theories, and in particular use it to recover the 11D R^4 coupling of M-theory from ABJM theory.
05-02-2018 05-09-2018 2016-2017
Date Name Title/Abstract 01-25-17 Sam Gershman, Harvard Center for Brain Science, Department of Psychology Title: Spectral graph theory of cognitive maps
Abstract: The concept of a “cognitive map” has played an important role in neuroscience and psychology. A cognitive map is a representation of the environment that supports navigation and decision making. A longstanding question concerns the precise computational nature of this map. I offer a new mathematical foundation for the cognitive map, based on ideas at the intersection of spectral graph theory and reinforcement learning. Empirical data from neural recordings and behavioral experiments supports this theory.
02-01-17 Sean Eddy, Harvard Department of Molecular and Cellular Biology Title: Biological sequence homology searches: the future of deciphering the past Abstract: Computational recognition of distant common ancestry of biological sequences is a key to studying ancient events in molecular evolution.The better our sequence analysis methods are, the deeper in evolutionary time we can see. A major aim in the field is to improve the resolution of homology recognition methods by building increasingly realistic, complex, parameter-rich models. I will describe current and future research in homology search algorithms based on probabilistic inference methods, using hidden Markov models(HMMs) and stochastic context-free grammars (SCFGs). We make these methods available in the HMMER and Infernal software from my laboratory, in collaboration with database teams at the EuropeanBioinformatics Institute in the UK.
02-08-17 Matthew Headrick, Brandeis University Title: Quantum entanglement, classical gravity, and convex programming: New connections Abstract: In recent years, developments from the study of black holes and quantum gravity have revealed a surprising connection between quantum entanglement and classical general relativity. The theory of convex programming, applied in the differential-geometry setting, turns out to be useful for understanding what’s behind this correspondence. We will describe these developments, giving the necessary background in quantum information theory and convex programming along the way.
02-15-17 Masahito Yamazaki, IMPU Title: Geometry of 3-manifolds and Complex Chern-Simons Theory Abstract: The geometry of 3-manifolds has been a fascinating subject in mathematics. In this talk I discuss a “quantization” of 3-manifold geometry, in the language of complex Chern-Simons theory. This Chern-Simons theory in turn is related to the physics of 30dimensional supersymmetric field theories through the so-called 3d/3d correspondence, whose origin can be traced back to a mysterious theory on the M5-branes. Along the way I will also comment on the connection with a number of related topics, such as knot theory, hyperbolic geometry, quantum dilogarithm and cluster algebras.
02-22-17 Steven Rayan, University of Saskatchewan Title: Higgs bundles and the Hitchin system
Abstract: I will give an informal introduction to the Hitchin system, an object lying at the crossroads of geometry and physics. As a moduli space, the Hitchin system parametrizes semistable Higgs bundles on a Riemann surface up to equivalence. From this point of view, the Hitchin map and spectral curves emerge. We’ll use these to form an impression of what the moduli space “looks like”. I will also outline the appearances of the Hitchin system in dynamics, hyperkaehler geometry, and mirror symmetry.
03-01-17 Jun Liu, Harvard University Title: Expansion of biological pathways by integrative Genomics Abstract: The number of publicly available gene expression datasets has been growing dramatically. Various methods had been proposed to predict gene co-expression by integrating the publicly available datasets. These methods assume that the genes in the query gene set are homogeneously correlated and consider no gene-specific correlation tendencies, no background intra-experimental correlations, and no quality variations of different experiments. We propose a two-step algorithm called CLIC (CLustering by Inferred Co-expression) based on a coherent Bayesian model to overcome these limitations. CLIC first employs a Bayesian partition model with feature selection to partition the gene set into disjoint co-expression modules (CEMs), simultaneously assigning posterior probability of selection to each dataset. In the second step, CLIC expands each CEM by scanning the whole reference genome for candidate genes that were not in the input gene set but co-expressed with the genes in this CEM. CLIC is capable of integrating over thousands of gene expression datasets to achieve much higher coexpression prediction accuracy compared to traditional co-expression methods. Application of CLIC to ~1000 annotated human pathways and ~6000 poorly characterized human genes reveals new components of some well-studied pathways and provides strong functional predictions for some poorly characterized genes. We validated the predicted association between protein C7orf55 and ATP synthase assembly using CRISPR knock-out assays.
Based on the joint work with Yang Li and the Vamsi Mootha lab.
03-08-17 Gabor Lippner, Northeastern University Title: Evolution of cooperation in structured populations Abstract: Understanding how the underlying structure affects the evolution of a population is a basic, but difficult, problem in the evolutionary dynamics. Evolutionary game theory, in particular, models the interactions between individuals as games, where different traits correspond to different strategies. It is one of the basic approaches to explain the emergence of cooperative behavior in Darwinian evolution.
In this talk I will present new results about the model where the population is represented by an interaction network. We study the likelihood of a random mutation spreading through the entire population. The main question is to understand how the network influences this likelihood. After introducing the model, I will explain how the problem is connected to the study of meeting times of random walks on graphs, and based on this connection, outline a general method to analyze the model on general networks.03-15-17 Spring Break: No session 03-22-17 Gunther Uhlmann, University of Washington Abstract: We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times ofwaves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also applications in optics and medical imaging among others.The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.We will also describe some recent results, joint with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.03-29-17 Leslie Greengard, Courant Institute Title: Inverse problems in acoustic scattering and cryo-electron microscopy Abstract: A variety of problems in image reconstruction give rise to large-scale, nonlinear and non-convex optimization problems. We will show how recursive linearization combined with suitable fast solvers are bringing such problems within practical reach, with an emphasis on acoustic scattering and protein structure determination via cryo-electron microscopy.
NOTE: This talk will begin at 4:00pm
04-05-17 Gongjie Li, Harvard University Title: Unveiling the Origin of Planetary Systems by Dynamical and Statistical Approaches Abstract: The unexpected diversity of observed extrasolar planetary systems has posed new challenges to our classical understanding of planetary formation. A lot of these challenges can be addressed by a deeper understanding of the dynamics in planetary systems, which will also allow us to construct more accurate planetary formation theories consistent with observations. In this talk, I will first explain the origin of counter orbiting planets using a new dynamical mechanism I discovered, which also has wide implications in other astrophysical systems, such as the enhancement of tidal disruption rates near supermassive black hole binaries. In addition, I will discuss the architectural properties of circumbinary planetary systems from selection biases using statistical methods, and infer the origin of such systems.
04-12-17 Shlomo Razamat, Israel Institute of Technology Title: Complicated four-dimensional physics and simple mathematics Abstract: We will discuss SCFTs in four dimensions obtained from compactifications of six dimensional models. We will discuss the relation of the partition functions, specifically the supersymmetric index, of the SCFTs to certain special functions, and argue that the partition functions are expected to be naturally expressed in terms of eigenfunctions of generalizations of Ruijsenaars-Schneider models. We will discuss how the physics of the compactifications implies various precise mathematical identities involving the special functions, most of which are yet to be proven.
04-19-17 Cumrun Vafa, Harvard University Title: String Swampland Abstract: In this talk I review the idea behind identification of the string swampland. In particular I discuss the weak gravity conjecture as one such criterion and explain a no-go theorem for non-supersymmetric AdS/CFT holography.
04-27-17 Mehran Kardar, MIT Title: Levitation by Casimir forces in and out of equilibrium Abstract: Equilibrium fluctuation-induced forces are abundant in nature, ranging from quantum electrodynamic (QED) Casimir and van der Waals forces, to their thermal analogs in fluctuating soft matter. Repulsive Casimir forces have been proposed for a variety of shapes and materials. A generalization of Earnshaw’s theorem constrains the possibility of levitation by Casimir forces in equilibrium. The scattering formalism, which forms the basis of this proof, can be used to study fluctuation-induced forces for different materials, diverse geometries, both in and out of equilibrium. Conformal field theory methods suggest that critical (thermal) Casimir forces are not subject to a corresponding constraint.
Note: This talk will begin at 3:00pm
05-02-17 Simona Cocco, Laboratoire de Physique Statistique de l’ENS Title: Reverse modeling of protein sequence data: from graphical models to structural and functional predictions Body: A fundamental yet largely open problem in biology and medicine is to understand the relationship between the amino-acid sequence of a protein and its structure and function. Protein databases such as Pfam, which collect, align, and classify protein sequences into families containing
similar (homologous) sequences are growing at a fast pace thanks to recent advances in sequencing technologies. What kind of information about the structure and function of proteins can be obtained from the statistical distribution of sequences in a protein family? To answer this question I will describe recent attempts to infer graphical models able to reproduce the low-order statistics of protein sequence data, in particular amino acid conservation and covariation. I will also review how those models
have led to substantial progress in protein structural and functional
predictions.Note: This talk will begin at 4:00pm
05-03-17 Xue-Mei Li, University of Warwick Title: Perturbation to conservation law and stochastic averaging Abstract: A deterministic or random system with a conservation law is often used to
approximate dynamics that are also subjected to smaller deterministic or random influences. Consider for example dynamical descriptions for Brownian motions and singular perturbed operators arising from rescaled Riemmannian metrics. In both cases the conservation laws, which are maps with values in a manifold, are used to separate the slow and fast variables. We discuss stochastic averaging and diffusion creation arising from these contexts. Our overarching question is to describe stochastic dynamics associated with the convergence of Riemannian manifolds and metric spaces.Note: This talk will be held in the Science Center, Room 507
05-10-17 05-17-17 Kwok Wai Chan, Chinese University of Hong Kong Title: Scattering diagrams from asymptotic analysis on Maurer-Cartan equations Abstract: In 2005, a program was set forth by Fukaya aiming at investigating SYZ mirror symmetry by asymptotic analysis on Maurer-Cartan equations. In this talk, I will explain some results which implement part of Fukaya’s program. More precisely, I will show how semi-classical limits of Maurer-Cartan solutions give rise naturally to consistent scattering diagrams, which are known to encode Gromov-Witten data on the mirror side and have played an important role in the works of Kontsevich-Soibelman and Gross-Siebert on the reconstruction problem in mirror symmetry. This talk is based on joint work with Conan Leung and Ziming Ma, which was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK14302015).
05-24-17 NO COLLOQUIUM 05-31-17 Peter Michor, University of Vienna Title: Geometry of shape spaces and diffeomorphism groups and some of their uses Abstract: This talk is devoted to shape spaces, Riemannian metrics on them, their geodesics and distance functions, and some of their uses, mainly in computational anatomy. The simplest Riemannian metrics have vanishing geodesic distance, so one has to use, for example, higher order Sobolev metrics on shape spaces. These have curvature, which complicates statistics on these spaces.
Date Name Title 09-09-16 Bong Lian, Brandeis
Title: Riemann-Hilbert Problem and Period Integrals
Abstract: Period integrals of an algebraic manifolds are certain special functions that describe, among other things, deformations of the variety. They were originally studied by Euler, Gauss and Riemann, who were interested in analytic continuation of these objects. In this lecture, we will discuss a number of long-standing problems on period integrals in connection with mirror symmetry and Calabi-Yau geometry. We will see how the theory of D-modules have led us to solutions and insights into some of these problems.
09-14-16 Sze-Man Ngai, Georgia Southern University Title: The multifractal formalism and spectral asymptotics of self-similar measures with overlaps Abstract: Self-similar measures form a fundamental class of fractal measures, and is much less understood if they have overlaps. The multifractal formalism, if valid, allows us to compute the Hausdorff dimension of the multifractal components of the measure through its Lq-spectrum. The asymptotic behavior of the eigenvalue counting function for the associated Laplacians is closely related to the multifractal structure of the measure. Throughout this talk, the infinite Bernoulli convolution associated with the golden ratio will be used as a basic example to describe some of the results.
09-21-16 Prof. L. Mahadevan, Harvard SEAS Title: “Morphogenesis: Biology, Physics and Mathematics” Abstract: A century since the publication of Darcy Thompson’s classic “On growth and form,” his vision has finally begun to permeate into the fabric of modern biology. Within this backdrop, I will discuss some simple questions inspired by the onset of form in biology wherein mathematical models and computations, in close connection with experiments allow us to begin unraveling the physical basis for morphogenesis in the context of examples such as tendrils, leaves, guts, and brains. I will also try and indicate how these problems enrich their roots, creating new questions in mathematics, physics, and biology.
09-28-16 Hong Liu, MIT Title: A new theory of fluctuating hydrodynamics Despite its long and glorious history, hydrodynamics has so far been formulated mostly at the level of equations of motion, which is inadequate for capturing fluctuations. In a fluid, however, fluctuations occur spontaneously and continuously, at both the quantum and statistical levels, the understanding of which is important for a wide variety of physical problems. Another unsatisfactory aspect of the current formulation of hydrodynamics is that the equations of motion are constrained by various phenomenological conditions on the solutions, which need to be imposed by hand. One of such constraints is the local second law of thermodynamics, which plays a crucial role, yet whose physical origin has been obscure.
We present a new theory of fluctuating hydrodynamics which incorporates fluctuations systematically and reproduces all the phenomenological constraints from an underlying Z_2 symmetry. In particular, the local second law of thermodynamics is derived. The theory also predicts new constraints which can be considered as nonlinear generalizations of Onsager relations. When truncated to Gaussian noises, the theory recovers various nonlinear stochastic equations.
Curiously, to describe thermal fluctuations of a classical fluid consistently one needs to introduce anti-commuting variables and the theory exhibits an emergent supersymmetry.
10-05-16 Alexander Logunov, Tel-Aviv University
Title: Zeroes of harmonic functions and Laplace eigenfunctions Abs: Nadirashvili conjectured that for any non-constant harmonic function in R^3 its zero set has infinite area. This question was motivated by the Yau conjecture on zero sets of Laplace eigenfunctions. Both conjectures can be treated as an attempt to control the zero set of a solution of elliptic PDE in terms of growth of the solution. For holomorhpic functions such kind of control is possible only from one side: there is a plenty of holomorphic functions that have no zeros. While for a real-valued harmonic function on a plane the length of the zero set can be estimated (locally) from above and below by the frequency, which is a characteristic of growth of the harmonic function. We will discuss the notion of frequency, its properties and applications to zero sets in the higher dimensional case, where the understanding is far from being complete.
10-12-16 Conan Nai Chung Leung, CUHK Title: Coisotropic A-branes and their SYZ transform
Abstract: “Kapustin introduced coisotropic A-branes as the natural boundary condition for strings in A-model, generalizing Lagrangian branes and argued that they are indeed needed to for homological mirror symmetry. I will explain in the semiflat case that the Nahm transformation along SYZ fibration will transform fiberwise Yang-Mills holomorphic bundles to coisotropic A-branes. This explains SYZ mirror symmetry away from the large complex structure limit.”
10-19-16 Vaughan Jones, UC Berkeley Title: Are the Thompson groups any good as a model for Diff(S^1)? Abstract. The Thompson groups are by definition groups of piecewise linear
diffeomorphisms of the circle. A result of Ghys-Sergiescu says that a Thompson group can
be conjugated to a group of smooth diffeomorphisms. That’s the good news.
The bad news is that there is an important central extension of Diff(S^1) which requires a certain amount of smoothness for its definition. And Ghys-Sergiescu show that, no matter how the Thompson groups are embedded in Diff(S^1), the restriction of the central extension splits. Is it possible to obtain central extensions of the Thompson groups by any
procedure analogous to the constructions of the central extension of Diff(S^1)?
I will define all the players in this game, explain this question in detail,and present some failed attempts to answer it.10-26-16 Henry Cohn, Microsoft
Sums of squares, correlation functions, and exceptional geometric structures
Some exceptional structures such as the icosahedron or E_8 root system have remarkable optimality properties in settings such as packing, energy minimization, or coding. How can we understand and prove their optimality? In this talk, I’ll interweave this story with two other developments in recent mathematics (without assuming familiarity with either): how semidefinite optimization and sums of squares have expanded the scope of optimization, and how representation theory has shed light on higher correlation functions for particle systems.
11-02-16 Christian Borgs, Microsoft
Title: Graphon processes and limits of sparse graph sequences
Abstract: The theory of graph limits for dense graphs is by now well established, with graphons describing both the limit of a sequence of deterministic graphs, and a model for so-called exchangeable random graphs. Here a graphon is a function defined over a “feature space’’ equipped with some probability measure, the measure describing the distribution of features for the nodes, and the graphon describing the probability that two nodes with given features form a connection. While there are rich models of sparse random graphs based on graphons, they require an additional parameter, the edge density, whose dependence on the size of the graph has either to be postulated as an additional function, or considered as an empirical observed quantity not described by the model.
In this talk I describe a new model, where the underlying probability space is replaced by a sigma-finite measure space, leading to both a new random model for exchangeable graphs, and a new notion of graph limits. The new model naturally produces a graph valued stochastic process indexed by a continuous time parameter, a “graphon process”, and describes graphs which typically have degree distributions with long tails, as observed in large networks in real life.
11-09-16 TIME CHANGE: 4PM
Norden E. Huang, National Central University, (Taiwan)
Title: On Holo-Hilbert Spectral Analysis Traditionally, spectral analysis is defined as transform the time domain data to frequency domain. It is achieved through integral transforms based on additive expansions of a priori determined basis, under linear and stationary assumptions. For nonlinear processes, the data can have both amplitude and frequency modulations generated by intra-wave and inter-wave interactions involving both additive and nonlinear multiplicative processes. Under such conditions, the additive expansion could not fully represent the physical processes resulting from multiplicative interactions. Unfortunately, all existing spectral analysis methods are based on additive expansions, based either on a priori or adaptive bases. While the adaptive Hilbert spectral analysis could accommodate the intra-wave nonlinearity, the inter-wave nonlinear multiplicative mechanisms that include cross-scale coupling and phase lock modulations are left untreated. To resolve the multiplicative processes, we propose a full informational spectral representation: The Holo-Hilbert Spectral Analysis (HHSA), which would accommodate all the processes: additive and multiplicative, intra-mode and inter-mode, stationary and non-stationary, linear and nonlinear interactions, through additional dimensions in the spectrum to account for both the variations in frequency and amplitude modulations (FM and AM) simultaneously. Applications to wave-turbulence interactions and other data will be presented to demonstrate the usefulness of this new spectral representation.
11-16-16 Tristan Collins, Harvard University TIME CHANGE: 3:30PM
Title: Restricted volumes and finite time singularities of the Kahler-Ricci flow
Abstract: I will discuss the relationship between restricted volumes, as defined algebraically or analytically, and the finite time singularities of the Kahler-Ricci flow. This is joint work with Valentino Tosatti.
11-22-16 TUESDAY TIME CHANGE: 4-5PM
Xiangfeng Gu, Stonybrook
Title: Differential Geometric Methods for Engineering Applications
Abstract: With the development of virtual reality and augmented reality, many challenging problems raised in engineering fields. Most of them are with geometric nature, and can be explored by modern geometric means. In this talk, we introduce our approaches to solve several such kind of problems: including geometric compression, shape classification, surface registration, cancer detection, facial expression tracking and so on, based on surface Ricci flow and optimal mass transportation.
11-30-16 TIME CHANGE: 4:20PM
Sharad Ramanathan, Harvard MCB & SEAS
Title: Finding co-ordinate systems to monitor the development of mammalian embryos 12-07-16 Valentino Tosatti, Northwestern
Title: Metric limits of hyperkahler manifolds
Abstract: I will discuss a proof of a conjecture of Kontsevich-Soibelman and Gross-Wilson about the behavior of unit-diameter Ricci-flat Kahler metrics on hyperkahler manifolds (fibered by holomorphic Lagrangian tori) near a large complex structure limit. The collapsed Gromov-Hausdorff limit is a special Kahler metric on a half-dimensional complex projective space, away from a singular set of Hausdorff codimension at least 2. The resulting picture is also compatible with the Strominger-Yau-Zaslow mirror symmetry. This is joint work with Yuguang Zhang.
12-14-16 2015-2016
Workshop on Dynamics, Randomness, and Control in Molecular and Cellular Networks
On November 12-14, 2019 the CMSA will be hosting a workshop on Dynamics, Randomness, and Control in Molecular and Cellular Networks. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Biological cells are the fundamental units of life, and predictive modeling of cellular dynamics is essential for understanding a myriad of biological processes and functions. Rapid advances in technologies have made it possible for biologists to measure many variables and outputs from complex molecular and cellular networks with various inputs and environmental conditions. However, such advances are far ahead of the development of mathematical theory, models and methods needed to secure a deep understanding of how high-level robust behaviors emerge from the interactions in complex structures, especially in dynamic and stochastic environments. This workshop will bring together mathematicians and biological scientists involved in developing mathematical theories and methods for understanding, predicting and controlling dynamic behavior of molecular and cellular networks. Particular emphasis will be placed on efforts directed towards discovering underlying biological principles that govern function, adaptation and evolution, and on the development of associated mathematical theories.
Organizers: Jeremy Gunawardena (Harvard) and Ruth Williams (University of California, San Diego)
A limited amount of funding is available to help in defraying the travel costs of early career researchers, women, and underrepresented minorities, participating the workshop. Early career researchers are researchers who received their Ph.D. in 2014 or later, or who are Ph.D. students expecting to complete their Ph.D. by the end of 2020.
To apply, please send a CV, a statement of why you wish to attend, and, if you are a grad student, a letter of support from your advisor to Sarah LaBauve at slabauve@math.harvard.edu
All applications received by 5pm, EDT, October 28, 2019 will receive full consideration.
Speakers:
- David Anderson, University of Wisconsin | Slides
- James Collins, MIT
- Domitilla Del Vecchio, MIT | Slides
- Olga K. Dudko, UC San Diego
- Massimiliano Esposito, University of Luxembourg | Slides
- John Fricks, Arizona State University | Slides
- Heather Harrington, University of Oxford
- Joe Howard, Yale University
- Krešimir Josić, University of Houston
- Samuel Kou, Harvard University
- Tom Kurtz, University of Wisconsin | Slides
- Andrew Murray, Harvard University
- Antonis Papachristodoulou, University of Oxford
- Johan Paulsson, Harvard University
- Lea Popovic, Concordia University
- Sharad Ramanathan, Harvard University
- Eduardo Sontag, Northeastern University
- Jörg Stelling, ETH Zurich | Slides
- Pieter Rein ten Wolde, AMOLF | Slides
Videos from the workshop can be found in the Youtube playlist.
Learning from health data in the million genome era
On November 1, 2019 the CMSA will be hosting a conference organized by Seven Bridges Genomics. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Projects currently underway around the world are collecting detailed health and genomic data from millions of volunteers. In parallel, numerous healthcare systems have announced commitments to integrate genomic data into the standard of care for select patients. These data have the potential to reveal transformative insights into health and disease. However, to realize this promise, novel approaches are required across the full life cycle of data analysis. This symposium will include discussion of advanced statistical and algorithmic approaches to draw insights from petabyte scale genomic and health data; success stories to date; and a view towards the future of clinical integration of genomics in the learning health system.
Speakers:
- Heidi Rehm, Ph.D.
Chief Genomics Officer, MGH; Professor of Pathology, MGH, BWH & Harvard Medical School; Medical Director, Broad Institute Clinical Research Sequencing Platform. - Saiju Pyarajan, Ph.D.
Director, Centre for Data and Computational Sciences,VABHS, and Department of Medicine, BWH and HMS - Tianxi Cai, Sci.D
John Rock Professor of Population and Translational Data Sciences, Department of Biostatistics, Harvard School of Public Health - Susan Redline, M.D., M.P.H
Farrell Professor of Sleep MedicineHarvard Medical School, Brigham and Women’s Hospital and Beth Israel Deaconess Medical Center - Avinash Sahu, Ph.D.
Postdoctoral Research Fellow, Dana Farber Cancer Institute, Harvard School of Public Health - Peter J. Park, Ph.D.
Professor of Biomedical Informatics, Department of Biomedical Informatics, Harvard Medical School - David Roberson
Community Engagement Manager, Seven Bridges
Registration & Schedule
Rank-Based Independence Testing in Near Linear Time
Speaker: Chaim Even-Zohar (Alan Turing Institute, London)
Title: Rank-Based Independence Testing in Near Linear Time
Abstract: In 1948 Hoeffding proposed a nonparametric test that detects dependence between two continuous random variables (X,Y), based on the ranking of n paired samples (Xi,Yi). The computation of this commonly-used test statistic requires O(n log n) time. Hoeffding’s test is consistent against any dependent probability density f(x,y), but can be fooled by other bivariate distributions with continuous margins. Variants of this test with stronger consistency have been considered in works by Blum, Kiefer, and Rosenblatt, Yanagimoto, and Bergsma and Dassios, and others. The so far best known algorithms to compute them have required quadratic time.
We present an algorithm that computes these improved tests in time O(n log n). It is based on a new combinatorial approach for counting pattern occurrences in a given permutation, which we call corner tree formulas, and will be explained in the talk.Joint work with Calvin Leng.
Symmetry types in QFT and the CRT theorem
Title: Symmetry types in QFT and the CRT theorem
Abstract: I will discuss ideas around symmetry and Wick rotation contained in joint work with Mike Hopkins (https://arxiv.org/abs/1604.06527). This includes general symmetry types for relativistic field theories and their Wick rotation. I will then indicate how the basic CRT theorem works for general symmetry types, focusing on the case of the pin groups. In particular, I expand on a subtlety first flagged by Greaves-Thomas.
Spacetime and Quantum Mechanics Master Class Workshop
As part of the program on Spacetime and Quantum Mechanics, Total Positivity and Motives, the CMSA will host a “Master Class Workshop” on October 28-30, 2019. Each day of the workshop will feature an intensive full day of pedagogical lectures, with the aim of bringing actively interested but non-expert physicists and mathematicians up to speed on the featured topics.
Everyone is welcome to attend the lectures.
The master class workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Photos of the event
Organizers:
- Nima Arkani-Hamed (IAS)
- Lauren Williams (Harvard) | Slides 1 | Slides 2 | Slides 3
- Alex Postnikov (MIT)
- Thomas Lam (Michigan)
02-03-2017 CMSA Members’ Seminar
Hansol Hong, Harvard
Title: Homological Mirror Functors
Abstract: I will first give a brief introduction to mirror symmetry, which intertwines symplectic geometry and complex geometry of a pair of Kahler manifolds, and explain mirror construction using formal deformation of a Lagrangian submanifold. We will see that counting of holomorphic discs bounding Lagrangian naturally gives rise to a mirror space (Landau-Ginzburg model) and a functor from Fukaya category to its mirror matrix factorization category. I will mainly focus on one specific example to give a concrete description of the construction.
Applications of instantons, sphalerons and instanton-dyons in QCD
Title: Applications of instantons, sphalerons and instanton-dyons in QCD
Abstract: I start with a general map of gauge topology, including monopoles, instantons and instanton-dyons. Then comes reminder of the “topological landscape”, the minimal energy gauge field configurations, as a function of Chern-Simons number Ncs and r.m.s. size. It includes “valleys” at integer Ncs separated by mountain ridges. The meaning of instantons, instanton-antiinstanton “streamlines” or thimbles, and sphalerons are reminded, together with some proposal to produce sphalerons at LHC and RHIC.
Applications of instanton ensembles, as a model of QCD vacuum, are mostly related to their fermionic zero modes and t’Hooft effective Lagrangian, which explains explicit and spontaneous breaking of chiral symmetries. Recent applications are related with hadronic wave functions, at rest and in the light front (LFWFs). Two application would be spin-dependent forces and the so called “flavor asymmetry of antiquark sea” of the nucleons. At temperatures comparable to deconfinement transition, instantons get split into constituents called instanton-dyons. Studies of their ensemble explains both deconfinement and chiral transitions, in ordinary and deformed QCD.
Oscillations in the thermal conductivity of a spin liquid*
Title: Oscillations in the thermal conductivity of a spin liquid*
Abstract: The layered honeycomb magnet alpha-RuCl3 orders below 7 K in a zigzag phase in zero field. An in-plane magnetic field H||a suppresses the zigzag order at 7 Tesla, leaving a spin-disordered phase widely believed to be a quantum spin liquid (QSL) that extends to ~12 T. We have observed oscillations in the longitudinal thermal conductivity Kxx vs. H from 0.4 to 4 K. The oscillations are periodic in 1/H (with a break-in-slope at 7 T). The amplitude function is maximal in the QSL phase (7 –11.5 T). I will describe a benchmark for crystalline disorder, the reproducibility and intrinsic nature of the oscillations, and discuss implications for the QSL state. I will also show detailed data on the thermal Hall conductivity Kxy measured from 0.4 K to 10 K and comment on recent half-quantization results.
*Czajka et al., Nature Physics 17, 915 (2021).
Collaborators: Czajka, Gao, Hirschberger, Lampen Kelley, Banerjee, Yan, Mandrus and Nagler.
Line defects in CFTs: Renormalization group flows and semiclassical limits
Title: Line defects in CFTs: Renormalization group flows and semiclassical limits
Abstract: I will discuss line defects in d-dimensional Conformal Field Theories (CFTs). In the first part of the talk, I will argue that the ambient CFT places nontrivial constraints on Renormalization Group (RG) flows on such line defects. I will show that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. In the second part of the talk, I will present some applications. In particular, I will discuss impurities with large isospin S for some O(3) symmetric theories in the epsilon expansion. For sufficiently large S diagrammatic perturbation theory breaks down, and these are studied in a semiclassical expansion at fixed epsilon S.
Quantum Information Workshop
Please note, this workshop has been postponed to a later date. Details will be posted to this page when they are available.
The CMSA will host a workshop on Quantum Information. This workshop will be held virtually using Zoom.
The workshop on Quantum information is organized by Mikhail Lukin, Horng-Tzer Yau, and Norman Yao.
More information to follow.
A tour of categorical symmetry
Title: A tour of categorical symmetry
Abstract: I will discuss some perspectives on symmetry coming from the study of topological defects in quantum field theory. I will argue that we should take topological defects themselves to define the symmetries of QFT. This gives us a view of the “category of QFTs”. I will describe some examples of these “categorical symmetries”, their applications, and some open problems.
Integrability and chaos of 1+1d chiral edge states
Speaker: Biao Lian (Princeton)
Title: Integrability and chaos of 1+1d chiral edge states
Abstract: I will talk about the integrability and chaos of 1+1d interacting chiral edge states, which may arise on the edge of 2+1d topological phases. We show that integrable chiral Luttinger liquid is not always a good low energy description of the edge states, and marginal interactions can significantly affect their spectrum and integrability. We first study N identical chiral Majorana fermion modes with random 4-fermion interactions, where we show that the system undergoes a transition from integrable to quantum chaotic as N increases. The large N limit defines a chiral SYK model where the Lyapunov exponent in the out-of-time-ordered correlation can be solved analytically. I will also present a chiral SY model consisting of N interacting SU(M)_1 WZW models, which host anyons and exhibits similar quantum chaos for Abelian anyons. Lastly, I will talk about the analytical and numerical study of the 4/3 FQH edge theory, which shows unusual behavior in its integrability.
Anomaly resolution via decomposition
Speaker: Eric Sharpe (Virginia Tech)
Title: Anomaly resolution via decomposition
Abstract: In this talk we will discuss a method of anomaly resolution due to Wang-Wen-Witten in the special case of (1+1) dimensional theories. Briefly, for our purposes, Wang-Wen-Witten argued that an ill-defined anomalous orbifold [X/G] could be resolved by extending G to a larger group and adding suitable phases. We analyze this process from the perspective of decomposition, a property of (1+1)-dimensional theories with “one-form symmetries” first described in 2006. Examples of such theories include orbifolds with trivially-acting subgroups, of which the extensions of [X/G] are examples. After a review of decomposition, we will see that decomposition implies that in (1+1) dimensions, the Wang-Wen-Witten procedure results in orbifolds that are equivalent to disjoint unions of orbifolds of X by explicitly nonanomalous subgroups of G.
General Relativity Seminar, Wednesdays
The Seminar on General Relativity will take place every Wednesday from 12pm – 1pm in CMSA Building, 20 Garden Street, G10.
The list of speakers is below and will be updated as details are confirmed.
Date Name Title 04-06-2016 Mihalis Dafermos (Princeton) The black hole stability problem: the inside story 04-13-2016 Felix Finster, University of Regensburg Linear stability of Kerr black holes 04-20-2016 Paul Chesler, Harvard Physics Numerical relativity in asymptotically anti-de Sitter spacetime 04-27-2016 Andy Strominger (Harvard Physics) & Mihalis Dafermos (Princeton University) The Scattering Problem in General Relativity 05-04-2016 Robert Penna, MIT BMS invariance and the membrane paradigm 05-11-2016 Piotr T. Chruściel, University of Vienna Gluing things in general relativity 05-18-2016 Achilleas Porfyriadis, Harvard Physics Gravitational waves from the Kerr/CFT correspondence 05-25-2016 Scott Hughes, MIT The gravitational-wave event GW150914: What we learned, and how we learned it CMSA Math-Science Literature Lecture: Immersions of manifolds and homotopy theory
Ralph Cohen (Stanford University)
Title: Immersions of manifolds and homotopy theory
Abstract: The interface between the study of the topology of differentiable manifolds and algebraic topology has been one of the richest areas of work in topology since the 1950’s. In this talk I will focus on one aspect of that interface: the problem of studying embeddings and immersions of manifolds using homotopy theoretic techniques. I will discuss the history of this problem, going back to the pioneering work of Whitney, Thom, Pontrjagin, Wu, Smale, Hirsch, and others. I will discuss the historical applications of this homotopy theoretic perspective, going back to Smale’s eversion of the 2-sphere in 3-space. I will then focus on the problems of finding the smallest dimension Euclidean space into which every n-manifold embeds or immerses. The embedding question is still very much unsolved, and the immersion question was solved in the 1980’s. I will discuss the homotopy theoretic techniques involved in the solution of this problem, and contributions in the 60’s, 70’s and 80’s of Massey, Brown, Peterson, and myself. I will also discuss questions regarding the best embedding and immersion dimensions of specific manifolds, such has projective spaces. Finally, I will end by discussing more modern approaches to studying spaces of embeddings due to Goodwillie, Weiss, and others. This talk will be geared toward a general mathematical audience.
Talk chair: Michael Hopkins
The Inside View: Raymarching and the Thurston Geometries
On Wednesday, December 16 at 12:00 p.m. EST, WAM and CMSA will host a holiday seminar featuring Sabetta Matsumoto, Georgia Institute of Technology who will present The Inside View: Raymarching and the Thurston Geometries.
The properties of euclidean space seem natural and obvious to us, to the point that it took mathematicians over two thousand years to see an alternative to Euclid’s parallel postulate. The eventual discovery of hyperbolic geometry in the 19th century shook our assumptions, revealing just how strongly our native experience of the world blinded us from consistent alternatives, even in a field that many see as purely theoretical. Non-euclidean spaces are still seen as unintuitive and exotic, but with direct immersive experiences we can get a better intuitive feel for them. The latest wave of virtual reality hardware, in particular the HTC Vive, tracks both the orientation and the position of the headset within a room-sized volume, allowing for such an experience. We create realtime rendering to explore the three-dimensional geometries of the Thurston/Perelman geometrization theorem. In this talk, we use the “inside view” of each manifold to try to understand its geometry and what life might be like on the inside. Joint work with Rémi Coulon, Henry Segerman and Steve Trettel.
Register to access this event here
Cosmic Road to New Physics
The CMSA will host a 3-day workshop on cosmological signatures of fundamental physics. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
The next decade will see a wealth of new cosmological data, which can lead to new insights for fundamental physics. This upcoming data will span the entire history of the cosmos, from the era prior to big-bang nucleosynthesis to the inner Galactic structure today, including the intervening eras of recombination and cosmic dawn. Often, beyond-standard-model (BSM) physics will leave imprints in more than one of these eras. Thus, it is timely to gather experts in BSM physics across the entire cosmic history to exchange ideas and develop joint and powerful probes of new physics. For this program, it will be crucial to have an overlap of particle physicists, astrophysicists and cosmologists. There are a number of tools and techniques being actively developed across these disciplines. The workshop aims to provide a platform for efficient exchange of these new ideas.
The first day we will discuss sub-Galactic probes, including Gaia data and gravitational waves. The second day we will cover cosmological probes, such as the cosmic microwave background and the 21-cm line. The third day we will discuss early Universe probes, such as inflation and phase transitions. Every day the meeting will begin with a pedagogical blackboard talk plus an overview talk, followed by about 4 talks on more specific topics.
Organizers:
Scientific Advisory:
- Lisa Randall
- Matt Reece
- Shing-Tung Yau
Speakers:
- Yacine Ali-Haimoud, NYU
- Mustafa Amin, Rice University
- Xingang Chen, Harvard University
- Francis-Yan Cyr-Racine, University of New Mexico
- Francesca Chadha-Day, University of Cambridge
- Jiji Fan, Brown University
- Daniel Grin, Institute for Advanced Study
- Anson Hook, University of Maryland
- Junwu Huang, Perimeter Institute
- Hongwan Liu, NYU
- Gustavo Marques-Tavares, University of Maryland
- Guilherme Pimentel, University of Amsterdam
- Tracy Slatyer, MIT
- Lian-Tao Wang, University of Chicago
Compactification for cluster varieties without frozen variables of finite type
CMSA, 20 Garden Street, Cambridge, MA 02138 USASpeaker: Man-Wai Cheung
Title: Compactification for cluster varieties without frozen variables of finite type
Abstract: Cluster varieties are blow up of toric varieties. They come in pairs $(A,X)$, with $A$ and $X$ built from dual tori. Compactifications of $A$, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties while the compactifications of X, studied by Fock and Goncharov, generalize the fan construction. The conjecture is that the $A$ and the $X$ cluster varieties are mirrors to each other. Together with Tim Magee, we have shown that there exists a positive polytope for the type $A$ cluster varieties which give us a hint to the Batyrev–Borisov construction.
Special Lecture Series on Donaldson-Thomas and Gromov-Witten Theories
From March 8 to April 19, the Center of Mathematical Sciences and Applications will be hosting a special lecture series on Donaldson-Thomas and Gromov-Witten Theories. Artan Sheshmani (QGM Aarhus and CMSA Harvard) will give eight talks on the topic on Wednesdays and Fridays from 9:00-10:30 am, which will be recorded and promptly available on CMSA’s Youtube Channel.
2019 Ding Shum Lecture
On October 22, 2019, the CMSA will be hosting our third annual Ding Shum lecture. This year’s lecture will be a talk on “Election Security” by Ronald L. Rivest (MIT). The lecture will take place from 4:30-5:30pm in Science Center, Hall A.
Ronald L. Rivest is an Institute Professor at the Massachusetts Institute of Technology. He is a member of the Electrical Engineering and Computer Science Department and the Computer Science and Artificial Intelligence Laboratory (CSAIL) and a founder of the Cryptography and Information Security research group within CSAIL. His research has been in the areas of algorithms, machine learning, cryptography, and election security, for which he has received multiple awards, including: the ACM Turing Award (with Adleman and Shamir), the BBVA Frontiers of Knowledge Award, National Inventor’s Hall of Fame membership, and the Marconi Prize.
Prof. Rivest is also well-known as a co-author of the textbook “Introduction to Algorithms” (with Cormen, Leiserson, and Stein), and as a co-inventor of the RSA public-key cryptosystem (with Adleman and Shamir). He is a co-founder of RSA and of Verisign.He has served on the Technical Guidelines Development Committee (advisory to the Election Assistance Commission), in charge of the Security subcommittee. He is a member of the CalTech/MIT Voting Technology Project, on the Board of Verified Voting, and an advisor to the Electronic Privacy Information Center. Additionally, he has served on the Technical Guidelines Development Committee (advisory to the Election Assistance Commission), as a member of the CalTech/MIT Voting Technology Project, and as an advisor to the Electronic Privacy Information Center.
Last year featured Eric Maskin, who spoke on “How to Improve Presidential Elections: the Mathematics of Voting.” The first Ding Shum lecture took place on October 10, 2017, featuring Leslie Valiant on “Learning as a Theory of Everything.”
This event is made possible by the generous funding of Ding Lei and Harry Shum.
Noncommutative Analysis, Computational Complexity, and Quantum Information
On October 16-18, 2019 the CMSA will be hosting a workshop on Noncommutative Analysis, Computational Complexity, and Quantum Information.
This workshop will focus on linking three different rapidly developing areas: noncommutative real algebraic geometry (RAG), theory of computation and quantum information theory. This mix of overlapping but independently developing topics should lead to a stimulating flow of tools and important problems into several disciplines. Given the different communities there will be an emphasis on tutorials and making the lectures broadly understandable.
The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by Boaz Barak, Bill Helton, Pablo Parrilo, Tselil Schramm.
Please register here
Speakers:
- Jason Altschuler, MIT | Video
- Boaz Barak, Harvard | Video
- Ankit Garg, Microsoft Research | Slides | Video
- David Gosset, University of Waterloo | Video
- Aram Harrow, MIT | Video
- Igor Klep, University of Ljubljana
- Salma Kuhlmann, Universität Konstanz | Video
- Scott McCullough, University of Florida | Slides
- Ion Nechita, Laboratoire de Physique Théorique | Slides | Video
- Rafael Oliveira, University of Toronto | Video
- Vern Paulsen, University of Waterloo | Video
- Suvrit Sra, MIT | Video
- Victor Vinnikov, Ben Gurion University | Video
- Jurij Volčič, Texas A&M University | Slides | Video
- Adam Bene Watts, MIT
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Conference on Differential Geometry, Calabi-Yau theory and General Relativity: A conference in honor of the 70th Birthday of Shing-Tung Yau
1 Oxford Street, Cambridge MA 02138On May 2-5, 2019 the Harvard Mathematics Department hosted a Conference on Differential Geometry, Calabi-Yau Theory and General Relativity: A conference in honor of the 70th Birthday of Shing-Tung Yau. The conference was held in the Science Center, Lecture Hall C.
Organizers:
- Horng-Tzer Yau (Harvard)
- Wilfried Schmid (Harvard)
- Clifford Taubes (Harvard)
- Cumrun Vafa (Harvard)
Speakers:
- Lydia Bieri, University of Michigan
- Tristan Collins, MIT
- Simon Donaldson, Imperial College
- Fan Chung Graham, UC San Diego
- Nigel Hitchin, Oxford University
- Jun Li, Stanford University
- Kefeng Liu, UCLA
- Chiu-Chu Melissa Liu, Columbia University
- Alina Marian, Northeastern University
- Xenia de la Ossa, Oxford University
- Duong H. Phong, Columbia University
- Richard Schoen, UC Irvine
- Andrew Strominger, Harvard University
- Nike Sun, MIT
- Clifford Taubes, Harvard University
- Chuu-Lian Terng, UC Irvine
- Valentino Tosatti, Northwestern University
- Karen Uhlenbeck, University of Texas
- Cumrun Vafa, Harvard University
- Mu Tao Wang, Columbia University
- Edward Witten, IAS
- Stephen Yau, Tsinghua University, P.R. China
CMSA Math-Science Literature Lecture: A personal story of the 4D Poincare conjecture
Michael Freedman (Microsoft – Station Q)
Title: A personal story of the 4D Poincare conjecture
Abstract: The proof of PC4 involved the convergence of several historical streams. To get started: high dimensional manifold topology (Smale), a new idea on how to study 4-manifolds (Casson), wild “Texas” topology (Bing). Once inside the proof: there are three submodules: Casson towers come to life (in the sense of reproduction), a very intricate explicit shrinking argument (provided by Edwards), and the “blind fold” shrinking argument (which in retrospect is in the linage of Brown’s proof of the Schoenflies theorem). Beyond those mentioned: Kirby, Cannon, Ancel, Quinn, and Starbird helped me understand my proof. I will discuss the main points and how they fit together.
Talk Chair: Peter Kronheimer
CMSA Math-Science Literature Lecture: From string theory and Moonshine to vertex algebras
Bong Lian (Brandeis)
Title: From string theory and Moonshine to vertex algebras
Abstract: This is a brief survey of the early historical development of vertex algebras, beginning in the seventies from Physics and Representation Theory. We shall also discuss some of the ideas that led to various early formulations of the theory’s foundation, and their relationships, as well as some of the subsequent and recent developments. The lecture is aimed at a general audience.
CMSA Math-Science Literature Lecture: Four-dimensional topology
Ciprian Manolescu (Stanford)
Title: Four-dimensional topology
Abstract: I will outline the history of four-dimensional topology. Some major events were the work of Donaldson and Freedman from 1982, and the introduction of the Seiberg-Witten equations in 1994. I will discuss these, and then move on to what has been done in the last 20 years, when the focus shifted to four-manifolds with boundary and cobordisms. Floer homology has led to numerous applications, and recently there have also been a few novel results (and proofs of old results) using Khovanov homology. The talk will be accessible to a general mathematical audience.
Conference on Algebraic Geometry, Representation theory and Mathematical Physics
From April 29 to May 1, 2019 the CMSA will be hosting a Conference on Algebraic Geometry, Representation theory and Mathematical Physics. This workshop is organized by Bong Lian (Brandeis) and Artan Sheshmani (CMSA) . The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Videos
Speakers:
- Dan Abramovich, Brown
- Roman Bezrukavnikov, MIT
- Fedor Bogomolov, NYU
- Qile Chen, Boston College
- Dawei Chen, Boston College
- Alexander Efimov, Moscow
- Pavel Etingof, MIT
- Maksym Fedorchuk, Boston College
- Dennis Gaitsgory, Harvard
- Amin Gholampour, Maryland
- Brendan Hassett, Brown
- Ludmil Katzarkov, Miami & Moscow
- Si Li, Tsinghua
- Andrei Negut, MIT
- Yuri Tschinkel, NYU
- Wei Zhang, MIT
Monday, April 29
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 10:00am Wei Zhang, MIT Title: The arithmetic fundamental lemma for diagonal cycles Abstract: I’ll recall the Gross–Zagier theorem and a high dimensional generalization, the arithmetic Gan-Gross-Prasad conjecture, which relates the height pairing of arithmetic diagonal cycles on certain shimura varieties to the first order derivative of certain L-functions. The arithmetic fundamental lemma conjecture arises from the relative trace formula approach to this conjecture. I will recall the statement of the arithmetic fundamental lemma and outline a proof.
10:00 – 10:30am Break 10:30 – 11:30am Yuri Tschinkel, NYU Title: Equivariant birational geometry and modular symbols Abstract: We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups (joint with M. Kontsevich and V. Pestun).
11:30 – 1:30pm Lunch 1:30 – 2:30pm Alexander Efimov, Moscow Title: Torsionness for regulators of canonical extensions Abstract: I will sketch a generalization of the results of Iyer and Simpson arXiv:0707.0372 to the general case of a normal-crossings divisor at infinity.
2:30 – 3:00pm Break 3:00 – 4:00pm Amin Gholampour, Maryland Title: Euler Characteristics of punctual quot schemes on threefolds Abstract: Let F be a homological dimension 1 torsion free sheaf on a nonsingular quasi-projective threefold. The first cohomology of the derived dual of F is a 1-dimension sheaf G supported on the singular locus of F. We prove a wall-crossing formula relating the generating series of the Euler characteristics of Quot(F, n) and Quot(G,n), where Quot(-,n) denotes the quot scheme of length n quotients. We will use this relation in studying the Euler characteristics of the moduli spaces of stable torsion free sheaves on nonsingular projective threefolds. This is a joint work with Martijn Kool.
4:00 – 4:30pm Break 4:30 – 5:30pm Maksym Fedorchuck, BC Title: Stability of one-parameter families of weighted hypersurfaces Abstract: We define a notion of stability for fibrations over a curve with generic fibers being weighted hypersurfaces (in some weighted projective space) generalizing Kollár’s stability for families of hypersurfaces in a projective space. The stability depends on a choice of an effective line bundle on the parameter space of weighted hypersurfaces and different choices pick out different birational model of the total space of the fibration. I will describe enumerative geometry that goes into understanding these stability conditions, and, if time permits, examples where this machinery can be used to produce birational models with good properties. Joint work with Hamid Ahmadinezhad and Igor Krylov.
Tuesday, April 30
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 10:00am Brendan Hassett, Brown Title: Rationality for geometrically rational threefolds Abstract: We consider rationality questions for varieties over non-closed fields that become rational over an algebraic closure, like smooth complete intersections of two quadrics. (joint with Tschinkel)
10:00 – 10:30am Break 10:30 – 11:30am Dennis Gaitsgory, Harvard Title: The Fundamental Local Equivalence in quantum geometric Langlands Abstract: The Fundamental Local Equivalence is statement that relates the q-twisted Whittaker category of the affine Grassmannian for the group G and the category of modules over the Langlands dual “big” quantum group. The non-triviaiity of the statement lies is the fact that the relationship between the group and its dual is combinatorial, so to prove the FLE one needs to express both sides in combinatorial terms. In the talk we will indicate the proof of a related statement for the “small” quantum group. The combinatorial link is provided by the category of factorization modules over a certain factorization algebra, which in itself is a geometric device that concisely encodes the root data.
11:30 – 1:00pm Lunch 1:00- 2:00pm Andrei Negut, MIT Title: AGT relations in geometric representation theory Abstract: I will survey a program that seeks to translate the Alday-Gaiotto-Tachikawa correspondence (between gauge theory on R^4 and conformal field theory) into the language of algebraic geometry. The objects of study become moduli spaces of sheaves on surfaces, and the goal is to connect them with the W-algebra of type gl_n.
2:00 – 2:15pm Break 2:15 – 3:15pm Dan Abramovich, Brown Title: Resolution in characteristic 0 using weighted blowing up Abstract: Given a variety $X$, one wants to blow up the worst singular locus, show that it gets better, and iterate until the singularities are resolved.
Examples such as the whitney umbrella show that this iterative process cannot be done by blowing up smooth loci – it goes into a loop.
We show that there is a functorial way to resolve varieties using \emph{weighted} blowings up, in the stack-theoretic sense. To an embedded variety $X \subset Y$ one functorially assigns an invariant $(a_1,\ldots,a_k)$, and a center locally of the form $(x_1^{a_1} , \ldots , x_k^{a_k})$, whose stack-theoretic weighted blowing up has strictly smaller invariant under the lexicographic order.
This is joint work with Michael Tëmkin (Jerusalem) and Jaroslaw Wlodarczyk (Purdue), a side product of our work on functorial semistable reduction. A similar result was discovered by G. Marzo and M. McQuillan.
3:15 – 3:30pm Break 3:30 – 4:30pm Fedor Bogomolov, NYU Title: On the base of a Lagrangian fibration for a compact hyperkahler manifold. Abstract: In my talk I will discuss our proof with N. Kurnosov that the base of such fibration for complex projective manifold hyperkahler manifold of dimension $4$ is always a projective plane $P^2$. In fact we show that the base of such fibration can not have a singular point of type $E_8$. It was by the theorem of Matsushita and others that only quotient singularities can occur and if the base is smooth then the it is isomorphic to $P^2$. The absence of other singularities apart from $E_8$ has been already known and we show that $E-8$ can not occur either. Our method can be applied to other types of singularities for the study of Lagrangian fibrations in higher dimensions More recently similar result was obtained by Huybrechts and Xu.
4:30 – 4:45pm Break 4:45 – 5:45pm Dawei Chen, BC Title: Volumes and intersection theory on moduli spaces of Abelian differentials Abstract: Computing volumes of moduli spaces has significance in many fields. For instance, Witten’s conjecture regarding intersection numbers on moduli spaces of Riemann surfaces has a fascinating connection to the Weil-Petersson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. In this talk I will introduce an analogue of Witten’s intersection numbers on moduli spaces of Abelian differentials to compute the Masur-Veech volumes induced by the flat metric associated with Abelian differentials. This is joint work with Moeller, Sauvaget, and Zagier (arXiv:1901.01785).
Wednesday, May 1
Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 10:00am Pavel Etingof, MIT Title: Short star-products for filtered quantizations Abstract: PDF
This is joint work with Eric Rains and Douglas Stryker.
10:00 – 10:30am Break 10:30 – 11:30am Roman Bezrukavnikov, MIT Title: Stability conditions and representation theory Abstract: I will recall the concept of real variation of stabilities (introduced in my work with Anno and Mirkovic)
and its relation to modular Lie algebra representations. I will also address a potential generalization of that picture
to modular representations of affine Lie algebras related to the classical limit of geometric Langlands duality and its local counterpart.11:30 – 11:45am Break 11:45 – 12:45pm Qile Chen, BC Title: Counting curves in critical locus via logarithmic compactification Abstract: An R-map consists of a pre-stable map to possibly non-GIT quotient together with sections of certain spin bundles. The moduli of R-maps are in general non-compact. When the target of R-maps is equipped with a super-potential W with compact critical locus, using Kiem-Li cosection localization it has been proved by many authors in various settings that the virtual cycle of R-maps can be represented by the cosection localized virtual cycle which is supported on the proper locus consisting of R-maps in the critical locus of W. Though the moduli of R-maps is equipped with a natural torus action by scaling of the spin bundles, the non-compactness of the R-maps moduli makes such powerful torus action useless.
In this talk, I will introduce a logarithmic compactification of the moduli of R-maps using certain modifications of stable logarithmic maps. The logarithmic moduli space carries a canonical virtual cycle from the logarithmic deformation theory. In the presence of a super-potential with compact critical locus, it further carries a reduced virtual cycle. We prove that (1) the reduced virtual cycle of the compactification can be represented by the cosection localized virtual cycle; and (2) the difference of the canonical and reduced virtual cycles is another reduced virtual cycle supported along the logarithmic boundary. As an application, one recovers the Gromov-Witten invariants of the critical locus as the invariants of logarithmic R-maps of its ambient space in an explicit form. The latter can be calculated using the spin torus action.
This is a joint work with Felix Janda and Yongbin Ruan.
12:45 – 2:30pm Lunch 2:30 – 3:30pm Si Li, Tsinghua Title: Semi-infinite Hodge structure: from BCOV theory to Seiberg-Witten geometry Abstract: I will explain how the semi-infinite Hodge theory extends Kodaira-Spencer gravity (Bershadsky-Cecotti-Ooguri-Vafa theory of B-twisted closed topological string field theory) into a full solution of Batalin-Vilkovisky master equation. This allows us to formulate quantum B-model via a rigorous BV quantization method and construct integrable hierarchies arising naturally from the background symmetry. In the second part of the talk, I will explain the recent discovery of the connection between K.Saito’s primitive form and 4d N=2 Seiberg-Witten geometry arising from singularity theory.
3:30 – 4:00pm Break 4:00 – 5:00pm Ludmil Katzarkov, Moscow Title: PDE’s non commutative motives and HMS. Abstract: In this talk we will discuss the theory of central manifolds and the new structures in geometry it produces. Application to Bir. Geometry will be discussed.
Workshop on Mirror Symmetry and Stability
This three-day workshop will take place at Harvard University on March 18-20, 2019 in Science Center room 507. The main topic will be stability conditions in homological mirror symmetry. This workshop is funded by the Simons Collaboration in Homological Mirror Symmetry.
Organizers: Denis Auroux, Yu-Wei Fan, Hansol Hong, Siu-Cheong Lau, Bong Lian, Shing-Tung Yau, Jingyu Zhao
Speakers:
Dylan Allegretti (Sheffield)
Tristan Collins (MIT)
Naoki Koseki (Tokyo)
Chunyi Li (Warwick)
Jason Lo (CSU Northridge)
Emanuele Macrì (NEU & IHES)
Genki Ouchi (Riken iTHEMS)
Pranav Pandit (ICTS)
Laura Pertusi (Edinburgh)
Jacopo Stoppa (SISSA)
Alex Takeda (UC Berkeley)
Xiaolei Zhao (UC Santa Barbara)More details will be added later.
Visit the event page for more information.
Stochastic PDE as scaling limits of interacting particle systems
Abstract: Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models. In this talk, I will illustrate how this challenge can be overcome by elucidating the probabilistic connections between models of different levels of detail. These connections explain how stochastic partial differential equations (SPDE) arise naturally from particle models.
I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics.
The Festina Lente Bound
Abstract: I will explain what the Festina Lente bound means and where it comes from. Then I discuss its possible implications for phenomenology, both top-down and bottom-up.
CMSA Math-Science Literature Lecture: Deep Networks from First Principles
Yi Ma (University of California, Berkeley)
Title: Deep Networks from First Principles
Abstract: In this talk, we offer an entirely “white box’’ interpretation of deep (convolution) networks from the perspective of data compression (and group invariance). In particular, we show how modern deep layered architectures, linear (convolution) operators and nonlinear activations, and even all parameters can be derived from the principle of maximizing rate reduction (with group invariance). All layers, operators, and parameters of the network are explicitly constructed via forward propagation, instead of learned via back propagation. All components of so-obtained network, called ReduNet, have precise optimization, geometric, and statistical interpretation. There are also several nice surprises from this principled approach: it reveals a fundamental tradeoff between invariance and sparsity for class separability; it reveals a fundamental connection between deep networks and Fourier transform for group invariance – the computational advantage in the spectral domain (why spiking neurons?); this approach also clarifies the mathematical role of forward propagation (optimization) and backward propagation (variation). In particular, the so-obtained ReduNet is amenable to fine-tuning via both forward and backward (stochastic) propagation, both for optimizing the same objective. This is joint work with students Yaodong Yu, Ryan Chan, Haozhi Qi of Berkeley, Dr. Chong You now at Google Research, and Professor John Wright of Columbia University.
Talk chair: Harry Shum
On singular Hilbert schemes of points
Abstract: It is well known that the Hilbert schemes of points on smooth surfaces are smooth. In higher dimensions the Hilbert schemes of points are in general singular. In this talk we will present some examples and conjectures on the local structures of the Hilbert scheme of points on $\mathbb{P}^3$. As an application we study a conjecture of Wang-Zhou on the Euler characteristics of the tautological sheaves on Hilbert schemes of points.
A mirror theorem for GLSMs
Abstract: A gauged linear sigma model (GLSM) consists roughly of a complex vector space V, a group G acting on V, a character \theta of G, and a G-invariant function w on V. This data defines a GIT quotient Y = [V //_\theta G] and a function on that quotient. GLSMs arise naturally in a number of contexts, for instance as the mirrors to Fano manifolds and as examples of noncommutative crepant resolutions. GLSMs provide a broad setting in which it is possible to define an enumerative curve counting theory, simultaneously generalizing FJRW theory and the Gromov-Witten theory of hypersurfaces. Despite a significant effort to rigorously define the enumerative invariants of a GLSM, very few computations of these invariants have been carried out. In this talk I will describe a new method for computing generating functions of GLSM invariants. I will explain how these generating functions arise as derivatives of generating functions of Gromov-Witten invariants of Y.
Cytoskeletal Energetics and Energy Metabolism
Abstract: Life is a nonequilibrium phenomenon. Metabolism provides a continuous flux of energy that dictates the form and function of many subcellular structures. These subcellular structures are active materials, composed of molecules which use chemical energy to perform mechanical work and locally violate detailed balance. One of the most dramatic examples of such a self-organizing structure is the spindle, the cytoskeletal based assembly which segregates chromosomes during cell division. Despite its central role, very little is known about the nonequilibrium thermodynamics of active subcellular matter, such as the spindle. In this talk, I will describe ongoing work from my lab aimed at understanding the flows of energy which drive the nonequilibrium behaviors of the cytoskeleton in vitro and in vivo.
Simons Collaboration Workshop, April 5-7, 2018
The CMSA will be hosting a three-day Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory on April 5-7, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Please click here to register for this event. We have space for up to 30 registrants on a first come, first serve basis.
We may be able to provide some financial support for grad students and postdocs interested in this event. If you are interested in funding, please send a letter of support from your mentor to Hansol Hong at hansol84@gmail.com.
Confirmed Speakers:
- Jacob Bourjaily (Niels Bohr Institute)
- Mandy Cheung (Havard University)
- Tristan Collins (Harvard University)
- Yoosik Kim (Boston University)
- Yu-Shen Lin (Harvard University)
- Cheuk-Yu Mak (Cambridge University)
- Yu Pan (MIT)
- Mauricio Romo (Tsinghua University)
- Shu-Heng Shao (IAS)
- Zack Sylvan (Columbia University)
- Dmitry Vaintrob (IAS)
Exploring the Holographic Swampland
Abstract: I describe our work looking at `traditional’ scenarios of moduli stabilisation from a holographic perspective. This reveals some interesting structure that is not apparent from the top-down perspective. For vacua in the extreme regions of moduli space, such as LVS in type IIB or the DGKT flux vacua in type IIA, the dual moduli conformal dimensions reduce to fixed values – in a certain sense, the low-conformal dimension part of the CFT is unique and independent of the large number of flux choices. For the DGKT flux vacua these conformal dimensions are also integer, for reasons we do not understand.
What do bounding chains look like, and why are they related to linking numbers?
Abstract: Gromov-Witten invariants count pseudo-holomorphic curves on a symplectic manifold passing through some fixed points and submanifolds. Similarly, open Gromov-Witten invariants are supposed to count disks with boundary on a Lagrangian, but in most cases such counts are not independent of some choices as we would wish. Motivated by Fukaya’11, J. Solomon and S. Tukachinsky constructed open Gromov-Witten invariants in their 2016 papers from an algebraic perspective of $A_{\infty}$-algebras of differential forms, utilizing the idea of bounding chains in Fukaya-Oh-Ohta-Ono’06. On the other hand, Welschinger defined open invariants on sixfolds in 2012 that count multi-disks weighted by the linking numbers between their boundaries. We present a geometric translation of Solomon-Tukachinsky’s construction. From this geometric perspective, their invariants readily reduce to Welschinger’s.
Convexity of Charged Operators in CFTs and the Weak Gravity Conjecture
Abstract: In this talk I will introduce a particular formulation of the Weak Gravity Conjecture in AdS space in terms of the self-binding energy of a particle. The holographic CFT dual of this formulation corresponds to a certain convex-like structure for operators charged under continuous global symmetries. Motivated by this, we propose a conjecture that this convexity is a general property of all CFTs, not just those with weakly-curved gravitational duals. It is possible to test this in simple CFTs, the conjecture passes all the tests performed so far.
CMSA Math-Science Literature Lecture: My life and times with the sporadic simple groups
Robert Griess (University of Michigan)
Title: My life and times with the sporadic simple groups
Abstract: Five sporadic simple groups were proposed in 19th century and 21 additional ones arose during the period 1965-1975. There were many discussions about the nature of finite simple groups and how sporadic groups are placed in mathematics. While in mathematics grad school at University of Chicago, I became fascinated with the unfolding story of sporadic simple groups. It involved theory, detective work and experiments. During this lecture, I will describe some of the people, important ideas and evolution of thinking about sporadic simple groups. Most should be accessible to a general mathematical audience.
The many phases of a cell
Abstract: I will begin by introducing an emerging paradigm of cellular organization – the dynamic compartmentalization of biochemical pathways and molecules by phase separation into distinct and multi-phase condensates. Motivated by this, I will discuss two largely orthogonal problems, united by the theme of phase separation in multi-component and chemically active fluid mixtures.
1. I will propose a theoretical model based on Random-Matrix Theory, validated by phase-field simulations, to characterizes the rich emergent dynamics, compositions, and steady-state properties that underlie multi-phase coexistence in fluid mixtures with many randomly interacting components.
2. Motivated by puzzles in gene-regulation and nuclear organization, I will propose a role for how liquid-like nuclear condensates can be organized and regulated by the active process of RNA synthesis (transcription) and RNA-protein coacervation. Here, I will describe theory and simulations based on a Landau formalism and recent experimental results from collaborators.
Knot homology and sheaves on the Hilbert scheme of points on the plane.
Abstract: The knot homology (defined by Khovavov, Rozansky) provide us with a refinement of the knot polynomial knot invariant defined by Jones. However, the knot homology are much harder to compute compared to the polynomial invariant of Jones. In my talk I present recent developments that allow us to use tools of algebraic geometry to compute the homology of torus knots and prove long-standing conjecture on the Poincare duality the knot homology. In more details, using physics ideas of Kapustin-Rozansky-Saulina, in the joint work with Rozansky, we provide a mathematical construction that associates to a braid on n strands a complex of sheaves on the Hilbert scheme of n points on the plane. The knot homology of the closure of the braid is a space of sections of this sheaf. The sheaf is also invariant with respect to the natural symmetry of the plane, the symmetry is the geometric counter-part of the mentioned Poincare duality.
CMSA Math-Science Literature Lecture: Rationality questions in algebraic geometry
Joe Harris (Harvard)
Title: Rationality questions in algebraic geometry
Abstract: Over the course of the history of algebraic geometry, rationality questions — motivated by both geometric and arithmetic problems — have often driven the subject forward. The rationality or irrationality of cubic hypersurfaces in particular have led to the development of abelian integrals (dimension one), birational geometry (dimension two) and Hodge theory (dimension 3). But there remained much we didn’t understand about the condition of rationality, such as how it behaves in families. However, there has been recent progress: work of Hassett, Tschinkel, Pirutka and others, working with examples in dimension 4, showed that it is in general neither an open condition nor a closed one, but does behave well with respect to specialization. In this talk I’ll try to give an overview of the history of rationality and the current state of our knowledge.
Combinatorics & Complexity Seminar, Fridays
The seminar on Combinatorics and Complexity will be held every Friday from 1:00-4:00pm in CMSA Building, 20 Garden Street, Room G10.
The list of speakers for the upcoming academic year will be posted below and updated as details are confirmed. Titles and abstracts for the talks will be added as they are received.
Additional information on CMSA’s Combinatorics and Complexity program can be found here.
Date Name Title/Abstract 09-08-17 TBA 09-15-2017 TBA 09-22-17 TBA 09-29-17 TBA 10-06-17 TBA 10-13-2017 TBA 10-20-2017 TBA 10-27-2017 TBA 11-03-2017 TBA 11-10-2017 TBA 11-17-2017 TBA 11-24-2017 TBA 12-01-2017 TBA 12-08-2017 TBA Derived projectivizations of two-term complexes
Abstract: For a given two-term complex of vector bundles on a derived scheme (or stack), there are three natural ways to define its “derived projectivizations”: (i) as the derived base-change of the classical projectivization of Grothendieck; (ii) as the derived moduli parametrizing one-dimensional locally free quotients; (iii) as the GIT quotient of the total space by $\mathbb{G}_m$-action. In this talk, we first show that these three definitions are equivalent. Second, we prove a structural theorem about the derived categories of derived projectivizations and study the corresponding mutation theory. Third, we apply these results to various moduli situations, including the moduli of certain stable pairs on curves and the Hecke correspondences of one-point modification of moduli of stable sheaves on surfaces. If time allowed, we could also discuss the generalizations of these results to the derived Quot schemes of locally free quotients.
D-critical structure(s) on Quot schemes of points of Calabi-Yau 3-folds
Abstract: D-critical schemes and Artin stacks were introduced by Joyce in 2015, and play a central role in Donaldson-Thomas theory. They typically occur as truncations of (-1)-shifted symplectic derived schemes, but the problem of constructing the d-critical structure on a “DT moduli space” without passing through derived geometry is wide open. We discuss this problem, and new results in this direction, when the moduli space is the Hilbert (or Quot) scheme of points on a Calabi-Yau 3-fold. Joint work with Michail Savvas.
Drivers of Morphological Complexity
Abstract: During development, organisms interact with their natural habitats while undergoing morphological changes, yet we know little about how the interplay between developing systems and their environments impacts animal morphogenesis. Cnidaria, a basal animal lineage that includes sea anemones, corals, hydras, and jellyfish, offers unique insight into the development and evolution of morphological complexity. In my talk, I will introduce our research on “ethology of morphogenesis,” a novel concept that links the behavior of organisms to the development of their size and shape at both cellular and biophysical levels, opening new perspectives about the design principle of soft-bodied animals. In addition, I will discuss a fascinating feature of cnidarian biology. For humans, our genetic code determines that we will grow two arms and two legs. The same fate is true for all mammals. Similarly, the number of fins of a fish or legs and wings of an insect is embedded in their genetic code. I will describe how sea anemones defy this rule.
References
Anniek Stokkermans, Aditi Chakrabarti, Ling Wang, Prachiti Moghe, Kaushikaram Subramanian, Petrus Steenbergen, Gregor Mönke, Takashi Hiiragi, Robert Prevedel, L. Mahadevan, and Aissam Ikmi. Ethology of morphogenesis reveals the design principles of cnidarian size and shape development. bioRxiv 2021.08.19.456976Ikmi A, Steenbergen P, Anzo M, McMullen M, Stokkermans M, Ellington L, and Gibson M (2020). Feeding-dependent tentacle development in the sea anemone Nematostella vectensis. Nature communications, Sept 02; 11:4399
He S, Del Viso F, Chen C, Ikmi A, Kroesen A, Gibson MC (2018). An axial Hox code controls tissue segmentation and body patterning in Nematostella vectensis. Science, Vol. 361, Issue 6409, pp. 1377-1380.
Ikmi A, McKinney SA, Delventhal KM, Gibson MC (2014). TALEN and CRISPR/Cas9 mediated genome editing in the early-branching metazoan Nematostella vectensis. Nature communications. Nov 24; 5:5486.The Mirror Clemens-Schmid Sequence
Abstract: I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a “mirror P=W” conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting. This is joint work with Charles F. Doran.
Eppur si muovono: rotations in active matter
Abstract: Living matter relies on the self organization of its components into higher order structures, on the molecular as well as on the cellular, organ or even organism scale. Collective motion due to active transport processes has been shown to be a promising route for attributing fascinating order formation processes on these different length scales. Here I will present recent results on structure formation on actively transported actin filaments on lipid membranes and vesicles, as well as the cell migration induced structure formation in the developmental phase of mammary gland organoids. For both systems spherical structures with persistent collective rotations are observed.
9/28/2021 Combinatorics, Physics and Probability Seminar
Title: The hypersimplex and the m=2 amplituhedron
Abstract: I’ll discuss a curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). Along the way, we prove the sign-flip description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers (inspired by an analogous hypersimplex decomposition).
Second Annual STAR Lab Conference
The second annual STAR Lab conference is running 10/29/-10/30/2015 at the Harvard Business School. This event is co-sponsored by the Center of Mathematical Sciences and Applications.
For more information, please consult the event’s website.
Categorification and applications
Abstract: I will give a survey of the program of categorification for quantum groups, some of its recent development and applications to representation theory.
2016 Big Data Conference & Workshop
1 Oxford Street, Cambridge MA 02138! LOCATION CHANGE: The conference will be in Science Center Hall C on Tuesday, Aug.23, 2016.
The Center of Mathematical Sciences and Applications will be hosting a workshop on Big Data from August 12 – 21, 2016 followed by a two-day conference on Big Data from August 22 – 23, 2016.
Big Data Conference features many speakers from the Harvard Community as well as many scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics. This is the second conference on Big Data the Center will host as part of our annual events. The 2015 conference was a huge success.
The conference will be hosted at Harvard Science Center Hall A (Monday, Aug.22) & Hall C (Tuesday, Aug.23): 1 Oxford Street, Cambridge, MA 02138.
The 2016 Big Data conference is sponsored by the Center of Mathematical Sciences and Applications at Harvard University and the Alfred P. Sloan Foundation.
Conference Speakers:
- Jörn Boehnke, Harvard CMSA
- Joan Bruna, UC Berkeley [Video]
- Tamara Broderick, MIT [Video]
- Justin Chen, MIT [Video]
- Yiling Chen, Harvard University [Video]
- Amir Farbin, UT Arlington [Video]
- Doug Finkbeiner, Harvard University [Video]
- Andrew Gelman, Columbia University [Video]
- Nina Holden, MIT [Video]
- Elchanan Mossel, MIT
- Alex Peysakhovich, Facebook
- Alexander Rakhlin, University of Pennsylvania [Video]
- Neal Wadhwa, MIT [Video]
- Jun Yin, University of Wisconsin
- Harry Zhou, Yale University [Video]
Please click Conference Program for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Please click here for registration.
Conference Schedule:
August 22 – Day 1 8:30am Breakfast 8:55am Opening remarks 9:00am – 9:50am Yiling Chen, “Machine Learning with Strategic Data Sources” [Video] 9:50am – 10:40am Andrew Gelman, “Taking Bayesian Inference Seriously” [Video] 10:40am – 11:10am Break 11:10am – 12:00pm Harrison Zhou, “A General Framework for Bayes Structured Linear Models” [Video] 12:00pm – 1:30pm Lunch 1:30pm – 2:20pm Douglas Finkbeiner, “Mapping the Milky Way in 3D with star colors” [Video] 2:20pm – 3:10pm Nina Holden, “Sparse exchangeable graphs and their limits” [Video] 3:10pm – 3:40pm Break 3:40pm – 4:30pm Alex Peysakhovich, “How social science methods inform personalization on Facebook News Feed” [Video] 4:30pm – 5:20pm Amir Farbin, “Deep Learning in High Energy Physics” [Video] August 23 – Day 2 8:45am Breakfast 9:00am – 9:50am Joan Bruna Estrach, “Addressing Computational and Statistical Gaps with Deep Networks” [Video] 9:50am – 10:40am Justin Chen & Neal Wadhwa, “Smaller Than the Eye Can See: Big Engineering from Tiny Motions in Video” [Video] 10:40am – 11:10am Break 11:10am – 12:00pm Alexander Rakhlin, “How to Predict When Estimation is Hard: Algorithms for Learning on Graphs” [Video] 12:00pm – 1:30pm Lunch 1:30pm – 2:20pm Tamara Broderick, “Fast Quantification of Uncertainty and Robustness with Variational Bayes” [Video] 2:20pm – 3:10pm Elchanan Mossel, “Phylogenetic Reconstruction – a Rigorous Model of Deep Learning” 3:10pm – 3:40pm Break 3:40pm – 4:30pm Jörn Boehnke, “Amazon’s Price and Sales-rank Data: What can one billion prices on 150 thousand products tell us about the economy?” Workshop Participants:
Richard Freeman’s Group:
- Sen Chai, ESSEC
- Brock Mendel, Harvard University
- Raviv Muriciano-Goroff, Stanford University
- Sifan Zhou, CMSA
Scott Kominer’s Group:
- Bradly Stadie, UC Berkeley
- Neal Wadhwa, MIT [Video]
- Justin Chen
Christopher Rogan’s Group:
- Amir Farbin, UT Arlington [Video]
- Paul Jackson, University of Adelaide
For more information about the workshops, please reach out directly to the individual group leaders.
* This event is sponsored by CMSA Harvard University and the Alfred P. Sloan Foundation.
2/16/2021 Computer Science for Mathematicians
Speaker: Michael P. Kim (UC Berkeley)
Title: Outcome Indistinguishability
Abstract: Prediction algorithms assign numbers to individuals that are popularly understood as individual “probabilities” — e.g., what is the probability of 5-year survival after cancer diagnosis? — and which increasingly form the basis for life-altering decisions. The understanding of individual probabilities in the context of such unrepeatable events has been the focus of intense study for decades within probability theory, statistics, and philosophy. Building off of notions developed in complexity theory and cryptography, we introduce and study Outcome Indistinguishability (OI). OI predictors yield a model of probabilities that cannot be efficiently refuted on the basis of the real-life observations produced by Nature.
We investigate a hierarchy of OI definitions, whose stringency increases with the degree to which distinguishers may access the predictor in question. Our findings reveal that OI behaves qualitatively differently than previously studied notions of indistinguishability. First, we provide constructions at all levels of the hierarchy. Then, leveraging recently-developed machinery for proving average-case fine-grained hardness, we obtain lower bounds on the complexity of the more stringent forms of OI. The hardness result provides scientific grounds for the political argument that, when inspecting algorithmic risk prediction instruments, auditors should be granted oracle access to the algorithm, not simply historical predictions.
Joint work with Cynthia Dwork, Omer Reingold, Guy N. Rothblum, Gal Yona; to appear at STOC 2021.
Current Developments in Mathematics 2019
Friday, Nov. 22, 2019 1:30 pm – 5:20 pm
Saturday, Nov. 23, 2019 9:00 am – 5:00 pm
Harvard University Science Center, Hall C
Speakers:
· Svetlana Jitomirskaya (UC Irvine)
· Subash Khot (NYU)
· Jun Li (Stanford)
· André Neves (Chicago)
· Geordie Williamson (U Sidney)
Free and open to the public – registration is required.
Please register in advance online at www.math.harvard.edu/cdmConcluding Conference of the Special Program on Nonlinear Equations, April 8 – 10, 2016
The Center of Mathematical Sciences and Applications will be hosting a concluding conference on April 8-10, 2016 to accompany the year-long program on nonlinear equations. The conference will have 15 speakers and will be hosted at Harvard CMSA Building: Room G10 20 Garden Street, Cambridge, MA 02138
Speakers:
- Lydia Bieri (University of Michigan)
- Luis Caffarelli (University of Texas at Austin)
- Mihalis Dafermos (Princeton University)
- Camillo De Lellis (Universität Zürich)
- Pengfei Guan (McGill University)
- Slawomir Kolodziej (Jagiellonian University)
- Melissa Liu (Columbia University)
- Duong H. Phong (Columbia University)
- Richard Schoen (UC Irvine)
- Cliff Taubes (Harvard University)
- Blake Temple (UC Davis)
- Valentino Tosatti (Northwestern University)
- Tai-Peng Tsai (University of British Columbia)
- Mu-Tao Wang (Columbia University)
- Xu-jia Wang (Australian National University)
Please click NLE Conference Schedule with Abstracts for a downloadable schedule with talk abstracts.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Resturants.
Schedule:
April 8 – Day 1 8:30am Breakfast 8:45am Opening remarks 9:00am – 10:00am Camillo De Lellis, “A Nash Kuiper theorem for $C^{1,1:5}$ isometric immersions of disks“ 10:00am – 10:15am Break 10:15am – 11:15am Xu-Jia Wang, “Monge’s mass transport problem“ 11:15am – 11:30am Break 11:30am – 12:30pm Peng-Fei Guan, “The Weyl isometric embedding problem in general $3$ d Riemannian manifolds“ 12:30pm – 2:00pm Lunch 2:00pm – 3:00pm Blake Temple, “An instability in the Standard Model of Cosmology“ 3:00pm – 3:15pm Break 3:15pm – 4:15pm Lydia Bieri, “The Einstein Equations and Gravitational Radiation“ 4:15pm – 4:30pm Break 4:30pm – 5:30pm Valentino Tosatti, “Adiabatic limits of Ricci flat Kahler metrics“ April 9 – Day 2 8:45am Breakfast 9:00am – 10:00am D.H. Phong, “On Strominger systems and Fu-Yau equations” 10:00am – 10:15am Break 10:15am – 11:15am Slawomir Kolodziej, “Stability of weak solutions of the complex Monge-Ampère equation on compact Hermitian manifolds” 11:15am – 11:30am Break 11:30am – 12:30pm Luis Caffarelli, “Non local minimal surfaces and their interactions” 12:30pm – 2:00pm Lunch 2:00pm – 3:00pm Mihalis Dafermos, “The interior of dynamical vacuum black holes and the strong cosmic censorship conjecture in general relativity” 3:00pm – 3:15pm Break 3:15pm – 4:15pm Mu-Tao Wang, “The stability of Lagrangian curvature flows” 4:15pm – 4:30pm Break 4:30pm – 5:30pm Melissa Liu, “Counting curves in a quintic threefold” April 10 – Day 3 8:45am Breakfast 9:00am – 10:00am Rick Schoen, “Metrics of fixed area on high genus surfaces with largest first eigenvalue” 10:00am – 10:15am Break 10:15am – 11:15am Cliff Taubes, “The zero loci of Z/2 harmonic spinors in dimensions 2, 3 and 4” 11:15am – 11:30am Break 11:30am – 12:30pm Tai-Peng Tsai, “Forward Self-Similar and Discretely Self-Similar Solutions of the 3D incompressible Navier-Stokes Equations” * This event is sponsored by National Science Foundation (NSF) and CMSA Harvard University.
Random Matrix & Probability Theory Seminar
Beginning immediately, until at least December 31, all seminars will take place virtually, through Zoom.
In the 2020-2021 AY, the Random Matrix and Probability Theory Seminar will take place on select Wednesdays from 2:00 – 3:00pm virtually. This seminar is organized by Christian Brennecke (brennecke@math.harvard.edu ).
To learn how to attend this seminar, please fill out this form.
The schedule below will be updated as the details are confirmed.
Spring 2021:
Date Speaker Title/Abstract 3/31/2021 Philippe Sosoe, Cornell University Title: Fluctuation bounds for O’Connell-Yor type systems Abstract: The O’Connell-Yor polymer is a fundamental model of a polymer in a random environment. It corresponds to the positive temperature version of Brownian Last Passage percolation. Although much is known about this model thanks to remarkable algebraic structure uncovered by O’Connell, Yor and others, basic estimates for the behavior of the tails of the centered partition function for finite N that are available for zero temperature models are missing. I will present an iterative estimate to obtain strong concentration and localization bounds for the O’Connell-Yor polymer on an almost optimal scale N^{1/3+\epsilon}.
In the second part of the talk, I will introduce a system of interacting diffusions describing the successive increments of partition functions of different sizes. For this system, the N^{2/3} variance upper bound known for the OY polymer can be proved for a general class of interactions which are not expected to correspond to integrable models.
Joint work with Christian Noack and Benjamin Landon.
4/7/2021 Yue M. Lu, Harvard Title: Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Random Matrices
Abstract: In many problems in statistical learning, random matrix theory, and statistical physics, one needs to simulate dynamics on random matrix ensembles. A classical example is to use iterative methods to compute the extremal eigenvalues/eigenvectors of a (spiked) random matrix. Other examples include approximate message passing on dense random graphs, and gradient descent algorithms for solving learning and estimation problems with random initialization. We will show that all such dynamics can be simulated by an efficient matrix-free scheme, if the random matrix is drawn from an ensemble with translation-invariant properties. Examples of such ensembles include the i.i.d. Gaussian (i.e. the rectangular Ginibre) ensemble, the Haar-distributed random orthogonal ensemble, the Gaussian orthogonal ensemble, and their complex-valued counterparts.A “direct” approach to the simulation, where one first generates a dense n × n matrix from the ensemble, requires at least O(n^2) resource in space and time. The new algorithm, named Householder Dice (HD), overcomes this O(n^2) bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are O(nT) and O(n T^2), respectively, with T being the number of iterations. When T ≪ n, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint.Finally, the HD algorithm is not just a computational trick. I will show how its construction can serve as a simple proof technique for several problems in high-dimensional estimation4/14/2021 Canceled 4/16/2021
FridayPatrick Lopatto (IAS) Title: Fluctuations in local quantum unique ergodicity for generalized Wigner matrices Abstract: In a disordered quantum system, delocalization can be understood in many ways. One of these is quantum unique ergodicity, which was proven in the random matrix context by Bourgade and Yau. It states that for a given eigenvector and set of coordinates J, the mass placed on J by the eigenvector tends to N^{-1}|J|, the mass placed on those coordinates by the uniform distribution. Notably, this convergence holds for any size of J, showing that the eigenvectors distribute evenly on all scales.
I will present a result which establishes that the fluctuations of these averages are Gaussian on scales where |J| is asymptotically less than N, for generalized Wigner matrices with smooth entries. The proof uses new eigenvector observables, which are analyzed dynamically using the eigenvector moment flow and the maximum principle.
This is joint work with Lucas Benigni.
4/21/2021 Jean-Christophe Mourrat, Courant Institute, NYU Title: Mean-field spin glasses: beyond Parisi’s formula?
Abstract: Spin glasses are models of statistical mechanics encoding disordered interactions between many simple units. One of the fundamental quantities of interest is the free energy of the model, in the limit when the number of units tends to infinity. For a restricted class of models, this limit was predicted by Parisi, and later rigorously proved by Guerra and Talagrand. I will first show how to rephrase this result using an infinite-dimensional Hamilton-Jacobi equation. I will then present partial results suggesting that this new point of view may allow to understand limit free energies for a larger class of models, focusing in particular on the case in which the units are organized over two layers, and only interact across layers.Fall 2020:
Date Speaker Title/Abstract 9/9/2020 Yukun He (Zurich) Title: Single eigenvalue fluctuations of sparse Erdős–Rényi graphs Abstract: I discuss the fluctuations of individual eigenvalues of the adjacency matrix of the Erdös-Rényi graph $G(N,p)$. I show that if $N^{-1}\ll p \ll N^{-2/3}, then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. The main technical tool of the proof is a rigidity bound of accuracy $N^{-1-\varepsilon}p^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from previous works. Joint work with Antti Knowles.
10/14/2020 David Belius (University of Basel) Title: The TAP approach to mean field spin glasses Abstract: The Thouless-Anderson-Palmer (TAP) approach to the Sherrington-Kirkpatrick mean field spin glass model was proposed in one of the earliest papers on this model. Since then it has complemented subsequently elaborated methods in theoretical physics and mathematics, such as the replica method, which are largely orthogonal to the TAP approach. The TAP approach has the advantage of being interpretable as a variational principle optimizing an energy/entropy trade-off, as commonly encountered in statistical physics and large deviations theory, and potentially allowing for a more direct characterization of the Gibbs measure and its “pure states”. In this talk I will recall the TAP approach, and present preliminary steps towards a solution of mean field spin glass models entirely within a TAP framework.
10/28/2020 Giuseppe Genovese (University of Basel) Title: Non-convex variational principles for the RS free energy of restricted Boltzmann machines Abstract: From the viewpoint of spin glass theory, restricted Boltzmann machines represent a veritable challenge, as to the lack of convexity prevents us to use Guerra’s bounds. Therefore even the replica symmetric approximation for the free energy presents some challenges. I will present old and new results around the topic along with some open problems.
11/4/2020 Benjamin Landon (MIT) Title: Fluctuations of the spherical Sherrington-Kirkpatrick model Abstract: The SSK model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual SK model with Ising spins. Fluctuations of its observables may be related to quantities from random matrix theory using integral representations. In this informal talk we discuss some results on fluctuations of this model at critical temperature and with a magnetic field
11/11/2020
3:00 – 4:00pmLucas Benigni (University of Chicago) Title: Optimal delocalization for generalized Wigner matrices Abstract: We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.
11/18/2020 Simone Warzel (Technical University of Munich) Title: Hierarchical quantum spin glasses
Abstract: Hierarchical spin glasses such as the generalised random energy model are known to faithfully model typical energy landscapes in the classical theory of mean-field spin glasses. Their built-in hierarchical structure is known to emerge spontaneously in the spin-glass phase of, e.g., the Sherrington-Kirkpatrick model. In this talk, I will review recent results on the effects of a transversal magnetic field on such hierarchical quantum spin glasses.
In particular, I will present a formula of Parisi-type for their free energy which allows to make predictions about the phase diagram.12/2/2020 Sabine Jansen (LMU Munich) Title: Thermodynamics of a hierarchical mixture of cubes
Abstract: The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes of side-lengths 2^j, j\in \N_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).For information on previous seminars, click here
The schedule will be updated as details are confirmed.
Topological Insulators and Mathematical Science – Conference and Program
The CMSA will be hosting a conference on the subject of topological insulators and mathematical science on September 15-17. Seminars will take place each day from 2:00-7:00pm in Science Center Hall D, 1 Oxford Street, Cambridge, MA.
Homological Mirror Symmetry Seminar
The seminar series, Homological Mirror Symmetry, will be held on selected Thursdays from 2PM – 4pm in CMSA Building, 20 Garden Street, Room G10.
The list of speakers is below and will be updated as details are confirmed.
Date Name Title 09-15-16 09-22-16 Netanel Blaier, Brandeis “Intro to HMS.” Abstract: This is the first talk of the seminar series. We survey the statement of Homological Mirror Symmetry (introduced by Kontsevich in 1994) and some known results, as well as briefly discussing its importance, and the connection to other formulations of Mirror Symmetry and the SYZ conjecture. Following that, we will begin to review the definition of the A-side (namely, the Fukaya category) in some depth. No background is assumed! Also, in the last half hour, we will divide papers and topics among participants.
09-29-16 Netanel Blaier, Brandeis “Intro to HMS 2.” Abstract: In the second talk, we review (some) of the nitty-gritty details needed to construct a Fukaya categories. This include basic Floer theory, the analytic properties of J-holomorphic curves and cylinders, Gromov compactness and its relation to metric topology on the compactified moduli space, and Banach setup and perturbation schemes commonly used in geometric regularization. We then proceed to recall the notion of an operad, Fukaya’s differentiable correspondences, and how to perform the previous constructions coherently in order to obtain $A_\infty$-structures. We will try to demonstrate all concepts in the Morse theory ‘toy model’.
10-06-16 Hansol Hong, CMSA
Title: Homological mirror symmetry for elliptic curves Abstract:
We survey the proof of homological mirror symmetry by Polishchuk and Zaslow. Some of more recent methods to prove HMS for elliptic curves will be discussed also,
which use homological algebra techniques and formal deformation theory of Lagrangians etc.Notes
Notes (Baris)
10-13-16 Yu-Wei Fan, Harvard
Title: Semi-flat mirror symmetry and Fourier-Mukai transform
Abstract: We will review the semi-flat mirror symmetry setting in Strominger-Yau-Zaslow, and discuss the correspondence between special Lagrangian sections on the A-side and deformed Hermitian-Yang-Mills connections on the B-side using real Fourier-Mukai transform, following Leung-Yau-Zaslow.
10-20-16 Tim Large, MIT
Title: “Symplectic cohomology and wrapped Fukaya categories” Abstract: While mirror symmetry was originally conjectured for compact manifolds, the phenomenon applies to non-compact manifolds as well. In the setting of Liouville domains, a class of open symplectic manifolds including affine varieties, cotangent bundles and Stein manifolds, there is an A-infinity category called the wrapped Fukaya category, which is easier to define and often more amenable to computation than the original Fukaya category. In this talk I will construct it, along with symplectic cohomology (its closed-string counterpart), and compute some examples. We will then discuss how compactifying a symplectic manifold corresponds, on the B-side of mirror symmetry, to turning on a Landau-Ginzburg potential.
10-27-16 Philip Engel, Columbia
Title: Mirror symmetry in the complement of an anticanonical divisor”
According to the SYZ conjecture, the mirror of a Calabi-Yau variety can be constructed by dualizing the fibers of a special Lagrangian fibration. Following Auroux, we consider this rubric for an open Calabi-Yau variety X-D given as the complement of a normal crossings anticanonical divisor D in X. In this talk, we first define the moduli space of special Lagrangian submanfiolds L with a flat U(1) connection in X-D, and note that it locally has the structure of a Calabi-Yau variety. The Fukaya category of such Lagrangians is obstructed, and the degree 0 part of the obstruction on L defines a holomorphic function on the mirror. This “superpotential” depends on counts of holomorphic discs of Maslov index 2 bounded by L. We then restrict to the surface case, where there are codimension 1 “walls” consisting of Lagrangians which bound a disc of Maslov index 0. We examine how the superpotential changes when crossing a wall and discuss how one ought to “quantum correct” the complex structure on the moduli space to undo the discontinuity introduced by these discs.
11-03-16 Yusuf Baris Kartal, MIT
I will present Auroux-Katzarkov-Orlov’s proof of one side of the homological mirror symmetry for Del Pezzo surfaces. Namely I will prove their derived categories are equivalent to the categories of vanishing cycles for certain LG-models together with B-fields. I plan to show how the general B-field corresponds to non-commutative Del Pezzo surfaces and time allowing may mention HMS for simple degenerations of Del Pezzo surfaces. The tools include exceptional collections( and mutations for degenerate case), explicit description of NC deformations, etc.
11-10-16 No seminar this week 12-08-16 Lino Amorim, Boston University
Title: The Fukaya category of a compact toric manifold
Abstract: In this talk I will discuss the Fukaya category of a toric manifold following the work of Fukaya-Oh-Ohta-Ono. I will start with an overview of the general structure of the Fukaya category of a compact symplectic manifold. Then I will consider toric manifolds in particular the Fano case and construct its mirror.
Math Science Lectures in Honor of Raoul Bott: Freddy Cachazo
1 Oxford Street, Cambridge MA 02138On April 2-3, the CMSA will be hosting two lectures by Freddy Cachazo (Perimeter Institute) on “Geometry and Combinatorics in Particle Interactions.” This will be the first of the new annual Bott Math Science Lecture Series hosted by the CMSA.
The lectures will take place from 4:30-5:30pm in Science Center, Hall D.
09-22-2016 Homological Mirror Symmetry Seminar
References:
- D. Auroux, A beginner’s introduction to Fukaya categories. arXiv:1301.7056
- I. Smith, A symplectic prolegomenon. arXiv:1401.0269
- D. Auroux, “Topics in geometry: mirror symmetry”, Fall 2009 (MIT Math 18.969)
- Nick Sheridan’s IAS and Jussieu lectures.
- Sheel Gantara “Topics in symplectic topology”, Spring 2016 (Stanford Math 257B)
Quantum Cohomology, Nakajima Varieties and Quantum groups
During the Spring 2018 Semester Artan Sheshmani (QGM/CMSA) will be teaching a CMSA special lecture series on Quantum Cohomology, Nakajima Vareties and Quantum groups. The lectures will be held Tuesdays and Thursdays beginning January 25th, from 1:00 to 3:00pm in room G10, CMSA Building.
You can watch Prof. Sheshmani describe the series here.
The Syllabus is as follows:
Date……….. Topic Video/Audio 1-25-2018 Gromov-Witten invariants Definition, examples via algebraic geometry I
Video / Audio / Combined
*due to technical difficulties the audio and video are split for this lecture.2-01-2018 Gromov-Witten invariants Virtual Fundamental Class I (definition)
Video / Audio / Combined
*due to technical difficulties the audio and video are split for this lecture2-13-2018 Gromov-Witten invariants Virtual Fundamental Class II (computation in some cases)
2-15-2018 Computing GW invariants Three level GW classes
Genus zero invariants of the projective plane
2-20-2018 Quantum Cohomology Small Quantum Cohomology (Definition and Properties) I
2-22-2018 Quantum Cohomology Small Quantum Cohomology (Definition and Properties) II
2-27-2018 Quantum Cohomology Big Quantum Cohomology I
3-1-2018 Quantum Cohomology Big Quantum Cohomology II
GW potential
WDVV equation
3-6-2018 GW invariants via Quantum Cohomology The Quintic threefold case
The P^2 case
GW invariants via Quantum Cohomology Dubrovin (quantum) connection
Nakajima varieties -Algebraic and symplectic reduction
Nakajima varieties Quasi maps to Nakajima varieties
Quantum cohomology of Nakajima varieties Small Quantum Cohomology of Hilb^n (C2) I
Quantum cohomology of Nakajima varieties Small Quantum Cohomology of Hilb^n (C2) II
Quantum cohomology of Nakajima varieties Small Quantum Cohomology of Hilb^n (C2) III
Quantum cohomology of Nakajima varieties Big Quantum Cohomology of Hilb^n (C2) I
Quantum cohomology of Nakajima varieties Big Quantum Cohomology of Hilb^n (C2) II
Quantum cohomology of Nakajima varieties Big Quantum Cohomology of Hilb^n (C2) III
Quantum cohomology of Nakajima varieties Big Quantum Cohomology of Hilb^n (C2) IV
Existence of Canonical Metrics on Non-Kähler Geometry
On Wednesday September 9, CMSA director Prof. Shing-Tung Yau gave a lecture for the Simons foundation on “Existence of Canonical Metrics on Non-Kähler Geometry.“
In this lecture, Prof. Yau surveys the existence of canonical balanced metrics on non-Kähler complex manifolds through the Hull-Strominger system, which was motivated by string theory on compactifications. He discusses works by Jun Li of Fudan University in Shanghai, Ji-Xiang Fu of Fudan University, Ivan Smith of the University of Cambridge, Richard P. Thomas of Imperial College London, Tristan C. Collins of the Massachusetts Institute of Technology, French mathematician Émile Picard, Teng Fei of Rutgers University in Newark, New Jersey, Adam Jacob of the University of California, Davis, and Duong H. Phong of Columbia University.
More information about this talk can be found on the Simons Foundation website.
1/13/2022 Interdisciplinary Science Seminar
Title: A universal triangulation for flat tori
Abstract: A celebrated theorem of Nash completed by Kuiper implies that every smooth Riemannian surface has a C¹ isometric embedding in the Euclidean 3-space E³. An analogous result, due to Burago and Zalgaller, states that every polyhedral surface, obtained by gluing Euclidean triangles, has an isometric PL embedding in E³. In particular, this provides PL isometric embeddings for every flat torus (a quotient of E² by a rank 2 lattice). However, the proof of Burago and Zalgaller is partially constructive, relying on the Nash-Kuiper theorem. In practice, it produces PL embeddings with a huge number of vertices, moreover distinct for every flat torus. Based on a construction of Zalgaller and on recent works by Arnoux et al. we exhibit a universal triangulation with less than 10.000 vertices, admitting for any flat torus an isometric embedding that is linear on each triangle. Based on joint work with Florent Tallerie.
Hyperbolic Geometry and Quantum Invariants
Abstract: There are two very different approaches to 3-dimensional topology, the hyperbolic geometry following the work of Thurston and the quantum invariants following the work of Jones and Witten. These two approaches are related by a sequence of problems called the Volume Conjectures. In this talk, I will explain these conjectures and present some recent joint works with Ka Ho Wong related to or benefited from this relationship.
A Mathematical Language
Speaker: Thomas Hales, Univ. of Pittsburgh Dept. of Mathematics
Title: A Mathematical Language
Abstract: A controlled natural language for mathematics is an artificial language that is designed in an explicit way with precise computer-readable syntax and semantics. It is based on a single natural language (which for us is English) and can be broadly understood by mathematically literate English speakers. This talk will describe the design of a controlled natural language for mathematics that has been influenced by the Lean theorem prover, by TeX, and by earlier controlled natural languages. The semantics are provided by dependent type theory.
Special Lecture Series on Derived Algebraic/Differential Geometry
In the Spring 2019 Semester, the CMSA will be hosting a special lecture series on Derived algebraic/differential geometry run by Artan Sheshmani, with lectures given by Prof. Sheshmani and Dr. Dennis Borisov. The seminar will be held on Tuesdays and Thursdays from 3:00-4:30pm in CMSA, room G10.
Click here for reference material Schedule:
Section 1: Basic setting of derived geometry
The goal: To collect the minimum set of tools needed to do algebraic geometry in the derived context.
2/05/2019 Lecture 1: Model and с-categories Video 2/07/2019 Lecture 2: Grothendieck topologies and homotopy descent Video 2/12/2019 Lecture 3: Derived Artin stacks Video 2/14/2019 Lecture 4: Cotangent complexes Section 2: Loop spaces and differential forms
The goal: This is the algebraic heart of the course – here we learn the homological techniques that are needed for shifted symplectic forms.
2/19/2019 Lecture 5: De Rham complexes and S1-equivariant schemes (loop spaces) Video 2/21/2019 Lecture 6: Chern character Video 2/26/2019 Room G02
Lecture 7: Local structure of closed differential forms in the derived sense Part I Video 2/28/2019 Lecture 8: Local structure of closed differential forms in the derived sense Part II Video 3/05/2019 Lecture 9: Cyclic homology Video Section 3: Shifted symplectic structures
Goal: To see applications of the algebraic techniques from above in the geometric context of the actual moduli spaces.3/07/2019 Lecture 10: Definition and existence results Video 3/12/2019 Lecture 11: Lagrangians and Lagrangian fibrations Video 3/14/2019 Room G02
Lecture 12: Lagrangians and Lagrangian fibrations Video 3/26/2019 Lecture 13: Intersections of Lagrangians Video 3/28/2019 Room G02
Lecture 14: Examples and applications 2 (Part I) Video 4/02/2019 Lecture 15: Examples and applications 2 (Part II) Video Section 4: Uhlenbeck–Yau construction and correspondence
4/04/2019 Lecture 16: Examples and applications 2 (Part III) Video 4/09/2019 Room G02
Lecture 17: Uhlenbeck–Yau construction and correspondence Examples (Part I) Video CMSA Math-Science Literature Lecture: Quantum Groups
Pavel Etingof (MIT)
Title: Quantum Groups
Abstract: The theory of quantum groups developed in mid 1980s from attempts to construct and understand solutions of the quantum Yang-Baxter equation, an important equation arising in quantum field theory and statistical mechanics. Since then, it has grown into a vast subject with profound connections to many areas of mathematics, such as representation theory, the Langlands program, low-dimensional topology, category theory, enumerative geometry, quantum computation, algebraic combinatorics, conformal field theory, integrable systems, integrable probability, and others. I will review some of the main ideas and examples of quantum groups and try to briefly describe some of the applications.
CMSA Math-Science Literature Lecture: The ADHM construction of Yang-Mills instantons
Simon Donaldson (Stony Brook)
Title: The ADHM construction of Yang-Mills instantons
Abstract: In 1978 (Physics Letters 65A) Atiyah, Hitchin, Drinfeld and Manin (ADHM) described a construction of the general solution of the Yang-Mills instanton equations over the 4-sphere using linear algebra. This was a major landmark in the modern interaction between geometry and physics, and the construction has been the scene for much research activity up to the present day. In this lecture we will review the background and the original ADHM proof, using Penrose’s twistor theory and results on algebraic vector bundles over projective 3-space. As time permits, we will also discuss some further developments, for example, the work of Nahm on monopoles and connections to Mukai duality for bundles over complex tori.
AI and Theorem Proving
Speaker: Josef Urban, Czech Technical University
Title: AI and Theorem Proving
Abstract: The talk will discuss the main approaches that combine machine learning with automated theorem proving and automated formalization. This includes learning to choose relevant facts for “hammer” systems, guiding the proof search of tableaux and superposition automated provers by interleaving learning and proving (reinforcement learning) over large ITP libraries, guiding the application of tactics in interactive tactical systems, and various forms of lemmatization and conjecturing. I will also show some demos of the systems, and discuss autoformalization approaches such as learning probabilistic grammars from aligned informal/formal corpora, combining them with semantic pruning, and using neural methods to learn direct translation from Latex to formal mathematics.
Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models
Speaker: Jason Rute, CIBO Technologies
Title: Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models
Abstract: Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT ({\bf P}roof {\bf A}rtifact {\bf C}o-{\bf T}raining), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32% to 48%.
2018 Ding Shum Lecture
On October 24, 2018, the CMSA will be hosting our second annual Ding Shum lecture. This event was made possible by the generous funding of Ding Lei and Harry Shum. Last year featured Leslie Valiant, who spoke on “learning as a Theory of Everything.”
This year will feature Eric Maskin, who will speak on “How to Improve Presidential Elections: the Mathematics of Voting.” This lecture will take place from 5:00-6:00pm in Science Center, Hall D.
Pictures of the event can be found here.
Algebraic Geometry Seminar, Thursdays
This seminar will not be held in the Spring 2018 Semester.
The Algebraic Geometry Seminar will be every Thursday from 3pm-4pm in CMSA Building, 20 Garden Street, Room G10.
The schedule will be updated as details are confirmed.
Date Name Title/Abstract 09-14-17 Yu-Wei Fan (Harvard Math) Entropy of an autoequivalence on Calami-Yau manifolds
Abstract: We will recall the notion of entropy of an autoequivalence on triangulated categories, and provide counterexamples of a conjecture by Kikuta-Takahashi.
11-1-17 *5:00pm, G10*
Shamil Shakirov, Harvard Math Undulation invariants of plane curves
Abstract: “One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has singularities or other distinctive features of interest). A classical example of such a problem, described by Cayley and Salmon in 1852, is to determine whether or not a given plane curve of degree r > 3 has undulation points — the points where the tangent line meets the curve with multiplicity four. Cayley proved that there exists an invariant of degree (r – 3)(3 r – 2) that vanishes if and only if the curve has undulation points. We construct this invariant explicitly for quartics (r=4) as the determinant of a 21 times 21 matrix with polynomial entries, and we conjecture a generalization for r = 5
11-2-17 Alexander Moll, IHES Hilbert Schemes from Geometric Quantization of Dispersive Periodic Benjamin-Ono Waves
ABSTRACT: By Grojnowski and Nakajima, Fock spaces are cohomology rings of Hilbert scheme of points in the plane. On the other hand, by Pressley-Segal, Fock spaces are spaces of J-holomorphic functions on the loop space of the real line that appear in geometric quantization with respect to the Kähler structure determined by the Sobolev regularity s= -1/2 and the Hilbert transform J. First, we show that the classical periodic Benjamin-Ono equation is a Liouville integrable Hamiltonian system with respect to this Kähler structure. Second, we construct an integrable geometric quantization of this system in Fock space following Nazarov-Sklyanin and describe the spectrum explicitly after a non-trivial rewriting of our coefficients of dispersion \ebar = e_1 + e_2 and quantization \hbar = – e_1 e_2 that is invariant under e_2 <-> e_1. As a corollary of Lehn’s theorem, our construction gives explicit creation and annihilation operator formulas for multiplication by new explicit universal polynomials in the Chern classes of the tautological bundle in the equivariant cohomology of our Hilbert schemes, in particular identifying \ebar with the deformation parameter of the Maulik-Okounkov Yangian and \hbar with the handle-gluing element. Our key ingredient is a simple formula for the Lax operators as elliptic generalized Toeplitz operators on the circle together with the spectral theory of Boutet de Monvel and Guillemin. As time permits, we discuss the relation of dispersionless \ebar -> 0 and semi-classical \hbar \rightarrow 0 limits to Nekrasov’s BPS/CFT Correspondence.
11-9-17 TBD TBD 11-16-17 TBD TBD 11-23-17 TBD TBD 11-30-17 TBD TBD 12-7-17 TBD TBD 12-15-17 TBD TBD Random Matrix & Probability Theory Seminar (2016-2017)
CMSA, 20 Garden Street, Cambridge, MA 02138 USAThe random matrix and probability theory will be every Wednesday from 3pm-4pm in CMSA Building, 20 Garden Street, Room G10.
Working Conference on Applications of Random Matrix Theory to Data Analysis, January 9-13, 2017
The Center of Mathematical Sciences and Applications will be hosting a working Conference on Applications of Random Matrix Theory to Data Analysis, January 9-13, 2017. The conference will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
Participants:
Gerard Ben Arous, Courant Institute of Mathematical Sciences
Alex Bloemendal, Broad Institute
Arup Chakraburty, MIT
Zhou Fan, Stanford University
Alpha Lee, Harvard University
Matthew R. McKay, Hong Kong University of Science and Technology (HKUST)
David R. Nelson, Harvard University
Nick Patterson, Broad Institute
Marc Potters, Capital Fund management
Yasser Roudi, IAS
Tom Trogdon, UC Irvine
Organizers:
Michael Brenner, Lucy Colwell, Govind Menon, Horng-Tzer Yau
Please click Program for a downloadable schedule with talk abstracts.
Please note that breakfast & lunch will be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants should you need recommendations for dinner.
Schedule:
January 9 – Day 1 9:30am – 10:00am Breakfast & Opening remarks 10:00am – 11:00am Marc Potters, “Eigenvector overlaps and the estimation of large noisy matrices” 11:00am – 12:00pm Yasser Roudi 12:00pm – 2:00pm Lunch 2:00pm Afternoon Discussion January 10 – Day 2 8:30am – 9:00am Breakfast 9:00am – 10:00am Arup Chakraburty, “The mathematical analyses and biophysical reasons underlying why the prevalence of HIV strains and their relative fitness are simply correlated, and pose the challenge of building a general theory that encompasses other viruses where this is not true.” 10:00am – 11:00am Tom Trogdon, “On the average behavior of numerical algorithms” 11:00am – 12:00pm David R. Nelson, “Non-Hermitian Localization in Neural Networks” 12:00pm – 2:00pm Lunch 2:00pm Afternoon Discussion January 11 – Day 3 8:30am – 9:00am Breakfast 9:00am – 10:00am Nick Patterson 10:00am – 11:00am Lucy Colwell 11:00am – 12:00pm Alpha Lee 12:00pm – 2:00pm Lunch 2:00pm-4:00pm Afternoon Discussion 4:00pm Gerard Ben Arous (Public Talk), “Complexity of random functions of many variables: from geometry to statistical physics and deep learning algorithms“ January 12 – Day 4 8:30am – 9:00am Breakfast 9:00am – 10:00am Govind Menon 10:00am – 11:00am Alex Bloemendal 11:00am – 12:00pm Zhou Fan, “Free probability, random matrices, and statistics” 12:00pm – 2:00pm Lunch 2:00pm Afternoon Discussion January 13 – Day 5 8:30am – 9:00am Breakfast 9:00am – 12:00pm Free for Working 12:00pm – 2:00pm Lunch 2:00pm Free for Working * This event is sponsored by CMSA Harvard University.
Members’ Seminar
The CMSA Members’ Seminar will occur every Friday at 9:30am ET on Zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. Please email the seminar organizers to obtain a link. This year’s seminar is organized by Tianqi Wu. The Schedule will be updated below.
Previous seminars can be found here.
Spring 2021:
Date Speaker Title/Abstract 1/29/2021 Cancelled 2/5/2021 Itamar Shamir Title: Boundary CFT and conformal anomalies Abstract: Boundary and defects in quantum field theory play an important role in many recent developments in theoretical physics. I will discuss such objects in the setting of conformal field theories, focusing mainly on conformal anomalies. Boundaries or defects can support various kinds of conformal anomalies on their world volume. Perhaps the one which is of greatest theoretical importance is associated with the Euler density in even dimensions. I will show how this anomaly is related to the one point function of exactly marginal deformations and how it arises explicitly from various correlation functions.
2/12/2021 Louis Fan Title: Joint distribution of Busemann functions in corner growth models Abstract: The 1+1 dimensional corner growth model with exponential weights is a centrally important exactly solvable model in the Kardar-Parisi-Zhang class of statistical mechanical models. While significant progress has been made on the fluctuations of the growing random shape, understanding of the optimal paths, or geodesics, is less developed. The Busemann function is a useful analytical tool for studying geodesics. We present the joint distribution of the Busemann functions, simultaneously in all directions of growth, in terms of mappings that represent FIFO (first-in-first-out) queues. As applications of this description we derive a marked point process representation for the Busemann function across a single lattice edge and point out its implication on structure of semi-infinite geodesics. This is joint work with Timo Seppäläinen.
2/19/2021 Daniel Junghans Title: Control issues of the KKLT scenario in string theory Abstract: The simplest explanation for the observed accelerated expansion of the universe is that we live in a 4-dimensional de Sitter space. We analyze to which extent the KKLT proposal for the construction of such de Sitter vacua in string theory is quantitatively controlled. As our main finding, we uncover and quantify an issue which one may want to call the “singular-bulk problem”. In particular, we show that, generically, a significant part of the manifold on which string theory is compactified in the KKLT scenario becomes singular. This implies a loss of control over the supergravity approximation on which the construction relies.
2/26/2021 Tsung-Ju Lee Title: SYZ fibrations and complex affine structures Abstract: Strominger–Yau–Zaslow conjecture has been a guiding principle in mirror symmetry. The conjecture predicts the existence of special Lagrangian torus fibrations of a Calabi–Yau manifold near a large complex structure limit point. Moreover, the mirror is given by the dual fibrations and the Ricci-flat metric is obtained from the semi-flat metric with corrections from holomorphic discs whose boundaries lie in a special Lagrangian fiber. By a result of Collins–Jacob–Lin, the complement of a smooth elliptic curve in the projective plane admits a SYZ fibration. In this talk, I will explain how to compute the complex affine structure induced from this SYZ fibration and show that it agrees with the affine structure used in Carl–Pumperla–Siebert. This is based on a joint work with Siu-Cheong Lau and Yu-Shen Lin.
3/5/2021 Cancelled 3/11/2021 9:00pm ET
Ryan Thorngren Title: Symmetry protected topological phases, anomalies, and their classification Abstract: I will give an overview of some mathematical aspects of the subject of symmetry protected topological phases (SPTs), especially as their theory relates to index theorems in geometry, cobordism of manifolds, and group cohomology.
3/18/2021 9:00pm ET
Ryan Thorngren Title: Symmetry protected topological phases, anomalies, and their classification
Abstract: I will give an overview of some mathematical aspects of the subject of symmetry protected topological phases (SPTs), especially as their theory relates to index theorems in geometry, cobordism of manifolds, and group cohomology.3/26/2021 8:30am ET
Aghil Alaee Title: Rich extra dimensions are hidden inside black holes Abstract: In this talk, I present an argument that shows why it is difficult to see rich extra dimensions in the Universe.
4/2/2021
8:30am ETEnno Keßler Title: Super Stable Maps of Genus Zero Abstract: I will report on a supergeometric generalization of J-holomorphic curves. Supergeometry is a mathematical theory of geometric spaces with anti-commuting coordinates and functions which is motivated by the concept of supersymmetry from theoretical physics. Super J-holomorphic curves and super stable maps couple the equations of classical J-holomorphic curves with a Dirac equation for spinors and might, in the future, lead to a supergeometric generalization of Gromov-Witten invariants.
4/9/2021 Juven Wang Title: Ultra Unification Abstract: Strong, electromagnetic, and weak forces were unified in the Standard Model (SM) with spontaneous gauge symmetry breaking. These forces were further conjectured to be unified in a simple Lie group gauge interaction in the Grand Unification (GUT). In this work, we propose a theory beyond the SM and GUT by adding new gapped Topological Phase Sectors consistent with the nonperturbative global anomaly matching and cobordism constraints (especially from the baryon minus lepton number B − L and the mixed gauge-gravitational anomaly). Gapped Topological Phase Sectors are constructed via symmetry extension, whose low energy contains unitary topological quantum field theories (TQFTs): either 3+1d non-invertible TQFT (long-range entangled gapped phase), or 4+1d invertible or non-invertible TQFT (short-range or long-range entangled gapped phase), or right-handed neutrinos, or their combinations. We propose that a new high-energy physics frontier beyond the conventional 0d particle physics relies on the new Topological Force and Topological Matter including gapped extended objects (gapped 1d line and 2d surface operators or defects, etc., whose open ends carry deconfined fractionalized particle or anyonic string excitations). I will also fill in the dictionary between math, QFT, and condensed matter terminology, and elaborate more on the nonperturbative global anomalies of Z2, Z4, Z16 classes useful for beyond SM. Work is based on arXiv:2012.15860, arXiv:2008.06499, arXiv:2006.16996, arXiv:1910.14668.
4/16/2021 Sergiy Verstyuk Title: Deep learning methods for economics Abstract: The talk discusses some recent developments in neural network models and their applicability to problems in international economics as well as macro-via-micro economics. Along the way, interpretability of neural networks features prominently.
4/23/2021 Yifan Wang Title: Virtues of Defects in Quantum Field Theories Abstract: Defects appear ubiquitously in many-body quantum systems as boundaries and impurities. They participate inextricably in the quantum dynamics and give rise to novel phase transitions and critical phenomena. Quantum field theory provides the natural framework to tackle these problems, where defects define extended operators over sub-manifolds of the spacetime and enrich the usual operator algebra. Much of the recent progress in quantum field theory has been driven by the exploration of general structures in this extended operator algebra, precision studies of defect observables, and the implications thereof for strongly coupled dynamics. In this talk, I will review selected developments along this line that enhance our understanding of concrete models in condensed matter and particle physics, and that open new windows to nonperturbative effects in quantum gravity.
4/30/2021 Yun Shi Title: D-critical locus structure for local toric Calabi-Yau 3-fold Abstract: Donaldson-Thomas (DT) theory is an enumerative theory which produces a count of ideal sheaves of 1-dimensional subschemes on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will give a brief introduction to motivic DT theory following the definition of Bussi-Joyce-Meinhardt, in particular the role of d-critical locus structure in the definition of motivic DT invariant. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint work in progress with Sheldon Katz.
5/7/2021 Thérèse Yingying Wu Title: Topological aspects of Z/2Z eigenfunctions for the Laplacian on S^2 Abstract: In this talk, I will present recent work with C. Taubes on an eigenvalue problem for the Laplacian on the round 2-sphere associated with a configuration of an even number of distinct points on that sphere, denoted as C_2n. I will report our preliminary findings on how eigenvalues and eigenfunctions change as a function of the configuration space. I will also discuss how the compactification of C_2n is connected to the moduli space of algebraic curves (joint work with S.-T. Yau). There is a supergeometry tie-in too.
5/14/2021 Du Pei Title: Three applications of TQFTs Abstract: Topological quantum field theories (TQFTs) often serve as a bridge between physics and mathematics. In this talk, I will illustrate how TQFTs that arise in physics can help to shed light on 1) the quantization of moduli spaces 2) quantum invariants of 3-manifolds, and 3) smooth structures on 4-manifolds.
5/21/2021 Farzan Vafa Title: Active nematic defects and epithelial morphogenesis Abstract: Inspired by recent experiments that highlight the role of topological defects in morphogenesis, we develop a minimal framework to study the dynamics of an active curved surface driven by its nematic texture (a rank 2 symmetric traceless tensor). Allowing the surface to evolve via relaxational dynamics (gradient flow) leads to a theory linking nematic defect dynamics, cellular division rates, and Gaussian curvature. Regions of large positive (negative) curvature and positive (negative) growth are colocalized with the presence of positive (negative) defects, and cells accumulate at positive defects and are depleted at negative defects. We also show that activity stabilizes a bound $+1$ defect state by creating an incipient tentacle, while a bound $+1$ defect state surrounded by two $-1/2$ defects can create a stationary ring configuration of tentacles, consistent with experimental observations. The talk is based on a recent paper with L Mahadevan [arXiv:2105.0106].
Fall 2020:
Date Speaker Title/Abstract 9/11/2020 Moran Koren Title: Observational Learning and Inefficiencies in Waitlists Abstract: Many scarce resources are allocated through waitlists without monetary transfers. We consider a model, in which objects with heterogeneous qualities are offered to strategic agents through a waitlist in a first-come-first-serve manner. Agents, upon receiving an offer, accept or reject it based on both a private signal about the quality of the object and the decisions of agents ahead of them on the list. This model combines observational learning and dynamic incentives, two features that have been studied separately. We characterize the equilibrium and quantify the inefficiency that arises due to herding and selectivity. We find that objects with intermediate expected quality are discarded while objects with a lower expected quality may be accepted. These findings help in understanding the reasons for the substantial discard rate of transplant organs of various qualities despite the large shortage of organ supply.
9/18/2020 Michael Douglas Title: A talk in two parts, on strings and on computers and math Abstract: I am dividing my time between two broad topics. The first is string theory, mostly topics in geometry and compactification. I will describe my current work on numerical Ricci flat metrics, and list many open research questions. The second is computation and artificial intelligence. I will introduce transformer models (Bert,GPT) which have led to breakthroughs on natural language processing, describe their potential for helping us do math, and sketch some related theoretical problems.
9/25/2020 Cancelled – Math Science Lecture 10/2/2020 Cancelled – Math Science Lecture 10/9/2020 Wai Tong (Louis) Fan Title: Stochastic PDE as scaling limits of interacting particle systems Abstract: Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.
In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics. Joint work with Rick Durrett.10/16/2020 Tianqi Wu Title: Koebe circle domain conjecture and the Weyl problem in hyperbolic 3-space Abstract: In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. The Weyl problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 into the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture, the Weyl problem and the work of Alexandrov and Thurston on convex surfaces. This is a joint work with Feng Luo.
10/23/2020 Changji Xu Title: Random Walk Among Bernoulli Obstacles Abstract: Place an obstacle with probability $1 – p$ independently at each vertex of $\mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. This is called random walk among Bernoulli obstacles. The most prominent feature of this model is a strong localization effect: the random walk will be localized in a very small region conditional on the event that it survives for a long time. In this talk, we will discuss some recent results about the behaviors of the conditional random walk, in quenched, annealed, and biased settings.
10/30/2020 Michael Simkin Title: The differential equation method in Banach spaces and the $n$-queens problem Abstract: The differential equation method is a powerful tool used to study the evolution of random combinatorial processes. By showing that the process is likely to follow the trajectory of an ODE, one can study the deterministic ODE rather than the random process directly. We extend this method to ODEs in infinite-dimensional Banach spaces.
We apply this tool to the classical $n$-queens problem: Let $Q(n)$ be the number of placements of $n$ non-attacking chess queens on an $n \times n$ board. Consider the following random process: Begin with an empty board. For as long as possible choose, uniformly at random, a space with no queens in its row, column, or either diagonal, and place on it a queen. We associate the process with an abstract ODE. By analyzing the ODE we conclude that the process almost succeeds in placing $n$ queens on the board. Furthermore, we can obtain a complete $n$-queens placement by making only a few changes to the board. By counting the number of choices available at each step we conclude that $Q(n) \geq (n/C)^n$, for a constant $C>0$ associated with the ODE. This is optimal up to the value of $C$.11/6/2020 Kenji Kawaguchi Title: Deep learning: theoretical results on optimization and mixup Abstract: Deep neural networks have achieved significant empirical success in many fields, including the fields of computer vision, machine learning, and artificial intelligence. Along with its empirical success, deep learning has been theoretically shown to be attractive in terms of its expressive power. However, the theory of the expressive power does not ensure that we can efficiently find an optimal solution in terms of optimization, robustness, and generalization, during the optimization process of a neural network. In this talk, I will discuss some theoretical results on optimization and the effect of mixup on robustness and generalization.
11/13/2020 Omri Ben-Eliezer Title: Sampling in an adversarial environment Abstract: How many samples does one need to take from a large population in order to truthfully “represent” the population? While this cornerstone question in statistics is very well understood when the population is fixed in advance, many situations in modern data analysis exhibit a very different behavior: the population interacts with and is affected by the sampling process. In such situations, the existing statistical literature does not apply.
We propose a new sequential adversarial model capturing these situations, where future data might depend on previously sampled elements; we then prove uniform laws of large numbers in this adversarial model. The results, techniques, and applications reveal close connections to various areas in mathematics and computer science, including VC theory, discrepancy theory, online learning, streaming algorithms, and computational geometry.
Based on joint works with Noga Alon, Yuval Dagan, Shay Moran, Moni Naor, and Eylon Yogev.
11/20/2020 Charles Doran Title: The Calabi-Yau Geometry of Feynman Integrals Abstract: Over the past 30 years Calabi-Yau manifolds have proven to be the key geometric structures behind string theory and its variants. In this talk, I will show how the geometry and moduli of Calabi-Yau manifolds provide a new framework for understanding and computing Feynman integrals. An important organizational principle is provided by mirror symmetry, and specifically the DHT mirror correspondence. This is joint work with Andrey Novoseltsev and Pierre Vanhove.
Strings, knots and quivers
Abstract: I will discuss a recently discovered relation between quivers and knots, as well as – more generally – toric Calabi-Yau manifolds. In the context of knots this relation is referred to as the knots-quivers correspondence, and it states that various invariants of a given knot are captured by characteristics of a certain quiver, which can be associated to this knot. Among others, this correspondence enables to prove integrality of LMOV invariants of a knot by relating them to motivic Donaldson-Thomas invariants of the corresponding quiver, it provides a new insight on knot categorification, etc. This correspondence arises from string theory interpretation and engineering of knots in brane systems in the conifold geometry; replacing the conifold by other toric Calabi-Yau manifolds leads to analogous relations between such manifolds and quivers.
Hodge and Noether-Lefschetz Loci Seminar
In the Fall 2018 Semester the CMSA will be hosting a seminar on Hodge and Noether-Lefschetz loci, with lectures given by Hossein Movasati (IMPA). The seminar will occur weekly on Wednesday at 1:30 in room G10 of the CMSA.
The schedule below will be updated as talks are confirmed.
Date Title/Abstract 11/7/2018 Title: Hodge and Noether-Lefschetz loci Abstract: Hodge cycles are topological cycles which are conjecturally (the millennium Hodge conjecture) supported in algebraic cycles of a given smooth projective complex manifold. Their study in families leads to the notion of Hodge locus, which is also known as Noether-Lefschetz locus in the case of surfaces. The main aim of this mini course is to introduce a computational approach to the study of Hodge loci for hypersurfaces and near the Fermat hypersurface. This will ultimately lead to the verification of the variational Hodge conjecture for explicit examples of algebraic cycles inside hypersurfaces and also the verification of integral Hodge conjecture for examples of Fermat hypersurfaces. Both applications highly depend on computer calculations of rank of huge matrices. We also aim to review some classical results on this topic, such as Cattani-Deligne-Kaplan theorem on the algebraicity of the components of the hodge loci, Deligne’s absolute Hodge cycle theorem for abelian varieties etc.
In the theoretical side another aim is to use the available tools in algebraic geometry and construct the moduli space of projective varieties enhanced with elements in their algebraic de Rham cohomology ring. These kind of moduli spaces have been useful in mathematical physics in order to describe the generating function of higher genus Gromov-Witten invariants, and it turns out that the Hodge loci in such moduli spaces are well-behaved, for instance, they are algebraic leaves of certain holomorphic foliations. Such foliations are constructed from the underlying Gauss-Manin connection. This lectures series involves many reading activities on related topics, and contributions by participants are most welcome.
11/14/2018 Title: Integral Hodge conjecture for Fermat varieties Abstract: We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same. This is a joint work with E. Aljovin and R. Villaflor.
11/21/2018 Title: Periods of algebraic cycles Abstract: The tangent space of the Hodge locus at a point can be described by the so called infinitesimal variation of Hodge structures and the cohomology class of Hodge cycles. For hypersurfaces of dimension $n$ and degree $d$ it turns out that one can describe it without any knowledge of cohomology theories and in a fashion which E. Picard in 1900’s wanted to study integrals/periods. The data of cohomology class is replaced with periods of Hodge cycles, and explicit computations of these periods, will give us a computer implementable description of the tangent space. As an application of this we show that for examples of $n$ and $d$, the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space.
11/28/2018 Title: Periods of Complete Intersection Algebraic Cycles Speaker: Roberto Villaflor
Abstract: In order to compute periods of algebraic cycles inside even dimensional smooth degree d hypersurfaces of the projective space, we restrict ourselves to cycles supported in a complete intersection subvariety. When the description of the complete intersection is explicit, we can compute its periods, and furthermore its cohomological class. As an application, we can use this data to describe the Zariski tangent space of the corresponding Hodge locus, as the degree d part of some Artinian Gorenstein ideal of the homogeneous coordinate ring of the projective space. Using this description, we can show that for d>5, the locus of hypersurfaces containing two linear cycles, is a reduced component of the Hodge locus in the underlying parameter space.
12/05/2018 Room G02
Title: Some explicit Hodge cycles Abstract: Explicit examples of Hodge cycles are due to D. Mumford and A. Weil in the case of CM abelian varieties. In this talk, I will describe few other examples for the Fermat variety. Effective verification of the Hodge conjecture for these cycles is not known.
12/12/2018 Title: A conjectural Hodge locus for cubic tenfold Abstract: In this talk we will consider the difference of two linear algebraic cycles of dimension 5 inside a smooth cubic tenfold and such that the dimension of their intersection is 3. We will show some computer assisted evidences to the fact that the corresponding Hodge locus is bigger than the expected locus of algebraic deformations of the cubic tenfold together with its linear cycles. A similar discussion will be also presented for cubic six and eightfold, for which we will prove that the corresponding second and third order infinitesimal Hodge loci are smooth. The main ingredient is a computer implementation of power series of periods of hypersurfaces.
1/16/2019 Title: Algebraic BCOV anomaly equation Abstract: We introduce the moduli space T of non-rigid compact Calabi-Yau threefolds enhanced with differential forms and a Lie algebra of vector fields in T. This will be used in order to give a purely algebraic interpretation of topological string partition functions and the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation (joint work with M. Alim, E. Scheidegger, S.-T. Yau). We will also define similar moduli spaces for even dimensional Calabi-Yau varieties, where we have the notion of Hodge locus.
1/23/2019 Title: A new model for modular curves Abstract: One of the non-trivial examples of a Hodge locus is the modular curve X_0(N), which is due to isogeny of elliptic curves (a Hodge/algebraic cycle in the product of two elliptic curves). After introducing the notion of enhanced moduli of elliptic curves, I will describe a new model for X_0(N) in the weighted projective space of dimension 4 and with weights (2,3,2,3,1). I will also introduce some elements in the defining ideal of such a model.
The talk is based on the article arXiv:1808.01689.
1/30/2019 Title: Constant Yukawa couplings Abstract: In this talk I will first introduce algebraic Yukawa couplings for any moduli of enhanced Calabi-Yau n-folds. Then I will list many examples in support of the following conjecture. A moduli of Calabi-Yau n-folds is a quotient of a Hermitian symmetric domain (constructed from periods) by an arithmetic group if and only if the corresponding Yukawa couplings are constants.
2/6/2019 Title: Integrality properties of CY modular forms Abstract: The integrality of the coefficients of the mirror map is a central problem in the arithmetic of Calabi-Yau varieties and it has been investigated by Lian-Yau (1996, 1998), Hosono-Lian-Yau (1996), Zudilin (2002), Kontsevich-Schwarz-Vologodsky (2006) Krattenthaler-Rivoal (2010). The central tool in most of these works has been the so called Dwork method. In this talk we use this method and classify all hypergeometric differential equations with a maximal unipotent monodromy whose mirror map has integral coefficients.
We also give a computable condition on the parameters of a hypergeometric function which conjecturally computes all the primes which appear in the denominators of the coefficients of the mirror map. This is a joint work with Kh. Shokri.
2/13/2019 Title: Foliations and Hodge loci Abstract: In this talk I will introduce a holomorphic foliation in a larger parameter space attached to families of enhanced projective varieties. Irreducible components of the Hodge locus with constant periods are algebraic leaves of such a foliation. Under the hypothesis that these are all the algebraic leaves, we get the fact that such algebraic leaves are defined over the algebraic closure of the base field and that Hodge classes are weak absolute in the sense of C. Voisin.
References:
- M. Alim, H. Movasati, E. Scheidegger, S.-T. Yau. Gauss-Manin connection in disguise: Calabi-Yau threefolds, Comm. Math. Phys. 344, (2016), no. 3, 889-914.
- E. H. Cattani, P. Deligne, and A. G. Kaplan. On the locus of Hodge classes. Amer. Math. Soc., 8(2):483–506, 1995.
- B. Haghighat H. Movasati, S.-T. Yau. Calabi-Yau modular forms in limit: Elliptic fibrations, Communications in Number Theory and Physics, Vol. 11, Number 4, 879-912, 2017.
- H. Movasati, Modular and automorphic forms & beyond, Book under preparation, 2019.
- H. Movasati. A Course in Hodge Theory: with Emphasis on Multiple Integrals.Book submitted,2018.
- H. Movasati, On elliptic modular foliation, II, 2018
- H. Movasati, R. Villaflor Loyola, Periods of linear algebraic cycles,, 2018.
- H. Movasati, Gauss-Manin connection in disguise: Calabi-Yau modular forms, Surveys in Modern Mathematics, Vol 13, International Press, Boston.
- H. Movasati, Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci, Asian Journal of Mathematics, Vol.21, No. 3, pp. 463-482, 2017.
- C. Voisin. Hodge loci and absolute Hodge classes. Compos. Math., 143(4):945–958, 2007.
- C. Voisin. Hodge loci. Handbook of moduli. Vol. III, volume 26 of Adv. Lect. Math. (ALM)}, pages 507–546. Int. Press, Somerville, MA, 2013.
Data Analysis Workshop, April 4 – 8, 2016
The Center of Mathematical Sciences and Applications will be hosting a 5-day workshop on Data Analysis and related areas on April 4 – 8, 2016.
Workshop Locations:
April 4 – 7 (Monday ~ Thursday)
Room G10,
20 Garden Street, Cambridge, MA 02138April 8 (Friday)
EPS Faculty Lounge, Room 409, 4th floor, Hoffman Lab
20 Oxford Street, Cambridge, MA 02138Participants:
- Peter Hubyers (Harvard University)
- Eli Tziperman (Harvard University)
- Andrew Rhines (University of Washington)
- Karen McKinnon (UCAR)
- Douglas MacMartin (Caltech)
- Thomas Laepple (Alfred Wegener Institute)
- Yossi Ashkenazy (Ben-Gurion University)
- Marlene Kretschamer (Potsdam Institute for Climate Impact Research)
- Natesh Pillai (Harvard University)
- Judah Cohen (Atmospheric and Environmental Research)
- Cristian Proistosescu (Harvard University)
Please click Workshop Agenda for a downloadable agenda.
* This event is sponsored by CMSA Harvard University.
Anisotropy, biased pairing theory and applications
Abstract: Not so long ago, the relations between algebraic geometry and combinatorics were strictly governed by the former party, with results like log-concavity of the coefficients of the characteristic polynomial of matroids shackled by intuitions and techniques from projective algebraic geometry, specifically Hodge Theory. And so, while we proved analogues for these results, combinatorics felt subjugated to inspirations from outside of it.
In recent years, a new powerful technique has emerged: Instead of following the geometric statements of Hodge theory about signature, we use intuitions from the Hall marriage theorem, translated to algebra: once there, they are statements about self-pairings, the non-degeneracy of pairings on subspaces to understand the global geometry of the pairing. This was used to establish Lefschetz type theorems far beyond the scope of algebraic geometry, which in turn established solutions to long-standing conjectures in combinatorics.I will survey this theory, called biased pairing theory, and new developments within it, as well as new applications to combinatorial problems. Reporting on joint work with Stavros Papadaki, Vasiliki Petrotou and Johanna Steinmeyer.
Mini-workshop on SYZ and Homological Mirror Symmetry
The Center of Mathematical Sciences and Applications will be hosting a 4-day workshop on SYZ and Homological Mirror Symmetry and related areas on November 28 – December 2, 2016 at Harvard CMSA Building: Room G10, 20 Garden Street, Cambridge, MA 02138.
Organizers:
Bong Lian (Brandeis University), Siu-Cheong Lau (Boston University), Shing-Tung Yau (Harvard University)
Speakers:
- Conan Leung, Chinese University of Hong Kong
- Junwu Tu, University of Missouri
- Jingyu Zhao, Columbia University
- David Treumann, Boston College
- Hiro Lee Tanaka, Harvard University
- Fabian Haiden, Harvard University
- Hansol Hong, Harvard CMSA/Brandeis University
- Netanel Blaier, Harvard CMSA/Brandeis University
- Garret Alston, The University of Oklahoma
Please click Workshop Program for a downloadable schedule with talk abstracts.
Conference Schedule:
Monday, November 28 – Day 1 10:30am –11:30am Hiro Lee Tanaka, “Floer theory through spectra” Lunch 1:00pm – 2:30pm Fabian Haiden, “Categorical Kahler Geometry” 2:30pm-2:45pm Break 2:45pm – 4:15pm Fabian Haiden, “Categorical Kahler Geometry” 4:30pm – 5:15pm Garret Alston, “Potential Functions of Non-exact fillings” Tuesday, November 29 – Day 2 10:30am –11:30am Conan Leung, “Remarks on SYZ” Lunch 1:00pm – 2:30pm Jingyu Zhao, “Homological mirror symmetry for open manifolds and Hodge theoretic invariants” 2:30pm-2:45pm Break 2:45pm – 4:15pm Hiro Lee Tanaka, “Floer theory through spectra” 4:30pm – 5:15pm Hansol Hong, “Mirror Symmetry for punctured Riemann surfaces and gluing construction” Wednesday, November 30 – Day 3 10:30am –11:30am Junwu Tu, “Homotopy L-infinity spaces and mirror symmetry” Lunch 1:00pm – 2:30pm Jingyu Zhao, “Homological mirror symmetry for open manifolds and Hodge theoretic invariants” 2:30-2:45pm Break 2:45pm – 4:15pm David Treumann, “Invariants of Lagrangians via microlocal sheaf theory” Thursday, December 1 – Day 4 10:30am –11:30am David Treumann, “Some examples in three dimensions” Lunch 1:00pm – 2:30pm Junwu Tu, “Homotopy L-infinity spaces and mirror symmetry” 2:30-2:45pm Break 2:45pm – 3:30pm Netanel Blaier, “The quantum Johnson homomorphism, and the symplectic mapping class group of 3-folds” * This event is sponsored by the Simons Foundation and CMSA Harvard University.
2020-2021 Colloquium, Wednesdays
During the Spring 2021 semester, and until further notice, all seminars will take place virtually.
The 2020-2021 Colloquium will take place every Wednesday from 9:00 to 10:00am ET virtually, using zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. Please email the seminar organizers to obtain a link. This year’s colloquium will be organized by Wei Gu and Sergiy Verstyuk. The schedule below will be updated as speakers are confirmed.
To learn how to attend, please fill out this form.
Information on previous colloquia can be found here.
Spring 2021:
Date Speaker Title/Abstract 1/27/2021 Evelyn Tang (Max Planck Institute for Dynamics and Self-Organization) Title: Topology protects chiral edge currents in stochastic systems Abstract: Living systems can exhibit time-scales much longer than those of the underlying components, as well as collective dynamical behavior. How such global behavior is subserved by stochastic constituents remains unclear. I will present two-dimensional stochastic networks that consist of out-of-equilibrium cycles at the molecular scale and support chiral edge currents in configuration space. I will discuss the topological properties of these networks and their uniquely non-Hermitian features such as exceptional points and vorticity. As these emergent edge currents are associated to macroscopic timescales and length scales, simply tuning a small number of parameters enables varied dynamical phenomena including a global clock, stochastic growth and shrinkage, and synchronization.
2/3/2021 André Luiz de Gouvêa (Northwestern) Title: The Brave Nu World Abstract: Neutrinos are the least understood of the fundamental particles that make up the so-called Standard Model of Particle Physics. Measuring neutrino properties and identifying how they inform our understanding of nature at the smallest distant scales is among the highest priorities of particle physics research today. I will discuss our current understanding of neutrinos, concentrating on the observation of neutrino oscillations and neutrino masses, along with all the open questions that came of these discoveries from the end of the 20th century.
2/10/2021 Mykhaylo Shkolnikov (Princeton) Title: Probabilistic approach to free boundary problems and applications Abstract: We will discuss a recently developed probabilistic approach to (singular) free boundary problems, such as the supercooled Stefan problem. The approach is based on a new notion of solution, referred to as probabilistic, which arises naturally in the context of large system limits of interacting particle systems. In the talk, I will give an example of how such interacting particle systems arise in applications (e.g., finance), then obtain a solution of a free boundary problem in the large system limit, and discuss how this solution can be analyzed mathematically (thereby answering natural questions about the systemic risk in financial systems and neural synchronization in the brain). The talk is based on recent and ongoing joint works with Sergey Nadtochiy, Francois Delarue, Jiacheng Zhang and Xiling Zhang
2/17/2021
9:00 – 10:00PM ETC. Seshadhri (UC Santa Cruz) Title: Studying the (in)effectiveness of low dimensional graph embeddings Abstract: Low dimensional graph embeddings are a fundamental and popular tool used for machine learning on graphs. Given a graph, the basic idea is to produce a low-dimensional vector for each vertex, such that “similarity” in geometric space corresponds to “proximity” in the graph. These vectors can then be used as features in a plethora of machine learning tasks, such as link prediction, community labeling, recommendations, etc. Despite many results emerging in this area over the past few years, there is less study on the core premise of these embeddings. Can such low-dimensional embeddings effectively capture the structure of real-world (such as social) networks? Contrary to common wisdom, we mathematically prove and empirically demonstrate that popular low-dimensional graph embeddings do not capture salient properties of real-world networks. We mathematically prove that common low-dimensional embeddings cannot generate graphs with both low average degree and large clustering coefficients, which have been widely established to be empirically true for real-world networks. Empirically, we observe that the embeddings generated by popular methods fail to recreate the triangle structure of real-world networks, and do not perform well on certain community labeling tasks. (Joint work with Ashish Goel, Caleb Levy, Aneesh Sharma, and Andrew Stolman.)
2/24/2021 David Ben-Zvi (U Texas) Title: Electric-Magnetic Duality for Periods and L-functions Abstract: I will describe joint work with Yiannis Sakellaridis and Akshay Venkatesh, in which ideas originating in quantum field theory are applied to a problem in number theory.
A fundamental aspect of the Langlands correspondence — the relative Langlands program — studies the representation of L-functions of Galois representations as integrals of automorphic forms. However, the data that naturally index the period integrals (spherical varieties for G) and the L-functions (representations of the dual group G^) don’t seem to line up.
We present an approach to this problem via the Kapustin-Witten interpretation of the [geometric] Langlands correspondence as electric-magnetic duality for 4-dimensional supersymmetric Yang-Mills theory. Namely, we rewrite the relative Langlands program as duality in the presence of supersymmetric boundary conditions. As a result the partial correspondence between periods and L-functions is embedded in a natural duality between Hamiltonian actions of the dual groups.3/3/2021 9:00pm ET
Omer Tamuz (Caltech) Title: Monotone Additive Statistics Abstract: How should a random quantity be summarized by a single number? We study mappings from random variables to real numbers, focussing on those with the following two properties: (1) monotonicity with respect to first-order stochastic dominance, and (2) additivity for sums of independent random variables. This problem turns out to be connected to the following question: Under what conditions on the random variables X and Y does there exist an independent Z so that X + Z first-order stochastically dominates Y + Z?
(Joint work with Tobias Fritz, Xiaosheng Mu, Luciano Pomatto and Philipp Strack.)
3/10/2021 9:00pm ET
Piotr Indyk (MIT) Title: Learning-Based Sampling and Streaming Abstract: Classical algorithms typically provide “one size fits all” performance, and do not leverage properties or patterns in their inputs. A recent line of work aims to address this issue by developing algorithms that use machine learning predictions to improve their performance. In this talk I will present two examples of this type, in the context of streaming and sampling algorithms. In particular, I will show how to use machine learning predictions to improve the performance of (a) low-memory streaming algorithms for frequency estimation (ICLR’19), and (b) sampling algorithms for estimating the support size of a distribution (ICLR’21). Both algorithms use an ML-based predictor that, given a data item, estimates the number of times the item occurs in the input data set. (The talk will cover material from papers co-authored with T Eden, CY Hsu, D Katabi, S Narayanan, R Rubinfeld, S Silwal, T Wagner and A Vakilian.
3/17/2021
9:00pm ETChiu-Chu Melissa Liu (Columbia) Title: Topological Recursion and Crepant Transformation Conjecture Abstract: The Crepant Transformation Conjecture (CTC), first proposed by Yongbin Ruan and later refined/generalized by others, relates Gromov-Witten (GW) invariants of K-equivalent smooth varieties or smooth Deligne-Mumford stacks. We will outline a proof of all-genus open and closed CTC for symplectic toric Calabi-Yau 3-orbifolds based on joint work with Bohan Fang, Song Yu, and Zhengyu Zong. Our proof relies on the Remodeling Conjecture (proposed by Bouchard-Klemm-Marino-Pasquetti and proved in full generality by Fang, Zong and the speaker) relating open and closed GW invariants of a symplectic toric Calabi-Yau 3-orbifold to invariants of its mirror curve defined by Chekhov-Eynard-Orantin Topological Recursion.
3/24/2021 Weinan E (Princeton) Title: Machine Learning and PDEs Abstract: I will discuss two topics:
(1) Machine learning-based algorithms and “regularity” theory for very high dimensional PDEs;
(2) Formulating machine learning as PDE (more precisely, integral-differental equation) problems.3/31/2021 Thore Graepel (DeepMind/UCL) Title: From AlphaGo to MuZero – Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model Abstract: Constructing agents with planning capabilities has long been one of the main challenges in the pursuit of artificial intelligence. Tree-based planning methods have enjoyed huge success in challenging domains, such as chess and Go, where a perfect simulator is available. However, in real-world problems the dynamics governing the environment are often complex and unknown. In this work we present the MuZero algorithm which, by combining a tree-based search with a learned model, achieves superhuman performance in a range of challenging and visually complex domains, without any knowledge of their underlying dynamics. MuZero learns a model that, when applied iteratively, predicts the quantities most directly relevant to planning: the reward, the action-selection policy, and the value function. When evaluated on 57 different Atari games – the canonical video game environment for testing AI techniques, in which model-based planning approaches have historically struggled – our new algorithm achieved a new state of the art. When evaluated on Go, chess and shogi, without any knowledge of the game rules, MuZero matched the superhuman performance of the AlphaZero algorithm that was supplied with the game rules.
4/7/2021 Kui Ren (Columbia) Title: Inversion via Optimization: Revisiting the Classical Least-Squares Formulation of Inverse Problems Abstract: The classical least-squares formulation of inverse problems has provided a successful framework for the computational solutions of those problems. In recent years, modifications and alternatives have been proposed to overcome some of the disadvantages of this classical formulation in dealing with new applications. This talk intends to provide an (likely biased) overview of the recent development in constructing new least-squares formulations for model and data-driven solutions of inverse problems.
4/14/2021 Siu-Cheong Lau (Boston U) Title: An algebro-geometric formulation of computing machines Abstract: Neural network in machine learning has obvious similarity with quiver representation theory. The main gap between the two subjects is that network functions produced from two isomorphic quiver representations are not equal, due to the presence of non-linear activation functions which are not equivariant under the automorphism group. This violates the important math/physics principle that isomorphic objects should produce the same results. In this talk, I will introduce a general formulation using moduli spaces of framed modules of (noncommutative) algebra and fix this gap. Metrics over the moduli space are crucial. I will also explain uniformization between spherical, Euclidean and hyperbolic moduli.
4/21/2021 Vasco Carvalho (Cambridge) Title: The Economy as a Complex Production Network
Abstract: A modern economy is an intricately linked web of specialized production units, each relying on the flow of inputs from their suppliers to produce their own output, which in turn is routed towards other downstream units. From this production network vantage point we: (i) present the theoretical foundations for the role of such input linkages as a shock propagation channel and as a mechanism for transforming micro-level shocks into macroeconomic, economy-wide fluctuations (ii) selectively survey both empirical and simulation-based studies that attempt to ascertain the relevance and quantitative bite of this argument and (time permitting) (iii) discuss a range of domains where this networked production view is currently being extended to.4/28/2021 9:00 – 10:00pm ET
Shamit Kachru (Stanford) Title: K3 Metrics from String Theory Abstract: Calabi-Yau manifolds have played a central role in important developments in string theory and mathematical physics. Famously, they admit Ricci flat metrics — but the proof of that fact is not constructive, and the metrics remain mysterious. K3 is perhaps the simplest non-trivial compact Calabi-Yau space. In this talk, I describe two different methods of constructing (smooth, Ricci flat) K3 metrics, and a string theory duality which relates them. The duality re-sums infinite towers of disc instanton corrections via a purely classical infinite-dimensional hyperkahler quotient construction, which can be practically implemented.
Fall 2020:
Date Speaker Title/Abstract 9/23/2020 David Kazhdan (Hebrew University) Title: On Applications of Algebraic Combinatorics to Algebraic Geometry Abstract: I present a derivation of a number of results on morphisms of a high Schmidt’s rank from a result in Algebraic Combinatorics. In particular will explain the flatness of such morphisms and show their fibers have rational singularities.
10/7/2020 10:00am
Mariangela Lisanti (Princeton University) Title: Mapping the Milky Way’s Dark Matter Halo with Gaia Abstract: The Gaia mission is in the process of mapping nearly 1% of the Milky Way’s stars—-nearly a billion in total. This data set is unprecedented and provides a unique view into the formation history of our Galaxy and its associated dark matter halo. I will review results based on the most recent Gaia data release, demonstrating how the evolution of the Galaxy can be deciphered from the stellar remnants of massive satellite galaxies that merged with the Milky Way early on. This analysis is an inherently “big data” problem, and I will discuss how we are leveraging machine learning techniques to advance our understanding of the Galaxy’s evolution. Our results indicate that the local dark matter is not in equilibrium, as typically assumed, and instead exhibits distinctive dynamics tied to the disruption of satellite galaxies. The updated dark matter map built from the Gaia data has ramifications for direct detection experiments, which search for the interactions of these particles in terrestrial targets.
10/14/2020 Gil Kalai (Hebrew University and IDC Herzliya) Title: Statistical, mathematical, and computational aspects of noisy intermediate-scale quantum computers Abstract: Noisy intermediate-scale quantum (NISQ) Computers hold the key for important theoretical and experimental questions regarding quantum computers. In the lecture I will describe some questions about mathematics, statistics and computational complexity which arose in my study of NISQ systems and are related to
a) My general argument “against” quantum computers,
b) My analysis (with Yosi Rinott and Tomer Shoham) of the Google 2019 “quantum supremacy” experiment.
Relevant papers:
Yosef Rinott, Tomer Shoham and Gil Kalai, Statistical aspects of the quantum supremacy demonstration, https://gilkalai.files.
wordpress.com/2019/11/stat-quantum2.pdf
Gil Kalai, The Argument against Quantum Computers, the Quantum Laws of Nature, and Google’s Supremacy Claims, https://gilkalai.files.
wordpress.com/2020/08/laws-blog2.pdf
Gil Kalai, Three puzzles on mathematics, computations, and games, https://gilkalai.files.
wordpress.com/2019/09/main-pr.pdf10/21/2020 Marta Lewicka (University of Pittsburgh) Title: Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models Abstract: We propose results that relate the following two contexts:
(i) Given a Riemann metric G on a thin plate, we study the question of what is its closest isometric immersion, with respect to the distance measured by energies E^h which are modifications of the classical nonlinear three-dimensional elasticity.
(ii) We perform the full scaling analysis of E^h, in the context of dimension reduction as the plate’s thickness h goes to 0, and derive the Gamma-limits of h^{-2n}E^h for all n. We show the energy quantization, in the sense that the even powers 2n of h are the only possible ones (all of them are also attained).
For each n, we identify conditions for the validity of the corresponding scaling, in terms of the vanishing of Riemann curvatures of G up to appropriate orders, and in terms of the matched isometry expansions. Problems that we discuss arise from the description of elastic materials displaying heterogeneous incompatibilities of strains that may be associated with growth, swelling, shrinkage, plasticity, etc. Our results display the interaction of calculus of variations,
geometry and mechanics of materials in the prediction of patterns and shape formation.10/28/2020 Jonathan Heckman (University of Pennsylvania) Title: Top Down Approach to Quantum Fields Abstract: Quantum Field theory (QFT) is the common language of particle physicists, cosmologists, and condensed matter physicists. Even so, many fundamental aspects of QFT remain poorly understood. I discuss some of the recent progress made in understanding QFT using the geometry of extra dimensions predicted by string theory, highlighting in particular the special role of seemingly “exotic” higher-dimensional supersymmetric QFTs with no length scales known as six-dimensional superconformal field theories (6D SCFTs). We have recently classified all examples of such 6D SCFTs, and are now using this to extra observables from strongly correlated systems in theories with more than four spacetime dimensions, as well as in spacetimes with four or fewer spacetime dimensions. Along the way, I will also highlight the remarkable interplay between physical and mathematical structures in the study of such systems
11/4/2020
9:00pm ETSurya Ganguli (Stanford) Title: Weaving together machine learning, theoretical physics, and neuroscience through mathematics Abstract: An exciting area of intellectual activity in this century may well revolve around a synthesis of machine learning, theoretical physics, and neuroscience. The unification of these fields will likely enable us to exploit the power of complex systems analysis, developed in theoretical physics and applied mathematics, to elucidate the design principles governing neural systems, both biological and artificial, and deploy these principles to develop better algorithms in machine learning. We will give several vignettes in this direction, including: (1) determining the best optimization problem to solve in order to perform regression in high dimensions; (2) finding exact solutions to the dynamics of generalization error in deep linear networks; (3) developing interpretable machine learning to derive and understand state of the art models of the retina; (4) analyzing and explaining the origins of hexagonal firing patterns in recurrent neural networks trained to path-integrate; (5) delineating fundamental theoretical limits on the energy, speed and accuracy with which non-equilibrium sensors can detect signals
Selected References:
M. Advani and S. Ganguli, Statistical mechanics of optimal convex inference in high dimensions, Physical Review X, 6, 031034, 2016.
M. Advani and S. Ganguli, An equivalence between high dimensional Bayes optimal inference and M-estimation, NeurIPS, 2016.
A.K. Lampinen and S. Ganguli, An analytic theory of generalization dynamics and transfer learning in deep linear networks, International Conference on Learning Representations (ICLR), 2019.
H. Tanaka, A. Nayebi, N. Maheswaranathan, L.M. McIntosh, S. Baccus, S. Ganguli, From deep learning to mechanistic understanding in neuroscience: the structure of retinal prediction, NeurIPS 2019.
S. Deny, J. Lindsey, S. Ganguli, S. Ocko, The emergence of multiple retinal cell types through efficient coding of natural movies, Neural Information Processing Systems (NeurIPS) 2018.
B. Sorscher, G. Mel, S. Ganguli, S. Ocko, A unified theory for the origin of grid cells through the lens of pattern formation, NeurIPS 2019.
Y. Bahri, J. Kadmon, J. Pennington, S. Schoenholz, J. Sohl-Dickstein, and S. Ganguli, Statistical mechanics of deep learning, Annual Reviews of Condensed Matter Physics, 2020.
S.E. Harvey, S. Lahiri, and S. Ganguli, A universal energy accuracy tradeoff in nonequilibrium cellular sensing, https://arxiv.org/abs/2002.1056711/11/2020 Kevin Buzzard (Imperial College London) Title: Teaching proofs to computers Abstract: A mathematical proof is a sequence of logical statements in a precise language, obeying some well-defined rules. In that sense it is very much like a computer program. Various computer tools have appeared over the last 50 years which take advantage of this analogy by turning the mathematical puzzle of constructing a proof of a theorem into a computer game. The newest tools are now capable of understanding some parts of modern research mathematics. In spite of this, these tools are not used in mathematics departments, perhaps because they are not yet capable of telling mathematicians *something new*.
I will give an overview of the Lean theorem prover, showing what it can currently do. I will also talk about one of our goals: using Lean to make practical tools which will be helpful for future researchers in pure mathematics.11/18/2020 Jose A. Scheinkman (Columbia) Title: Re-pricing avalanches Abstract: Monthly aggregate price changes exhibit chronic fluctuations but the aggregate shocks that drive these fluctuations are often elusive. Macroeconomic models often add stochastic macro-level shocks such as technology shocks or monetary policy shocks to produce these aggregate fluctuations. In this paper, we show that a state-dependent pricing model with a large but finite number of firms is capable of generating large fluctuations in the number of firms that adjust prices in response to an idiosyncratic shock to a firm’s cost of price adjustment. These fluctuations, in turn, cause fluctuations in aggregate price changes even in the absence of aggregate shocks. (Joint work with Makoto Nirei.)
11/25/2020 10:45am
Eric J. Heller (Harvard) Title: Branched Flow Abstract: In classical and quantum phase space flow, there exists a regime of great physical relevance that is belatedly but rapidly generating a new field. In evolution under smooth, random, weakly deflecting but persistent perturbations, a remarkable regime develops, called branched flow. Lying between the first cusp catastrophes at the outset, leading to fully chaotic statistical flow much later, lies the visually beautiful regime of branched flow. It applies to tsunami wave propagation, freak wave formation, light propagation, cosmic microwaves arriving from pulsars, electron flow in metals and devices, sound propagation in the atmosphere and oceans, the large scale structure of the universe, and much more. The mathematical structure of this flow is only partially understood, involving exponential instability coexisting with “accidental” stability. The flow is qualitatively universal, but this has not been quantified. Many questions arise, including the scale(s) of the random medium, and the time evolution of manifolds and “fuzzy” manifolds in phase space. The classical-quantum (ray-wave) correspondence in this flow is only partially understood. This talk will be an introduction to the phenomenon, both visual and mathematical, emphasizing unanswered questions
12/2/2020 Douglas Arnold (U of Minnesota) Title: Preserving geometry in numerical discretization Abstract: An important design principle for numerical methods for differential equations is that the discretizations preserve key geometric, topological, and algebraic structures of the original differential system. For ordinary differential equations, such geometric integrators were developed at the end of the last century, enabling stunning computations in celestial mechanics and other applications that would have been impossible without them. Since then, structure-preserving discretizations have been developed for partial differential equations. One of the prime examples has been the finite element exterior calculus or FEEC, in which the structures to preserve are related to Hilbert complexes underlying the PDEs, the de Rham complex being a canonical example. FEEC has led to highly successful new numerical methods for problems in fluid mechanics, electromagnetism, and other applications which relate to the de Rham complex. More recently, new tools have been developed which extend the applications of FEEC far beyond the de Rham complex, leading to progress in discretizations of problems from solid mechanics, materials science, and general relativity.
12/9/2020 Manuel Blum and Lenore Blum (Carnegie Mellon) Title: What can Theoretical Computer Science Contribute to the Discussion of Consciousness? Abstract: The quest to understand consciousness, once the purview of philosophers and theologians, is now actively pursued by scientists of many stripes. We study consciousness from the perspective of theoretical computer science. This is done by formalizing the Global Workspace Theory (GWT) originated by cognitive neuroscientist Bernard Baars and further developed by him, Stanislas Dehaene, and others. We give a precise formal definition of a Conscious Turing Machine (CTM), also called Conscious AI, in the spirit of Alan Turing’s simple yet powerful definition of a computer. We are not looking for a complex model of the brain nor of cognition but for a simple model of (the admittedly complex concept of) consciousness.
After formally defining CTM, we give a formal definition of consciousness in CTM. We then suggest why the CTM has the feeling of consciousness. The reasonableness of the definitions and explanations can be judged by how well they agree with commonly accepted intuitive concepts of human consciousness, the range of related concepts that the model explains easily and naturally, and the extent of the theory’s agreement with scientific evidenceJDG 2017 Conference, April 28 – May 2, 2017
In celebration of the Journal of Differential Geometry’s 50th anniversary, the Harvard Math Department will be hosting the Tenth Conference on Geometry and Topology (JDG 2017) from April 28 – May 2, 2017.
Registration and additional information on the conference can be found at http://abel.harvard.edu/jdg/index.html.
Confirmed Speakers
- Mina Aganagic, UC Berkeley
- Denis Auroux, UC Berkeley
- Caucher Birkar, University of Cambridge
- Huai-Dong Cao, Lehigh University
- Tristan Collins, Harvard University
- Camillo De Lellis, ETH Zurich
- Jean-Pierre Demailly, Grenoble Alpes University
- Simon Donaldson, Stony Brook University
- Dan Freed, University of Texas at Austin
- Kenji Fukaya, Stony Brook University
- David Gabai, Princeton University
- Larry Guth, Massachusetts Institute of Technology
- Richard Hamilton, Columbia University
- Yujiro Kawamata, University of Tokyo
- Frances Kirwan, Oxford University
- Blaine Lawson, Stony Brook University
- Jun Li, Stanford University
- Si Li, Tsinghua University
- Bong Lian, Brandeis University
- Chiu-Chu Melissa Liu, Columbia University
- Ciprian Manolescu, University of California, Los Angeles
- Fernando Marques, Princeton University
- William Meeks, University of Massachusetts Amherst
- William Minicozzi, Massachusetts Institute of Technology
- John Pardon, Princeton University
- Duong Phong, Columbia University
- Alena Pirutka, Courant Institute of New York University
- Richard Schoen, University of California, Irvine
- Artan Sheshmani, QGM Aarhus University/Harvard University
- Cliff Taubes, Harvard University
- Cumrun Vafa, Harvard University
- Mu-Tao Wang, Columbia University
- Shing-Tung Yau, Harvard University
- Steve Zelditch, Northwestern Univeristy
* This event is co-sponsored by Lehigh University and partially supported by the National Science Foundation.
Electric-magnetic duality and the Geometric Langlands duality
Title: Electric-magnetic duality and the Geometric Langlands duality
Abstract: I will give a pedagogical review of the connection between electric-magnetic duality and the Geometric Langlands duality.
Working Conference on Materials and Data Analysis, March 27-30, 2017
The Center of Mathematical Sciences and Applications will be hosting a 5-day working Conference on Materials and Data Analysis and related areas, March 27-30, 2017. The conference will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
Photos of the event can be found on CMSA’s Blog.
Participants:
- Ryan P. Adams, Harvard University
- Jörg Behler, University of Göttingen
- Kieron Burke, University of California, Irvine
- Lucy Colwell, University of Cambridge
- Gábor Csányi, University of Cambridge
- Ekin Doğuş Çubuk, Stanford University
- Leslie Greengard, Courant Institute of Mathematical Sciences, New York University
- Petros Koumoutsakos, Radcliffe Institute for Advanced Study, Harvard University
- Govind Menon, Brown University
- Evan Reed, Stanford University
- Patrick Riley, Google
- Matthias Rupp, Fitz Haber Institute of the Max Planck Society
- Sadasivan Shankar, Harvard University
- Dennis Sheberla, Harvard University
Schedule:
Monday, March 27
Time Speaker Title 8:30am – 9:00am Breakfast 9:00am – 10:00am Kieron Burke, University of California, Irvine Background in DFT and electronic structure calculations 10:00am – 11:00am Kieron Burke, University of California, Irvine The density functionals machines can learn
11:00am – 12:00pm Sadasivan Shankar, Harvard University A few key principles for applying Machine Learning to Materials (or Complex Systems) — Scientific and Engineering Perspectives Tuesday, March 28
Time Speaker Title 8:30am – 9:00am Breakfast 9:00am – 10:00am Ryan Adams, Harvard TBA 10:00am – 11:00am Gábor Csányi, University of Cambridge Interatomic potentials using machine learning: accuracy, transferability and chemical diversity
11:00am – 1:00pm Lunch Break 1:00pm – 2:00pm Evan Reed, Stanford University TBA Wednesday, March 29
Time Speaker Title 8:30am – 9:00am Breakfast 9:00am – 10:00am Patrick Riley, Google The Message Passing Neural Network framework and its application to molecular property prediction 10:00am – 11:00am Jörg Behler, University of Göttingen TBA 11:00am – 12:00pm Ekin Doğuş Çubuk, Stanford Univers TBA 4:00pm Leslie Greengard, Courant Institute Inverse problems in acoustic scattering and cryo-electron microscopy CMSA Colloquium
Thursday, March 30
Time Speaker Title 8:30am – 9:00am Breakfast 9:00am – 10:00am Matthias Rupp, Fitz Haber Institute of the Max Planck Society TBA 10:00am – 11:00am Petros Koumoutsakos, Radcliffe Institute for Advanced Study, Harvard TBA 11:00am – 1:00pm Lunch Break 1:00pm – 2:00pm Dennis Sheberla, Harvard University Rapid discovery of functional molecules by a high-throughput virtual screening Workshop on Discrete and Topological Models for Effective Field Theories, January 9-13, 2017
The Center of Mathematical Sciences and Applications will be hosting a Workshop on “Discrete and Topological Models for Effective Field Theories,” January 9-13, 2017. The workshop will be hosted in G02 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
Titles, abstracts and schedule will be provided nearer to the event.
Participants:
Dan Freed, UT Austin
Anton Kapustin, California Institute of Technology
Alexei Y. Kitaev, California Institute of Technology
Greg Moore, Rutgers University
Constantin Teleman, University of Oxford
Organizers:
Mike Hopkins, Shing-Tung Yau
* This event is sponsored by CMSA Harvard University.
Working Conference on Covariance Analysis in Biology, May 1-4, 2017
The Center of Mathematical Sciences and Applications will be hosting a working Conference on Covariance Analysis in Biology, May 1-4, 2017. The conference will be hosted in Room G10 of the CMSA Building located at 20 Garden Street, Cambridge, MA 02138.
This event is open and free. If you would like to attend, please register here to help us keep a headcount. A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Speakers:
Orr Ashenberg, Fred Hutchinson Cancer Research Center
John Barton, Massachusetts Institute of Technology
Simona Cocco, Laboratoire de Physique Statistique de l’ENS
Sean Eddy, Harvard University
Efthimios Kaxiras, Harvard University
Michael Laub, Massachusetts Institute of Technology
Debora S. Marks, Harvard University
Govind Menon, Brown University
Rémi Monasson, Laboratoire de Physique Théorique de l’ENS
Andrew Murray, Harvard University
Ilya Nemenman, Emory College
Chris Sander, Dana-Farber Cancer Institute, Harvard Medical School
Dave Thirumalai, University of Texas at Austin
Martin Weigt, IBPS, Université Pierre et Marie Curie
Matthieu Wyart, EPFL
More speakers will be confirmed soon.
Schedule:
(Please click here for a downloadable version of the schedule.)
Please note that the schedule for both days is currently tentative and is subject to change.
May 1, Monday
Time Speaker Topic 9:00-10:00am Sean Eddy TBA 10:00-11:00am Mike Laub TBA 11:00am-12:00pm Ilya Nemenman TBA May 2, TuesdayTime Speaker Topic 9:00-10:00am Orr Ashenberg TBA 10:00-11:00am Debora Marks TBA 11:00am-12:00pm Martin Weigt TBA 4:30pm-5:30pm Simona Cocco CMSA Colloquia May 3, WednesdayTime Speaker Topic 9:00-10:00am Andrew Murray TBA 10:00-11:00am Matthieu Wyart TBA 11:00am-12:00pm Rémi Monasson TBA May 4, ThursdayTime Speaker Topic 9:00-10:00am David Thirumalai TBA 10:00-11:00am Chris Sander TBA 11:00am-12:00pm John Barton TBA Organizers:
Michael Brenner, Lucy Colwell, Elena Rivas, Eugene Shakhnovich
* This event is sponsored by CMSA Harvard University.
A Celebration of Symplectic Geometry: 15 Years of JSG, June 5-6, 2017
In celebration of the Journal of Symplectic Geometry’s 15th anniversary, the Center of Mathematical Sciences and Applications will be hosting A Celebration of Symplectic Geometry: 15 Years of JSG on June 5-6, 2017.
To register for this event, please click here.
Confirmed speakers:
- Roger Casals, MIT
- Chen He, Northeastern University
- Yael Karshon, University of Toronto
- Ailsa Keating, Institute of Advanced Study
- Eckhard Meinrenken, University of Toronto
- Ana Rita Pires, Fordham University
- Sobhan Seyfaddini, Institute of Advanced Study
- Alejandro Uribe, University of Michigan
- Jonathan Weitsman, Northeastern University
The conference is co-organized by Denis Auroux and Victor Guillemin. Additional information on the conference will be announced closer to the event.
For a list of lodging options convenient to the Center, please see our recommended lodgings page.
Schedule:
The schedule for both days is currently tentative and is subject to change. A pdf version of the schedule can also be downloaded here.
June 5, Monday (Full day)
Time Speaker Topic 8:30am – 9:0am Breakfast 9:00am – 10:00am Jonathan Weitsman Title: On the geometric quantization of (some) Poisson manifolds 10:30am – 11:30am Eckhard Meinrenken Title: On Hamiltonian loop group spaces Abstract: Let G be a compact Lie group. We explain a construction of an LG-equivariant spinor module over any Hamiltonian loop group space with proper moment map. It may be regarded as its `canonical spin-c structure’. We show how to reduce to finite dimensions, resulting in actual spin-s structure on transversals, as well as twisted spin-c structures for the associated quasi-hamiltonian space. This is based on joint work with Yiannis Loizides and Yanli Song.
11:30am – 1:30pm Break 1:30pm – 2:30pm Ana Rita Pires Title: Infinite staircases in symplectic embedding problems Abstract: McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase related to the odd index Fibonacci numbers. Infinite staircases have been shown to exist also in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid E(2,3). I will describe how we use ECH capacities, lattice point counts and Ehrhart theory to show that infinite staircases exist for these and a few other target manifolds, as well as to conjecture that these are the only such target manifolds. This is a joint work with Cristofaro-Gardiner, Holm and Mandini.
3:00pm – 4:00pm Sobhan Seyfaddini Title: Rigidity of conjugacy classes in groups of area-preserving homeomorphisms Abstract: Motivated by understanding the algebraic structure of groups of area-preserving homeomorphims F. Beguin, S. Crvoisier, and F. Le Roux were lead to the following question: Can the conjugacy class of a Hamiltonian homeomorphism be dense? We will show that one can rule out existence of dense conjugacy classes by simply counting fixed points. This is joint work with Le Roux and Viterbo.
4:30pm – 5:30pm Roger Casals Title: Differential Algebra of Cubic Graphs
Abstract: In this talk we will associate a combinatorial dg-algebra to a cubic planar graph. This algebra is defined by counting binary sequences, which we introduce, and we shall provide explicit computations and examples. From there we study the Legendrian surfaces behind these constructions, including Legendrian surgeries, the count of Morse flow trees involved in contact homology, and the relation to microlocal sheaves. Time permitting, I will explain a connection to spectral networks.VideoJune 6, Tuesday (Full day)
Time Speaker Topic 8:30am – 9:00am Breakfast 9:00am – 10:00am Alejandro Uribe Title: Semi-classical wave functions associated with isotropic submanifolds of phase space Abstract: After reviewing fundamental ideas on the quantum-classical correspondence, I will describe how to associate spaces of semi-classical wave functions to isotropic submanifolds of phase space satisfying a Bohr-Sommerfeld condition. Such functions have symbols that are symplectic spinors, and they satisfy a symbol calculus under the action of quantum observables. This is the semi-classical version of the Hermite distributions of Boutet the Monvel and Guillemin, and it is joint work with Victor Guillemin and Zuoqin Wang. I will inlcude applications and open questions.
10:30am – 11:30am Alisa Keating Title: Symplectomorphisms of exotic discs Abstract: It is a theorem of Gromov that the group of compactly supported symplectomorphisms of R^4, equipped with the standard symplectic form, is contractible. While nothing is known in higher dimensions for the standard symplectic form, we show that for some exotic symplectic forms on R^{4n}, for all but finitely n, there exist compactly supported symplectomorphisms that are smoothly non-trivial. The principal ingredients are constructions of Milnor and Munkres, a symplectic and contact version of the Gromoll filtration, and Borman, Eliashberg and Murphy’s work on existence of over-twisted contact structures. Joint work with Roger Casals and Ivan Smith.
11:30am – 1:30pm Break 1:30pm – 2:30pm Chen He Title: Morse theory on b-symplectic manifolds Abstract: b-symplectic (or log-symplectic) manifolds are Poisson manifolds equipped with symplectic forms of logarithmic singularity. Following Guillemin, Miranda, Pires and Scott’s introduction of Hamiltonian group actions on b-symplectic manifolds, we will survey those classical results of Hamiltonian geometry to the b-symplectic case.
3:00pm – 4:00pm Yael Karshon Title: Geometric quantization with metaplectic-c structures Abstract: I will present a variant of the Kostant-Souriau geometric quantization procedure that uses metaplectic-c structures to incorporate the “half form correction” into the prequantization stage. This goes back to the late 1970s but it is not widely known and it has the potential to generalize and improve upon recent works on geometric quantization.
2017 Big Data Conference
1 Oxford Street, Cambridge MA 02138The Center of Mathematical Sciences and Applications will be hosting a conference on Big Data from August 18 – 19, 2017, in Hall D of the Science Center at Harvard University.
The Big Data Conference features many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics. This is the third conference on Big Data the Center will host as part of our annual events, and is co-organized by Richard Freeman, Scott Kominers, Jun Liu, Horng-Tzer Yau and Shing-Tung Yau.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Please note that lunch will not be provided during the conference, but a map of Harvard Square with a list of local restaurants can be found by clicking Map & Restaurants.
Confirmed Speakers:
- Mohammad Akbarpour, Stanford University
- Albert-László Barabási, Northeastern University
- Noureddine El Karoui, University of California, Berkeley
- Ravi Jagadeesan, Harvard University
- Lucas Janson, Harvard University
- Tracy Ke, University of Chicago
- Tze Leung Lai, Stanford University
- Annie Liang, University of Pennsylvania
- Marena Lin, Harvard University
- Nikhil Naik, Harvard University
- Alex Peysakhovich, Facebook
- Natesh Pillai, Harvard University
- Jann Spiess, Harvard University
- Bradly Stadie, Open AI, University of California, Berkeley
- Zak Stone, Google
- Hau-Tieng Wu, University of Toronto
- Sifan Zhou, Xiamen University
Following the conference, there will be a two-day workshop from August 20-21. The workshop is organized by Scott Kominers, and will feature:
- Jörn Boehnke, Harvard University
- Nikhil Naik, Harvard University
- Bradly Stadie, Open AI, University of California, Berkeley
Conference Schedule
A PDF version of the schedule below can also be downloaded here.
August 18, Friday (Full day)
Time Speaker Topic 8:30 am – 9:00 am Breakfast 9:00 am – 9:40 am Mohammad Akbarpour Title: Information aggregation in overlapping generations and the emergence of experts Abstract: We study a model of social learning with “overlapping generations”, where agents meet others and share data about an underlying state over time. We examine under what conditions the society will produce individuals with precise knowledge about the state of the world. There are two information sharing regimes in our model: Under the full information sharing technology, individuals exchange the information about their point estimates of an underlying state, as well as their sources (or the precision of their signals) and update their beliefs by taking a weighted average. Under the limited information sharing technology, agents only observe the information about the point estimates of those they meet, and update their beliefs by taking a weighted average, where weights can depend on the sequence of meetings, as well as the labels. Our main result shows that, unlike most social learning settings, using such linear learning rules do not guide the society (or even a fraction of its members) to learn the truth, and having access to, and exploiting knowledge of the precision of a source signal are essential for efficient social learning (joint with Amin Saberi & Ali Shameli).
9:40 am – 10:20 am Lucas Janson Title: Model-Free Knockoffs For High-Dimensional Controlled Variable Selection Abstract: Many contemporary large-scale applications involve building interpretable models linking a large set of potential covariates to a response in a nonlinear fashion, such as when the response is binary. Although this modeling problem has been extensively studied, it remains unclear how to effectively control the fraction of false discoveries even in high-dimensional logistic regression, not to mention general high-dimensional nonlinear models. To address such a practical problem, we propose a new framework of model-free knockoffs, which reads from a different perspective the knockoff procedure (Barber and Candès, 2015) originally designed for controlling the false discovery rate in linear models. The key innovation of our method is to construct knockoff variables probabilistically instead of geometrically. This enables model-free knockoffs to deal with arbitrary (and unknown) conditional models and any dimensions, including when the dimensionality p exceeds the sample size n, while the original knockoffs procedure is constrained to homoscedastic linear models with n greater than or equal to p. Our approach requires the design matrix be random (independent and identically distributed rows) with a covariate distribution that is known, although we show our procedure to be robust to unknown/estimated distributions. As we require no knowledge/assumptions about the conditional distribution of the response, we effectively shift the burden of knowledge from the response to the covariates, in contrast to the canonical model-based approach which assumes a parametric model for the response but very little about the covariates. To our knowledge, no other procedure solves the controlled variable selection problem in such generality, but in the restricted settings where competitors exist, we demonstrate the superior power of knockoffs through simulations. Finally, we apply our procedure to data from a case-control study of Crohn’s disease in the United Kingdom, making twice as many discoveries as the original analysis of the same data.
10:20 am – 10:50 am Break 10:50 pm – 11:30 pm Noureddine El Karoui Title: Random matrices and high-dimensional statistics: beyond covariance matrices Abstract: Random matrices have played a central role in understanding very important statistical methods linked to covariance matrices (such as Principal Components Analysis, Canonical Correlation Analysis etc…) for several decades. In this talk, I’ll show that one can adopt a random-matrix-inspired point of view to understand the performance of other widely used tools in statistics, such as M-estimators, and very common methods such as the bootstrap. I will focus on the high-dimensional case, which captures well the situation of “moderately” difficult statistical problems, arguably one of the most relevant in practice. In this setting, I will show that random matrix ideas help upend conventional theoretical thinking (for instance about maximum likelihood methods) and highlight very serious practical problems with resampling methods.
11:30 am – 12:10 pm Nikhil Naik Title: Understanding Urban Change with Computer Vision and Street-level Imagery Abstract: Which neighborhoods experience physical improvements? In this work, we introduce a computer vision method to measure changes in the physical appearances of neighborhoods from time-series street-level imagery. We connect changes in the physical appearance of five US cities with economic and demographic data and find three factors that predict neighborhood improvement. First, neighborhoods that are densely populated by college-educated adults are more likely to experience physical improvements. Second, neighborhoods with better initial appearances experience, on average, larger positive improvements. Third, neighborhood improvement correlates positively with physical proximity to the central business district and to other physically attractive neighborhoods. Together, our results illustrate the value of using computer vision methods and street-level imagery to understand the physical dynamics of cities.
(Joint work with Edward L. Glaeser, Cesar A. Hidalgo, Scott Duke Kominers, and Ramesh Raskar.)
12:10 pm – 12:25 pm Video #1 Data Science Lightning Talks 12:25 pm – 1:30 pm Lunch 1:30 pm – 2:10 pm Tracy Ke Title: A new SVD approach to optimal topic estimation Abstract: In the probabilistic topic models, the quantity of interest—a low-rank matrix consisting of topic vectors—is hidden in the text corpus matrix, masked by noise, and Singular Value Decomposition (SVD) is a potentially useful tool for learning such a low-rank matrix. However, the connection between this low-rank matrix and the singular vectors of the text corpus matrix are usually complicated and hard to spell out, so how to use SVD for learning topic models faces challenges.
We overcome the challenge by revealing a surprising insight: there is a low-dimensional simplex structure which can be viewed as a bridge between the low-rank matrix of interest and the SVD of the text corpus matrix, and which allows us to conveniently reconstruct the former using the latter. Such an insight motivates a new SVD-based approach to learning topic models.
For asymptotic analysis, we show that under a popular topic model (Hofmann, 1999), the convergence rate of the l1-error of our method matches that of the minimax lower bound, up to a multi-logarithmic term. In showing these results, we have derived new element-wise bounds on the singular vectors and several large deviation bounds for weakly dependent multinomial data. Our results on the convergence rate and asymptotical minimaxity are new. We have applied our method to two data sets, Associated Process (AP) and Statistics Literature Abstract (SLA), with encouraging results. In particular, there is a clear simplex structure associated with the SVD of the data matrices, which largely validates our discovery.
2:10 pm – 2:50 pm Albert-László Barabási Title: Taming Complexity: From Network Science to Controlling Networks Abstract: The ultimate proof of our understanding of biological or technological systems is reflected in our ability to control them. While control theory offers mathematical tools to steer engineered and natural systems towards a desired state, we lack a framework to control complex self-organized systems. Here we explore the controllability of an arbitrary complex network, identifying the set of driver nodes whose time-dependent control can guide the system’s entire dynamics. We apply these tools to several real networks, unveiling how the network topology determines its controllability. Virtually all technological and biological networks must be able to control their internal processes. Given that, issues related to control deeply shape the topology and the vulnerability of real systems. Consequently unveiling the control principles of real networks, the goal of our research, forces us to address series of fundamental questions pertaining to our understanding of complex systems.
2:50 pm – 3:20 pm Break 3:20 pm – 4:00 pm Marena Lin Title: Optimizing climate variables for human impact studies Abstract: Estimates of the relationship between climate variability and socio-economic outcomes are often limited by the spatial resolution of the data. As studies aim to generalize the connection between climate and socio-economic outcomes across countries, the best available socio-economic data is at the national level (e.g. food production quantities, the incidence of warfare, averages of crime incidence, gender birth ratios). While these statistics may be trusted from government censuses, the appropriate metric for the corresponding climate or weather for a given year in a country is less obvious. For example, how do we estimate the temperatures in a country relevant to national food production and therefore food security? We demonstrate that high-resolution spatiotemporal satellite data for vegetation can be used to estimate the weather variables that may be most relevant to food security and related socio-economic outcomes. In particular, satellite proxies for vegetation over the African continent reflect the seasonal movement of the Intertropical Convergence Zone, a band of intense convection and rainfall. We also show that agricultural sensitivity to climate variability differs significantly between countries. This work is an example of the ways in which in-situ and satellite-based observations are invaluable to both estimates of future climate variability and to continued monitoring of the earth-human system. We discuss the current state of these records and potential challenges to their continuity.
4:00 pm – 4:40 pm Alex Peysakhovich Title: Building a cooperator Abstract: A major goal of modern AI is to construct agents that can perform complex tasks. Much of this work deals with single agent decision problems. However, agents are rarely alone in the world. In this talk I will discuss how to combine ideas from deep reinforcement learning and game theory to construct artificial agents that can communicate, collaborate and cooperate in productive positive sum interactions.
4:40 pm – 5:20 pm Tze Leung Lai Title: Gradient boosting: Its role in big data analytics, underlying mathematical theory, and recent refinements Abstract: We begin with a review of the history of gradient boosting, dating back to the LMS algorithm of Widrow and Hoff in 1960 and culminating in Freund and Schapire’s AdaBoost and Friedman’s gradient boosting and stochastic gradient boosting algorithms in the period 1999-2002 that heralded the big data era. The role played by gradient boosting in big data analytics, particularly with respect to deep learning, is then discussed. We also present some recent work on the mathematical theory of gradient boosting, which has led to some refinements that greatly improves the convergence properties and prediction performance of the methodology.
August 19, Saturday (Full day)
Time Speaker Topic 8:30 am – 9:00 am Breakfast 9:00 am – 9:40 am Natesh Pillai Title: Accelerating MCMC algorithms for Computationally Intensive Models via Local Approximations Abstract: We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach introduces local approximations of these models into the Metropolis–Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler’s exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the exact posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. Our theoretical results reinforce the key observation underlying this article: when the likelihood has some local regularity, the number of model evaluations per Markov chain Monte Carlo (MCMC) step can be greatly reduced without biasing the Monte Carlo average. Numerical experiments demonstrate multiple order-of-magnitude reductions in the number of forward model evaluations used in representative ordinary differential equation (ODE) and partial differential equation (PDE) inference problems, with both synthetic and real data.
9:40 am – 10:20 am Ravi Jagadeesan Title: Designs for estimating the treatment effect in networks with interference Abstract: In this paper we introduce new, easily implementable designs for drawing causal inference from randomized experiments on networks with interference. Inspired by the idea of matching in observational studies, we introduce the notion of considering a treatment assignment as a quasi-coloring” on a graph. Our idea of a perfect quasi-coloring strives to match every treated unit on a given network with a distinct control unit that has identical number of treated and control neighbors. For a wide range of interference functions encountered in applications, we show both by theory and simulations that the classical Neymanian estimator for the direct effect has desirable properties for our designs. This further extends to settings where homophily is present in addition to interference.
10:20 am – 10:50 am Break 10:50 am – 11:30 am Annie Liang Title: The Theory is Predictive, but is it Complete? An Application to Human Generation of Randomness Abstract: When we test a theory using data, it is common to focus on correctness: do the predictions of the theory match what we see in the data? But we also care about completeness: how much of the predictable variation in the data is captured by the theory? This question is difficult to answer, because in general we do not know how much “predictable variation” there is in the problem. In this paper, we consider approaches motivated by machine learning algorithms as a means of constructing a benchmark for the best attainable level of prediction. We illustrate our methods on the task of predicting human-generated random sequences. Relative to a theoretical machine learning algorithm benchmark, we find that existing behavioral models explain roughly 15 percent of the predictable variation in this problem. This fraction is robust across several variations on the problem. We also consider a version of this approach for analyzing field data from domains in which human perception and generation of randomness has been used as a conceptual framework; these include sequential decision-making and repeated zero-sum games. In these domains, our framework for testing the completeness of theories provides a way of assessing their effectiveness over different contexts; we find that despite some differences, the existing theories are fairly stable across our field domains in their performance relative to the benchmark. Overall, our results indicate that (i) there is a significant amount of structure in this problem that existing models have yet to capture and (ii) there are rich domains in which machine learning may provide a viable approach to testing completeness (joint with Jon Kleinberg and Sendhil Mullainathan).
11:30 am – 12:10 pm Zak Stone Title: TensorFlow: Machine Learning for Everyone Abstract: We’ve witnessed extraordinary breakthroughs in machine learning over the past several years. What kinds of things are possible now that weren’t possible before? How are open-source platforms like TensorFlow and hardware platforms like GPUs and Cloud TPUs accelerating machine learning progress? If these tools are new to you, how should you get started? In this session, you’ll hear about all of this and more from Zak Stone, the Product Manager for TensorFlow on the Google Brain team.
12:10 pm – 1:30 pm Lunch 1:30 pm – 2:10 pm Jann Spiess Title: (Machine) Learning to Control in Experiments Abstract: Machine learning focuses on high-quality prediction rather than on (unbiased) parameter estimation, limiting its direct use in typical program evaluation applications. Still, many estimation tasks have implicit prediction components. In this talk, I discuss accounting for controls in treatment effect estimation as a prediction problem. In a canonical linear regression framework with high-dimensional controls, I argue that OLS is dominated by a natural shrinkage estimator even for unbiased estimation when treatment is random; suggest a generalization that relaxes some parametric assumptions; and contrast my results with that for another implicit prediction problem, namely the first stage of an instrumental variables regression.
2:10 pm – 2:50 pm Bradly Stadie Title: Learning to Learn Quickly: One-Shot Imitation and Meta Learning Abstract: Many reinforcement learning algorithms are bottlenecked by data collection costs and the brittleness of their solutions when faced with novel scenarios.
We will discuss two techniques for overcoming these shortcomings. In one-shot imitation, we train a module that encodes a single demonstration of a desired behavior into a vector containing the essence of the demo. This vector can subsequently be utilized to recover the demonstrated behavior. In meta-learning, we optimize a policy under the objective of learning to learn new tasks quickly. We show meta-learning methods can be accelerated with the use of auxiliary objectives. Results are presented on grid worlds, robotics tasks, and video game playing tasks.2:50 pm – 3:20 pm Break 3:20 pm – 4:00 pm Hau-Tieng Wu Title: When Medical Challenges Meet Modern Data Science Abstract: Adaptive acquisition of correct features from massive datasets is at the core of modern data analysis. One particular interest in medicine is the extraction of hidden dynamics from a single observed time series composed of multiple oscillatory signals, which could be viewed as a single-channel blind source separation problem. The mathematical and statistical problems are made challenging by the structure of the signal which consists of non-sinusoidal oscillations with time varying amplitude/frequency, and by the heteroscedastic nature of the noise. In this talk, I will discuss recent progress in solving this kind of problem by combining the cepstrum-based nonlinear time-frequency analysis and manifold learning technique. A particular solution will be given along with its theoretical properties. I will also discuss the application of this method to two medical problems – (1) the extraction of a fetal ECG signal from a single lead maternal abdominal ECG signal; (2) the simultaneous extraction of the instantaneous heart/respiratory rate from a PPG signal during exercise; (3) (optional depending on time) an application to atrial fibrillation signals. If time permits, the clinical trial results will be discussed.
4:00 pm – 4:40 pm Sifan Zhou Title: Citing People Like Me: Homophily, Knowledge Spillovers, and Continuing a Career in Science Abstract: Forward citation is widely used to measure the scientific merits of articles. This research studies millions of journal article citation records in life sciences from MEDLINE and finds that authors of the same gender, the same ethnicity, sharing common collaborators, working in the same institution, or being geographically close are more likely (and quickly) to cite each other than predicted by their proportion among authors working on the same research topics. This phenomenon reveals how social and geographic distances influence the quantity and speed of knowledge spillovers. Given the importance of forward citations in academic evaluation system, citation homophily potentially put authors from minority group at a disadvantage. I then show how it influences scientists’ chances to survive in the academia and continue publishing. Based on joint work with Richard Freeman.
To view photos and video interviews from the conference, please visit the CMSA blog.
Math Science Lectures in Honor of Raoul Bott: Mina Aganagic
1 Oxford Street, Cambridge MA 02138On April 9 and 10, 2019 the CMSA hosted two lectures by Mina Aganagic (UC Berkeley). This was the second annual Math Science Lecture Series held in honor of Raoul Bott.
The lectures took place in Science Center, Hall C
“Two math lessons from string theory”
Lecture 1: April 9, 2019
Title: “Lesson on Integrability”
Abstract: The quantum Knizhnik-Zamolodchikov (qKZ) equation is a difference generalization of the famous Knizhnik-Zamolodchikov (KZ) equation. The problem to explicitly capture the monodromy of the qKZ equation has been open for over 25 years. I will describe the solution to this problem, discovered jointly with Andrei Okounkov. The solution comes from the geometry of Nakajima quiver varieties and has a string theory origin.
Part of the interest in the qKZ monodromy problem is that its solution leads to integrable lattice models, in parallel to how monodromy matrices of the KZ equation lead to knot invariants. Thus, our solution of the problem leads to a new, geometric approach, to integrable lattice models. There are two other approaches to integrable lattice models, due to Nekrasov and Shatashvili and to Costello, Witten and Yamazaki. I’ll describe joint work with Nikita Nekrasov which explains how string theory unifies the three approaches to integrable lattice models.
Lecture 2: April 10, 2019
Title: “Lesson on Knot Categorification”
Abstract: An old problem is to find a unified approach to the knot categorification problem. The new string theory perspective on the qKZ equation I described in the first talk can be used to derive two geometric approaches to the problem.
The first approach is based on a category of B-type branes on resolutions of slices in affine Grassmannians. The second is based on a category of A-branes in a Landau-Ginzburg theory. The relation between them is two dimensional (equivariant) mirror symmetry. String theory also predicts that a third approach to categorification, based on counting solutions to five dimensional Haydys-Witten equations, is equivalent to the first two.
This talk is mostly based on joint work with Andrei Okounkov.
Information about last year’s Math Science Bott lecture can be found here.
Yip Annual Lecture
1 Oxford Street, Cambridge MA 02138On April 18, 2019 Harvard CMSA hosted the inaugural Yip lecture. The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip. This year’s speaker was Peter Galison (Harvard Physics).
The lecture was held from 4:00-5:00pm in Science Center, Hall A.
Duality String Seminar, Thursdays
The Duality String Seminar is held every Thursday at 4:15pm in Jefferson Lab, 453.
For details, please visit the website.
* The Duality String Seminar is sponsored by the Center of Mathematical Sciences and Applications’ Cheng Yu-Tong Fund, for Research at the Interface of Mathematics and Physics.
Why abstraction is the key to intelligence, and what we’re still missing
Abstract: This talk provides a personal perspective on the way forward towards more human-like and more intelligent artificial systems. Traditionally, symbolic and probabilistic methods have dominated the domains of concept formation, abstraction, and automated reasoning. More recently, deep learning-based approaches have led to significant breakthroughs, including successes in games and combinatorial search tasks. However, the resulting systems are still limited in scope and capabilities — they remain brittle, data-hungry, and their generalization capabilities are limited. We will address a set of questions: why is conceptual abstraction essential for intelligence? What is the nature of abstraction, and its relationship to generalization? What kind of abstraction can deep learning models generate, and where do they fail? What are the methods that are currently successful in generating strong conceptual abstraction? Finally, we will consider how to leverage a hybrid approach to reinforce the strength of different approaches while compensating for their respective weaknesses.
The complexity of matrix multiplication approached via algebraic geometry and representation theory.
Abstract: In 1968 V. Strassen discovered the way we usually multiply matrices is not the most efficient possible, and after considerable work by many authors, it is generally conjectured by computer scientists that as the size of matrices becomes large, it becomes almost as easy to multiply them as it is to add them. I will give a brief history of the problem, explain how this conjecture is naturally understood in the framework of classical algebraic geometry and representation theory, and conclude by describing recent advances using more sophisticated tools from algebraic geometry. For most of the talk, no knowledge of algebraic geometry or representation theory will be needed.
Noga Alon Public Talk, 9-7-17
Noga Alon (Tel Aviv University) will be giving a public talk on September 7, 2017,as part of the program on combinatorics and complexity hosted by the CMSA during AY17-18. The talk will be at 5:00pm in Askwith Hall, 13 Appian Way, Cambridge, MA.
Title: Graph Coloring: Local and Global
Abstract: Graph Coloring is arguably the most popular subject in Discrete Mathematics, and its combinatorial, algorithmic and computational aspects have been studied intensively. The most basic notion in the area, the chromatic number of a graph, is an inherently global property. This is demonstrated by the hardness of computation or approximation of this invariant as well as by the existence of graphs with arbitrarily high chromatic number and no short cycles. The investigation of these graphs had a profound impact on Graph Theory and Combinatorics. It combines combinatorial, probabilistic, algebraic and topological techniques with number theoretic tools. I will describe the rich history of the subject focusing on some recent results.
Jennifer Chayes Public Talk, 11-02-17
Jennifer Chayes (Microsoft Research) will be giving a public talk on November 02, 2017,as part of the program on combinatorics and complexity hosted by the CMSA during AY17-18. The talk will be at 5:00pm in Askwith Hall, 13 Appian Way, Cambridge, MA.
Title: Network Science: From the Online World to Cancer Genomics
Abstract: Everywhere we turn these days, we find that networks can be used to describe relevant interactions. In the high tech world, we see the Internet, the World Wide Web, mobile phone networks, and a variety of online social networks. In economics, we are increasingly experiencing both the positive and negative effects of a global networked economy. In epidemiology, we find disease spreading over our ever growing social networks, complicated by mutation of the disease agents. In biomedical research, we are beginning to understand the structure of gene regulatory networks, with the prospect of using this understanding to manage many human diseases. In this talk, I look quite generally at some of the models we are using to describe these networks, processes we are studying on the networks, algorithms we have devised for the networks, and finally, methods we are developing to indirectly infer network structure from measured data. I’ll discuss in some detail particular applications to cancer genomics, applying network algorithms to suggest possible drug targets for certain kinds of cancer.
CMSA Math-Science Literature Lecture: Indistinguishability Obfuscation: How to Hide Secrets within Software
Amit Sahai (UCLA)
Title: Indistinguishability Obfuscation: How to Hide Secrets within Software
Abstract: At least since the initial public proposal of public-key cryptography based on computational hardness conjectures (Diffie and Hellman, 1976), cryptographers have contemplated the possibility of a “one-way compiler” that translates computer programs into “incomprehensible” but equivalent forms. And yet, the search for such a “one-way compiler” remained elusive for decades.
In this talk, we look back at our community’s attempts to formalize the notion of such a compiler, culminating in our 2001 work with Barak, Goldreich, Impagliazzo, Rudich, Vadhan, and Yang, which proposed the notion of indistinguishability obfuscation (iO). Roughly speaking, iO requires that the compiled versions of any two equivalent programs (with the same size and running time) be indistinguishable to any efficient adversary. Leveraging the notion of punctured programming, introduced in our work with Waters in 2013, well over a hundred papers have explored the remarkable power of iO.
We’ll then discuss the intense effort that recently culminated in our 2020 work with Jain and Lin, finally showing how to construct iO in such a way that, for the first time, we can prove the security of our iO scheme based on well-studied computational hardness conjectures in cryptography.
Talk chair: Sergiy Verstyuk
2018 HMS Focused Lecture Series
As part of their CMSA visitation, HMS focused visitors will be giving lectures on various topics related to Homological Mirror Symmetry throughout the Spring 2018 Semester. The lectures will take place on Tuesdays and Thursdays in the CMSA Building, 20 Garden Street, Room G10.
The schedule will be updated below.
Date Speaker Title/Abstract January 23, 25, 30 and February 1 3-5pm
*Room G10*
Ivan Losev (Northeastern)
Title: BGG category O: towards symplectic duality Abstract: We will discuss a very classical topic in the representation theory of semisimple Lie algebras: the Bernstein-Gelfand-Gelfand (BGG) category O. Our aim will be to motivate and state a celebrated result of Beilinson, Ginzburg and Soergel on the Koszul duality for such categories, explaining how to compute characters of simple modules (the Kazhdan-Lusztig theory) along the way. The Koszul duality admits a conjectural generalization (Symplectic duality) that is a Mathematical manifestation of 3D Mirror symmetry. We will discuss that time permitting.
Approximate (optimistic) plan of the lectures:
1) Preliminaries and BGG category O.
2) Kazhdan-Lusztig bases. Beilinson-Bernstein localization theorem.
3) Localization theorem continued. Soergel modules.
4) Koszul algebras and Koszul duality for categories O.
Time permitting: other instances of Symplectic duality.
Prerequisites:
Semi-simple Lie algebras and their finite dimensional representation theory.
Some Algebraic geometry. No prior knowledge of category O/ Geometric
Representation theory is assumed.
February 27, and March 1
3-5pm
Colin Diemer (IHES)
Title: Moduli spaces of Landau-Ginzburg models and (mostly Fano) HMS. Abstract: Mirror symmetry as a general phenomenon is understood to take place near the large complex structure limit resp. large radius limit, and so implicitly involves degenerations of the spaces under consideration. Underlying most mirror theorems is thus a mirror map which gives a local identification of respective A-model and B-model moduli spaces. When dealing with mirror symmetry for Calabi-Yau’s the role of the mirror map is well-appreciated. In these talks I’ll discuss the role of moduli in mirror symmetry of Fano varieties (where the mirror is a Landau-Ginzburg (LG) model). Some topics I expect to cover are a general structure theory of moduli of LG models (follows Katzarkov, Kontsevich, Pantev), the interplay of the topology of LG models with autoequivalence relations in the Calabi-Yau setting, and the relationship between Mori theory in the B-model and degenerations of the LG A-model. For the latter topic we’ll focus on the case of del Pezzo surfaces (due to unpublished work of Pantev) and the toric case (due to the speaker with Katzarkov and G. Kerr). Time permitting, we may make some speculations on the role of LG moduli in the work of Gross-Hacking-Keel (in progress work of the speaker with T. Foster).
March 6 and 8 4-5pm
Adam Jacob (UC Davis)
Title: The deformed Hermitian-Yang-Mills equation Abstract: In this series I will discuss the deformed Hermitian-Yang-Mills equation, which is a complex analogue of the special Lagrangian graph equation of Harvey-Lawson. I will describe its derivation in relation to the semi-flat setup of SYZ mirror symmetry, followed by some basic properties of solutions. Later I will discuss methods for constructing solutions, and relate the solvability to certain geometric obstructions. Both talks will be widely accessible, and cover joint work with T.C. Collins and S.-T. Yau.
March 6, 8, 13, 15 3-4pm
Dmytro Shklyarov (TU Chemnitz)
Title: On categories of matrix factorizations and their homological invariants Abstract: The talks will cover the following topics:
1. Matrix factorizations as D-branes. According to physicists, the matrix factorizations of an isolated hypersurface singularity describe D-branes in the Landau-Ginzburg (LG) B-model associated with the singularity. The talk is devoted to some mathematical implications of this observation. I will start with a review of open-closed topological field theories underlying the LG B-models and then talk about their refinements.
2. Semi-infinite Hodge theory of dg categories. Homological mirror symmetry asserts that the “classical” mirror correspondence relating the number of rational curves in a CY threefold to period integrals of its mirror should follow from the equivalence of the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. The classical mirror correspondence can be upgraded to an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the assertion would be to try to attach similar Hodge-like data to abstract derived categories. I will talk about some recent results in this direction and illustrate the approach in the context of the LG B-models.
3. Hochschild cohomology of LG orbifolds. The scope of applications of the LG mod- els in mirror symmetry is significantly expanded once we include one extra piece of data, namely, finite symmetry groups of singularities. The resulting models are called orbifold LG models or LG orbifolds. LG orbifolds with abelian symmetry groups appear in mir- ror symmetry as mirror partners of varieties of general type, open varieties, or other LG orbifolds. Associated with singularities with symmetries there are equivariant versions of the matrix factorization categories which, just as their non-equivariant cousins, describe D-branes in the corresponding orbifold LG B-models. The Hochschild cohomology of these categories should then be isomorphic to the closed string algebra of the models. I will talk about an explicit description of the Hochschild cohomology of abelian LG orbifolds.
April 10 & 12 3-4pm
Mauricio Romo (IAS)
Title: Gauged Linear Sigma Models, Supersymmetric Localization and Applications Abstract: In this series of lectures I will review various results on connections between gauged linear sigma models (GLSM) and mathematics. I will start with a brief introduction on the basic concepts about GLSMs, and their connections to quantum geometry of Calabi-Yaus (CY). In the first lecture I will focus on nonperturbative results on GLSMs on closed 2-manifolds, which provide a way to extract enumerative invariants and the elliptic genus of some classes of CYs. In the second lecture I will move to nonperturbative results in the case where the worldsheet is a disk, in this case nonperturbative results provide interesting connections with derived categories and stability conditions. We will review those and provide applications to derived functors and local systems associated with CYs. If time allows we will also review some applications to non-CY cases (in physics terms, anomalous GLSMs).
April 17, 19, 26 3-5pm
Andrew Harder (University of Miami)
Title: Perverse sheaves of categories on surfaces Abstract: Perverse sheaves of categories on a Riemann surface S are systems of categories and functors which are encoded by a graphs on S, and which satisfy conditions that resemble the classical characterization of perverse sheaves on a disc.
I’ll review the basic ideas behind Kapranov and Schechtman’s notion of a perverse schober and generalize this to perverse sheaves of categories on a punctured Riemann surface. Then I will give several examples of perverse sheaves of categories in both algebraic geometry, symplectic geometry, and category theory. Finally, I will describe how one should be able to use related ideas to prove homological mirror symmetry for certain noncommutative deformations of projective 3-space.
May 15, 17 1-3pm
Charles Doran (University of Alberta)
Lecture One:
Title: Picard-Fuchs uniformization and Calabi-Yau geometryAbstract:Part 1: We introduce the notion of the Picard-Fuchs equations annihilating periods in families of varieties, with emphasis on Calabi-Yau manifolds. Specializing to the case of K3 surfaces, we explore general results on “Picard-Fuchs uniformization” of the moduli spaces of lattice-polarized K3 surfaces and the interplay with various algebro-geometric normal forms for these surfaces. As an application, we obtain a universal differential-algebraic characterization of Picard rank jump loci in these moduli spaces.Part 2: We next consider families with one natural complex structure modulus, (e.g., elliptic curves, rank 19 K3 surfaces, b_1=4 Calabi-Yau threefolds, …), where the Picard-Fuchs equations are ODEs. What do the Picard-Fuchs ODEs for such families tell us about the geometry of their total spaces? Using Hodge theory and parabolic cohomology, we relate the monodromy of the Picard-Fuchs ODE to the Hodge numbers of the total space. In particular, we produce criteria for when the total space of a family of rank 19 polarized K3 surfaces can be Calabi-Yau.Lecture Two:
Title: Calabi-Yau fibrations: construction and classification
Abstract:Part 1: Codimension one Calabi-Yau submanifolds induce fibrations, with the periods of the total space relating to those of the fibers and the structure of the fibration. We describe a method of iteratively constructing Calabi-Yau manifolds in tandem with their Picard-Fuchs equations. Applications include the tower of mirrors to degree n+1 hypersurfaces in P^n and a tower of Calabi-Yau hypersurfaces encoding the n-sunset Feynman integrals.
Part 2: We develop the necessary theory to both construct and classify threefolds fibered by lattice polarized K3 surfaces. The resulting theory is a complete generalization to threefolds of that of Kodaira for elliptic surfaces. When the total space of the fibration is a Calabi-Yau threefold, we conjecture a unification of CY/CY mirror symmetry and LG/Fano mirror symmetry by mirroring fibrations as Tyurin degenerations. The detailed classification of Calabi-Yau threefolds with certain rank 19 polarized fibrations provides strong evidence for this conjecture by matching geometric characteristics of the fibrations with features of smooth Fano threefolds of Picard rank 1.
2017 Ding Shum Lecture
1 Oxford Street, Cambridge MA 02138Leslie Valiant will be giving the inaugural talk of the Ding Shum Lectures on Tuesday, October 10 at 5:00 pm in Science Center Hall D, Cambridge, MA.
Learning as a Theory of Everything
Abstract: We start from the hypothesis that all the information that resides in living organisms was initially acquired either through learning by an individual or through evolution. Then any unified theory of evolution and learning should be able to characterize the capabilities that humans and other living organisms can possess or acquire. Characterizing these capabilities would tell us about the nature of humans, and would also inform us about feasible targets for automation. With this purpose we review some background in the mathematical theory of learning. We go on to explain how Darwinian evolution can be formulated as a form of learning. We observe that our current mathematical understanding of learning is incomplete in certain important directions, and conclude by indicating one direction in which further progress would likely enable broader phenomena of intelligence and cognition to be realized than is possible at present.
Constructions in combinatorics via neural networks
Abstract: Recently, significant progress has been made in the area of machine learning algorithms, and they have quickly become some of the most exciting tools in a scientist’s toolbox. In particular, recent advances in the field of reinforcement learning have led computers to reach superhuman level play in Atari games and Go, purely through self-play. In this talk I will give a very basic introduction to neural networks and reinforcement learning algorithms. I will also indicate how these methods can be adapted to the ““game” of trying to find a counterexample to a mathematical conjecture, and show some examples where this approach was successful.
New results in Supergravity via ML Technology
Abstract: The infrastructure built to power the Machine Learning revolution has many other uses beyond Deep Learning. Starting from a general architecture-level overview over the lower levels of Google’s TensorFlow machine learning library, we review how this has recently helped us to find all the stable vacua of SO(8) Supergravity in 3+1 dimensions, has allowed major progress on other related questions about M theory, and briefly discuss other applications in field theory and beyond.
Computer-Aided Mathematics and Satisfiability
Abstract: Progress in satisfiability (SAT) solving has made it possible to determine the correctness of complex systems and answer long-standing open questions in mathematics. The SAT solving approach is completely automatic and can produce clever though potentially gigantic proofs. We can have confidence in the correctness of the answers because highly trustworthy systems can validate the underlying proofs regardless of their size.
We demonstrate the effectiveness of the SAT approach by presenting some recent successes, including the solution of the Boolean Pythagorean Triples problem, computing the fifth Schur number, and resolving the remaining case of Keller’s conjecture. Moreover, we constructed and validated a proof for each of these results. The second part of the talk focuses on notorious math challenges for which automated reasoning may well be suitable. In particular, we discuss our progress on applying SAT solving techniques to the chromatic number of the plane (Hadwiger-Nelson problem), optimal schemes for matrix multiplication, an
Why explain mathematics to computers?
Abstract: A growing number of mathematicians are having fun explaining mathematics to computers using proof assistant softwares. This process is called formalization. In this talk I’ll describe what formalization looks like, what kind of things it teaches us, and how it could even turn out to be useful (in our usual sense of “useful”). This will not be a talk about foundations of mathematics, and I won’t assume any prior knowledge about formalization.
Langlands duality for 3 manifolds
Abstract: Langlands duality began as a deep and still mysterious conjecture in number theory, before branching into a similarly deep and mysterious conjecture of Beilinson and Drinfeld concerning the algebraic geometry of Riemann surfaces. In this guise it was given a physical explanation in the framework of 4-dimensional super symmetric quantum field theory by Kapustin and Witten. However to this day the Hilbert space attached to 3-manifolds, and hence the precise form of Langlands duality for them, remains a mystery.
In this talk I will propose that so-called “skein modules” of 3-manifolds give natural candidates for these Hilbert spaces at generic twisting parameter Psi , and I will explain a Langlands duality in this setting, which we have conjectured with Ben-Zvi, Gunningham and Safronov.
Intriguingly, the precise formulation of such a conjecture in the classical limit Psi=0 is still an open question, beyond the scope of the talk.
Fluid Dynamics Seminar
Beginning immediately, until at least April 30, all seminars will take place virtually, through Zoom. Links to connect can be found in the schedule below once they are created.
In the Spring 2019 Semester, the Center of Mathematical Sciences and Applications will be hosting a seminar on Fluid Dynamics. The seminar will take place on Wednesdays from 3:00-4:00pm in CMSA G10.
Spring 2020:
Date Speaker Title/Abstract 2/25/2020 Keaton Burns, MIT Title: Flexible spectral simulations of low-Mach-number astrophysical fluids Abstract: Fluid dynamical processes are key to understanding the formation and evolution of stars and planets. While the astrophysical community has made exceptional progress in simulating highly compressible flows, models of low-Mach-number stellar and planetary flows typically use simplified equations based on numerical techniques for incompressible fluids. In this talk, we will discuss improved numerical models of three low-Mach-number astrophysical phenomena: tidal instabilities in binary neutron stars, waves and convection in massive stars, and ice-ocean interactions in icy moons. We will cover the basic physics of these systems and how ongoing additions to the open-source Dedalus Project are enabling their efficient simulation in spherical domains with spectral accuracy, implicit timestepping, phase-field methods, and complex equations of state.
3/4/2020 G02
3/11/2020 3/18/2020 3/25/2020 4/1/2020 4/8/2020 G02 4/15/2020 4/22/2020 4/29/2020 G02
5/6/2020 5/13/2020 Fall 2019:
Date Speaker Title/Abstract 9/18/2019 Jiawei Zhuang (Harvard) Title: Simulation of 2-D turbulent advection at extreme accuracy with machine learning and differentiable programming Abstract: The computational cost of fluid simulations grows rapidly with grid resolution. With the recent slow-down of Moore’s Law, it can take many decades for 10x higher resolution grids to become affordable. To break this major barrier in high-performance scientific computing, we used a data-driven approach to learn an optimal numerical solver that can retain high-accuracy at much coarser grids. We applied this method to 2-D turbulent advection and achieved 4x effective resolution than traditional high-order flux-limited advection solvers. The machine learning component is tightly integrated with traditional finite-volume schemes and can be trained via an end-to-end differentiable programming framework. The model can achieve near-peak FLOPs on CPUs and accelerators via convolutional filters.
9/25/2019 Yantao Yang (Peking University) Title: Double diffusive convection and thermohaline staircases Abstract: Double diffusive convection (DDC), i.e. the buoyancy-driven flow with fluid density depending on two scalar components, is omnipresent in many natural and engineering environments. In ocean this is especially true since the seawater density is mainly determined by temperature and salinity. In upper water of both (sub-) tropical and polar oceans, DDC causes the intriguing thermohaline staircases, which consist of alternatively stacked convection layers and sharp interfaces with high gradients of temperature and salinity. In this talk, we will focus on the fingering DDC usually found in (sub-)tropical ocean, where the mean temperature and salinity decrease with depth. We numerically investigate the formation and the transport properties of finger structures and thermohaline staircases. Moreover, we show that multiple states exit for the exactly same global condition, and individual finger layers and finger layers within staircases exhibit very different transport behaviors.
10/2/2019 No talk 10/9/2019 Samuel Rudy (MIT) Title: Data-driven methods for discovery of partial differential equations and forecasting Abstract: A critical challenge in many modern scientific disciplines is deriving governing equations and forecasting models from data where derivation from first principals is intractable. The problem of learning dynamics from data is complicated when data is corrupted by noise, when only partial or indirect knowledge of the state is available, when dynamics exhibit parametric dependencies, or when only small volumes of data are available. In this talk I will discuss several methods for constructing models of dynamical systems from data including sparse identification for partial differential equations with or without parametric dependencies and approximation of dynamical systems governing equations using neural networks. Limitations of each approach and future research directions will also be discussed.
10/16/2019 No talk 10/23/2019 Kimee Moore (Harvard) Title: Using magnetic fields to investigate Jupiter’s fluid interior Abstract: The present-day interior structure of a planet is an important reflection of the formation and subsequent thermal evolution of that planet. However, despite decades of spacecraft missions to a variety of target bodies, the interiors of most planets in our Solar System remain poorly constrained. In this talk, I will discuss how actively generated planetary magnetic fields (dynamos) can provide important insights into the interior properties and evolution of fluid planets. Using Jupiter as a case study, I will present new results from the analysis of in situ spacecraft magnetometer data from the NASA Juno Mission (currently in orbit about Jupiter). The spatial morphology of Jupiter’s magnetic field shows surprising hemispheric asymmetry, which may be linked to the dissolution of Jupiter’s rocky core in liquid metallic hydrogen. I also report the first definitive detection of time-variation (secular variation) in a planetary dynamo beyond Earth. This time-variation can be explained by the advection of Jupiter’s magnetic field by the zonal winds, which places a lower bound on the velocity of Jupiter’s winds at depth. These results provide an important complement to other analysis techniques, as gravitational measurements are currently unable to uniquely distinguish between deep and shallow wind scenarios, and between solid and dilute core scenarios. Future analysis will continue to resolve Jupiter’s interior, providing broader insight into the physics of giant planets, with implications for the formation of our Solar System.
10/30/2019 No Talk 11/6/2019 Federico Toschi (Eindhoven University of Technology) Title: Deep learning and reinforcement learning for turbulence Abstract: This talk tells two stories.
Chapter 1: We investigate the capability of a state-of-the-art deep neural model at learning features of turbulent velocity signals. Deep neural network (DNN) models are at the center of the present machine learning revolution. The set of complex tasks in which they over perform human capabilities and best algorithmic solutions grows at an impressive rate and includes, but it is not limited to, image, video and language analysis, automated control, and even life science modeling. Besides, deep learning is receiving increasing attention in connection to a vast set of problems in physics where quantitatively accurate outcomes are expected. We consider turbulent velocity signals, spanning decades in Reynolds numbers, which have been generated via shell models for the turbulent energy cascade. Given the multi-scale nature of the turbulent signals, we focus on the fundamental question of whether a deep neural network (DNN) is capable of learning, after supervised training with very high statistics, feature extractors to address and distinguish intermittent and multi-scale signals. Can the DNN measure the Reynolds number of the signals? Which feature is the DNN learning?
Chapter 2: Thermally driven turbulent flows are common in nature and in industrial applications. The presence of a (turbulent) flow can greatly enhance the heat transfer with respect to its conductive value. It is therefore extremely important -in fundamental and applied perspective- to understand if and how it is possible to control the heat transfer in thermally driven flows. In this work, we aim at maintaining a Rayleigh–Bénard convection (RBC) cell in its conductive state beyond the critical Rayleigh number for the onset of convection. We specifically consider controls based on local modifications of the boundary temperature (fluctuations). We take advantage of recent developments in Artificial Intelligence and Reinforcement Learning (RL) to find -automatically- efficient non-linear control strategies. We train RL agents via parallel, GPU-based, 2D lattice Boltzmann simulations. Trained RL agents are capable of increasing the critical Rayleigh number of a factor 3 in comparison with state-of-the-art linear control approaches. Moreover, we observe that control agents are able to significantly reduce the convective flow also when the conductive state is unobtainable. This is achieved by finding and inducing complex flow fields.
11/13/2019 2:10pm
G02
Martin Lellep (Philipps University of Marburg, Germany) Title: Predictions of relaminarisation in turbulent shear flows using deep learning Abstract: Given the increasing performance of deep learning algorithms in tasks such as classification during the last years and the vast amount of data that can be generated in turbulence research, I present one application of deep learning to fluid dynamics in this talk. We train a deep learning machine learning model to classify if turbulent shear flow becomes laminar a certain amount of time steps ahead in the future. Prior to this, we use a 2D toy example to develop an understanding how the performance of the deep learning algorithm depends on hyper parameters and how to understand the errors. The performance of both algorithms is high and therefore opens up further steps towards the interpretation of the results in future work.
11/19/2019 Tuesday
3-4 pm
Pierce Hall 209, 29 Oxford Street
Detlef Lohse (University of Twente) Title: Rayleigh vs. Marangoni Abstract: In this talk I will show several examples of an interesting and surprising competition between buoyancy and Marangoni forces. First, I will introduce the audience to the jumping oil droplet – and its sudden death – in a density stratified liquid consisting of water in the bottom and ethanol in the top : After sinking for about a minute, before reaching the equilibrium the droplet suddenly jumps up thanks to the Marangoni forces. This phenomenon repeats about 30-50 times, before the droplet falls dead all the sudden. We explain this phenomenon and explore the phase space where it occurs. Next, I will focus on the evaporation of multicomponent droplets, for which the richness of phenomena keeps surprising us. I will show and explain several of such phenomena, namely evaporation-triggered segregation thanks to either weak solutal Marangoni flow or thanks to gravitational effects. The dominance of the latter implies that sessile droplets and pending droplets show very different evaporation behavior, even for Bond number << 1. I will also explain the full phase diagram in the Marangoni number vs Rayleigh number phase space, and show where Rayleigh convections rolls prevail, where Marangoni convection rolls prevail, and where they compete.
The research work shown in this talks combines experiments, numerical simulations, and theory. It has been done by and in collaboration with Yanshen Li, Yaxing Li, and Christian Diddens, and many others.
11/20/2019 Time: 3:00-3:35 pm Speaker: Haoran Liu
Title: Applications of Phase Field method: drop impact and multiphase turbulence
Abstract: Will a mosquito survive raindrop collisions? How the bubbles under a ship reduce the drag force? In nature and industry, flows with drops and bubbles exist everywhere. To understand these flows, one of the powerful tools is the direct numerical simulation (DNS). Among all the DNS methods, we choose the Phase Field (PF) method and develop some models based on it to simulate the complicated flows, such as flows with moving contact lines, fluid-structure interaction, ternary fluids and turbulence. In this talk, I will firstly introduce the advantages and disadvantages of PF method. Then, I will show its applications: drop impact on an object, compound droplet dynamics, water entry of an object and multiphase turbulence.
Time: 3:35-4:10 pm
Speaker: Steven Chong
Abstract: For Rayleigh-Bénard under geometrical confinement, under rotation or the double diffusive convection with the second scalar component stabilizing the convective flow, they seem to be the three different canonical models in turbulent flow. However, previous research coincidentally reported the scalar transport enhancement in these systems. The results are counter-intuitive because the higher efficiency of scalar transport is bought about by the slower flow. In this talk, I will show you a fundamental and unified perspective on such the global transport behavior observed in the seemingly different systems. We further show that the same view can be applied to the quasi-static magnetoconvection, and indeed the regime with heat transport enhancement has been found. The beauty of physics is to understand the seemingly unrelated phenomena by a simplified concept. Here we provide a simplified and generic view, and this concept could be potentially extended to other situations where the turbulent flow is subjected to an additional stabilization.
11/27/2019 12/4/2019 12/11/2019 See previous seminar information here.
Topological Aspects of Condensed Matter Seminar
As part of the Program on Topological Aspects of Condensed Matter, a weekly seminar will be held on Mondays from 10:00-11:30pm in CMSA room G10.
Date Speaker Title/Abstract 8/29/2018 Zeng-Cheng Gu Title: Towards a complete classification of symmetry protected topological phases for interacting fermions in three dimensions and a general group supercohomology theory Abstract: Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this talk, I will revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. I will show how to construct very general fixed point SPT wavefunctions for interacting fermion systems. I will also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions.
9/10/2018 Dominic Else, MIT Title: Phases and topology in periodically driven (Floquet) systems Abstract: I will give a pedagogical overview of new topological phenomena that occur in systems that are driven periodically in time (Floquet systems). As a warm-up, I will review new topological invariants in free-fermion Floquet systems. Then, I will discuss the richer physics that occurs in interacting Floquet phases, stabilized in systems with strong quenched disorder by many-body-localization (MBL). Finally, time permitting, I will explain how to realize interacting topological phenomena in a metastable (“pre-thermal”) regime of a clean system.
9/17/2018 Adrian Po, MIT Title: A modern solution to the old problem of symmetries in band theory Abstract: There are 230 space groups and 1,651 magnetic space groups in three dimensions. Thankfully, these are finite numbers, and one might go about solving all the possible ways free electrons represent them. This is a central question in the nine-decade-old band theory, which is long-thought to be solvable if only one had the time and patience to crank through all the cases. In this talk, I would describe how this problem can be solved efficiently from the modern perspective of band topology. As a by-product, we will describe a simple method to detect topologically nontrivial band insulators using only symmetry eigenvalues, which offers great computational advantage compared to the traditional, wave-function-based definitions of topological band invariants.
9/24/2018 Maxim Metlitski Title: Surface Topological Order and a new ‘t Hooft Anomaly of Interaction Enabled 3+1D Fermion SPTs Abstract: Symmetry protected topological (SPT) phases have attracted a lot of attention in recent years. A key property of SPTs is the presence of non-trivial surface states. While for 1+1D and 2+1D SPTs the boundary must be either symmetry broken or gapless, some 3+1D SPTs admit symmetric gapped surface states that support anyon excitation (intrinsic topological order). In all cases, the boundary of an SPT is anomalous – it cannot be recreated without the bulk; furthermore, the anomaly must “match” the bulk. I will review this bulk-boundary correspondence for 3d SPT phases of bosons with topologically ordered boundaries where it is fairly well understood. I will then proceed to describe recent advances in the understanding of strongly interacting 3+1D SPT phases of fermions and their topologically ordered surface states.
10/1/2018 Cancelled 10/9/2018 Tuesday
3:00-4:30pm
Sagar Vijay Title: Fracton Phases of Matter Abstract: Fracton phases are new kinds of highly-entangled quantum matter in three spatial dimensions that are characterized by gapped, point-like excitations (“fractons”) that are strictly immobile at zero temperature, and by degenerate ground-states that are locally indistinguishable. Fracton excitations provide an alternative to Fermi or Bose statistics in three spatial dimensions, and these states of matter are a gateway for exploring mechanisms for quantum information storage, and for studying “slow” dynamical behavior in the absence of disorder. I will review exactly solvable models for these phases, constructions of these states using well-studied two-dimensional topological phases, and a model in which the fracton excitations carry a protected internal degeneracy, which provides a natural generalization of non-Abelian anyons to three spatial dimensions. I will then describe recent advances in categorizing these states of matter using finite-depth unitary transformations.
10/15/2018 Ethan Lake Title: A primer on higher symmetries Abstract: The notion of a higher symmetry, namely a symmetry whose charged objects have a dimension greater than zero, is proving to be very useful for organizing our understanding of gauge theories and topological phases of matter. Just like regular symmetries, higher symmetries can be gauged, spontaneously broken, and can have anomalies. I will review these aspects of higher symmetries and motivate why beyond their conceptual utility, they are often an indispensable tool for making statements about dualities and phase diagrams of theories with gauge fields.
10/22/2018 Room G02
Yin-Chen He, Perimeter Title: Emergent QED3 and QCD3 in condensed matter system Abstract: QED3-Chern-Simons and QCD3-Chern-Simons theories are interesting critical theories in the 2+1 dimension. These theories are described by gapless Dirac fermions interacting with dynamical gauge fields (U(1), SU(N), U(N), etc.) with a possible Chern-Simon term. These theories have fundamental importance as it will flow to the 3D conformal field theories and have interesting dualities in the infrared. Various of condensed matter system are described by these critical theories. I will introduce several examples including the Dirac spin liquid in the frustrated magnets (kagome, triangular lattice), quantum phase transitions in the fractional quantum Hall systems and Kitaev materials.
10/29/2018 Dominic Williamson, Yale Title: Symmetry and topological order in tensor networks Abstract: I will present an overview of how topological states of matter with global symmetries can be described using tensor networks. First reviewing the classification of 1D symmetry-protected topological phases with matrix product states, before moving on to the description of 2D symmetry-enriched topological phases with projected-entangled pair states.
11/13/2018 Tuesday
3:00-4:30pm
Jason Alicea, Caltech Title: Time-crystalline topological superconductors 11/19/2018 X. G. Wen, MIT Title: A classification of 3+1D topological orders Abstract: I will discuss a classification of 3+1D topological orders in terms of fusion 2 category. The 3+1D topological orders can be divided into two classes: the ones without emergent fermions and the ones with emergent fermions. The 3+1D topological orders with emergent fermions can be further divided into two classes: the ones without emergent Majorana zero mode and the ones with emergent Majorana zero mode. I will present pictures to understand those 3+1D topological orders.
12/3/2018 *Room G02*
Claudio Chamon, Boston University Title: Many-body scar states with topological properties in 1D, 2D, and 3D. Abstract: We construct (some) exact excited states of a class of non-integrable quantum many-body Hamiltonians in 1D, 2D and 3D. These high energy many-body “scar” states have area law entanglement entropy, and display properties usually associated to gapped ground states of symmetry protected topological phases or topologically ordered phases of matter, including topological degeneracies.
12/10/2018 Room G02
Anders Sandvik, Boston University and Institute of Physics, CAS, Beijing Title: Quantum Monte Carlo simulations of exotic states in 2D quantum magnets Abstract: Some exotic ground states of 2D quantum magnets can be accessed through sign-free quantum Monte Carlo simulations of certain “designer Hamiltonians”. I will discuss recent examples within the J-Q family of models, where the standard Heisenberg exchange J on the square lattice is supplemented by multi-spin terms Q projecting correlated singlets, such that dimer (columnar valence-bond) order is favored. In addition to a possible deconfined quantum critical point separating the Neel and dimer phases, I will discuss recent work on a modified model where a rather strongly first-order transition between the Neel state and a plaquette-singlet-solid is associated with emergent O(4) symmetry up to length scales of at least 100 lattice spacings. This type of transition may be realized in SrCu2(BO3)2 under pressure. I will also discuss a random-singlet state obtained when randomness is introduced in a system with dimerized ground state. This type of state may be realized in some frustrated disordered quantum magnets.
1/8/2019 Lukasz Fidkowski, Univ. of Washington Title: Non-trivial quantum cellular automata in 3 dimensions Abstract: Motivated by studying the entanglement structure of certain symmetry protected topological phases, we construct a non-trivial quantum cellular automaton in a Hilbert space for a 3d lattice of spin 1/2 degrees of freedom. This is an operator which takes local operators to nearby local operators, but is not locally generated. We discuss implications for the classification of SPT phases in equilibrium and Floquet settings.
3/18/2019 Ari Turner, Technion Title: Trapping Excitations at Phantasmagoric Wave Vectors Abstract: This talk will explain some properties of the fracton state devised by Jeongwan Haah. A fracton state has excitations that are extremely localized–it is impossible for them to move (unlike Anderson localization, e.g.–Anderson localized excitations can move if there is an external field to provide energy). One can understand why in a simple way using “mod 2” Fourier analysis. I will explain this, and also introduce “finite fields”, which are the number systems one needs to define exponentials mod. 2.
4/1/2019 Yi-Zhuang You (UCSD) Title: Emergent Symmetry and Conserved Currents at Deconfined Quantum Critical Points Abstract: Noether’s theorem is one of the fundamental laws of physics, relating continuous symmetries and conserved currents. Here we explore the role of Noether’s theorem at the deconfined quantum critical point (DQCP), which is an exotic quantum phase transition beyond the Landau-Ginzburg-Wilson paradigm. It was expected that a larger continuous symmetry could emerge at the DQCP, which, if true, should lead to conserved current at low energy. By identifying the emergent current fluctuation in the spin excitation spectrum, we can quantitatively study the current-current correlation in large-scale quantum Monte Carlo simulations. Our results reveal the conservation of the emergent current, as signified by the vanishing anomalous dimension of the current operator, and hence provide supporting evidence for the emergent symmetry at the DQCP. We also extend our discussion of emergent conserved current to the recently proposed one-dimensional analog of DQCP and confirm the emergent O(2)xO(2) symmetry in that case. Finally, I will briefly discuss the significance of our findings in a potential realization of DQCP in the Shastry-Sutherland lattice material SrCu2(BO3)2.
4/8/2019 Adam Nahum (Oxford) Title: Emergent statistical mechanics of entanglement in random unitary circuits Abstract: I will talk about quantum-classical mappings for real-time observables in some simple many-body systems (random unitary circuits). Specifically I will discuss how (1) entanglement entropy growth and (2) two-point correlation functions in these systems can be related to partition functions for interacting random walks. If time permits I will mention a phase transition in the entanglement structure of a repeatedly measured quantum state.
4/16/2019 Lyman 425
1:30pm
Xie Chen (Calthech) Title: Foliated Fracton Order Abstract: The quantum information study of quantum codes and quantum memory has led to the discovery of a new class of exactly solvable lattice models called the fracton models. The fracton models are similar to the better understood topological models in that they also support fractional excitations and have stable ground state degeneracy. But it is also clear that the fracton models exist beyond the realm of conventional topological order due to their extensive ground state degeneracy and the restricted motion of their fractional excitations. In this talk, I will present a new framework, which we call the “foliated fracton order”, to capture the nontrivial nature of the order in a large class of fracton models. Such a framework not only clarifies the connection between various different models, but also points to the direction of search for interesting new features.
4/24/2019 10:30am
Michael Freedman (Microsoft Station Q) Title: Quantum cellular automata in higher dimensions Abstract: I’ll discuss Joint work with Matt Hastings on local endomorphisms of the operator algebra. We found these have a cohomological invariant similar to that of an incompressible flow.
4/26/2019 10:30am
Maissam Barkeshli (University of Maryland) Title: Relative anomalies in (2+1)D symmetry enriched topological states Abstract: It has recently been understood that some patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. In this talk I will explain some recent advances in our understanding of how to compute relative anomalies between different symmetry fractionalization classes in (2+1)D topological states. The theory applies to general types of symmetries, including symmetries that permute anyon types and space-time reflection symmetries. This allows us to compute anomalies for more general types of space-time reflection symmetries than previously known methods.
5/3/2019 Yuan-Ming Lu (Ohio State) Title: Spontaneous symmetry breaking from anyon condensation Abstract: In the context of quantum spin liquids, it is long known that the condensation of fractionalized excitations can inevitably break certain physical symmetries. For example, condensing spinons will usually break spin rotation and time reversal symmetries. We generalize these phenomena to the context of a generic continuous quantum phase transition between symmetry enriched topological orders, driven by anyon condensation. We provide two rules to determine whether a symmetry is enforced to break across an anyon condensation transition or not. Using a dimensional reduction scheme, we establish a mapping between these symmetry-breaking anyon-condensation transitions in two spatial dimensions, and deconfined quantum criticality in one spatial dimension.
5/9/2019 10:30am
Michael Zaletel (UC Berkeley) Title: Three-partite entanglement in CFTs and chiral topological orders Abstract: While the entanglement entropy provides an essentially complete description of two-partite entanglement, multi-partite entanglement is far richer, with a concomitant zoo of possible measures. This talk will focus on applications of one such measure, the “entanglement of purification,” in many-body systems. I will first present a holographic prescription for calculating it which we can compare with numerical calculations. Interestingly, we find that a 1+1D CFT on a ring contains a universal number of GHZ states for any tri-partition of the ring. Using this result I’ll conjecture a bulk entanglement diagnostic for 2+1D chiral orders, and solicit the audience’s help in proving or disproving it.
5/28/2019 10:30am
Masaki Oshikawa (U Tokyo) Title: Gauge invariance, polarization, and conductivity Abstract: The large gauge transformation on a quantum many-body system under a periodic boundary condition has had numerous applications including generalizations of Lieb-Schultz-Mattis theorem. It is also deeply related to the electric polarization in insulators. I will discuss an application to a scaling of the fluctuation of the polarization in conductors, and also to general constraints on the electric conductivity.
7/18/2019 Eslam Khalaf (Harvard) Title: Dynamical correlations in anomalous disordered wires
Abstract: In a (multichannel) disordered wire, classical diffusion at short times (large frequencies) gives way to Anderson localization at long times (small frequencies). I study what happens in a disordered wire with topologically protected channels, e.g. a wire with unequal number of left and right movers which is realizable at the edge of a Quantum Hall system. In this case, the classical dynamics are described by diffusion + drift, but it is unclear what the effect of quantum corrections in the long time (small frequency) limit is.The problem is described by a 0+1-dimensional supersymmetric (graded) non-linear sigma model with a topological WZW term and a scalar potential. The computation of the local dynamical correlations of this model is equivalent to finding the ground state (zero mode) of the Laplace-Beltrami operator on a symmetric superspace with specific scalar and vector potentials. Surprisingly, I find that this zero mode has a relatively simple explicit integral representation in the Wigner-Dyson symmetry classes which has no counterpart in the absence of supersymmetry. This leads to an exact mapping between the local correlation functions in this 0+1D theory and observables in a 0+0D chiral random matrix problem.The mapping is used to explicitly compute two simple dynamical observables: the diffusion probability of return and the correlation of local density of states. In the former, we find that the interference effects change the exponential decay expected from drift-diffusion to a power law decay. In the latter, we find that the local density of states exhibits statistical level attraction in contrast to the level repulsion expected in a a standard Anderson insulator. At the end, I discuss possible relationship to the recently developed framework of non-Hermitian topological systems.Spacetime and Quantum Mechanics Seminar
As part of the program on Spacetime and Quantum Mechanics, the CMSA will be hosting a weekly seminar on Thursdays at 2:30pm in room G10.
Date Speaker Title/Abstract 9/12/2019 Pasha Pylyavskyy (University of Minnesota) Title: Vector-relation configurations and plabic graphs 19/18/2019 2:00pm
G02
Francis Brown (University of Oxford) Title: Amplitudes, Polylogs and Moduli Spaces 9/19/2019 Chuck Doran (University of Alberta) Title: Calabi-Yau geometry of the N-loop sunset Feynman integrals Abstract: I will present an overview of the algebraic and transcendental features of the computation of N-loop sunset Feynman integrals.
Starting from the realization of arbitrary Feynman graph hypersurfaces as (generalized) determinantal varieties, we describe the Calabi-Yau subvarieties of permutohedral varieties that arise from the N-loop sunset Feynman graphs and some key features of their geometry and moduli.
These include: (1) an iterated fibration structure which allows one to “bootstrap” both periods and Picard-Fuchs equations from lower N cases; (2) specialization to and interpretation of coincident mass loci (“jump loci”) in moduli; (3) a significant generalization of the Griffiths-Dwork algorithm via “creative telescoping”; and (4) the realization of Calabi-Yau pencils as Landau-Ginzburg models mirror to weak Fano varieties.
Details of each of these will be discussed in later lectures this semester. This is joint work with Pierre Vanhove and Andrey Novoseltsev.
9/26/2019 Tomasz Taylor (Northeastern) Title: Celestial Amplitudes 10/3/2019 Simon Caron-Huot (McGill) Title: Poincare Duals of Feynman Integrals 10/10/2019 3:30pm
Yutin Huang (National Taiwan University) Title: Dualities of Planar Ising Networks and the Positive Orthogonal Grassmannian 10/15/2019 Tuesday
3:30pm
Sergey Fomin (Univ. of Michigan)
Title: “Morsifications and mutations” (joint work with P. Pylyavskyy, E. Shustin, and D. Thurston). 10/18/2019 Friday
G02
Sebastian Franco (The City College of New York) Title: Graded quivers, generalized dimer models, and topic geometry 10/31/2019 Junjie Rao (Albert Einstein Institute) Title: All-loop Mondrian Reduction of 4-particle Amplituhedron at Positive Infinity 11/1/2019 SC 232
1:30pm
George Lusztig (MIT) Title: Total positivity in Springer fibres 11/12/2019 Tuesday
G02
3:30pm
Pierpaolo Mastrolia (University of Padova)
Title: Feynman Integrals and Intersection Theory 11/14/2019 G02
Pierpaolo Mastrolia (University of Padova) Title: Feynman Integrals and Intersection Theory Pt. II 11/21/2019 Cristian Vergu (Niels Bohr Institute) Title: The Octagonal Alphabet 11/26/2019 Stephan Stieberger (IAS) Title: Strings on the Celestial Sphere 12/4/2019 Hadleigh Frost (Oxford) Title: BCJ numerators, $\mathcal{M}_{0,n}$, and ABHY Abstract: We relate the BCJ numerator Jacobi property to the classical fact that the top homology group of $\mathcal{M}_{0,n}$ is isomorphic to a component of the free Lie algebra. We describe ways to get BCJ numerators, and caution that the BCJ Jacobi property doesn’t imply the existence of what has been called a ‘kinematic algebra.’
12/5/2019 David Kosower (IAS) Title: From scattering amplitudes to classical observables 12/10/2019 Ramis Movassagh (MIT) Title: Highly entangled quantum spin chains: Exactly solvable counter-examples to the area law Abstract: In recent years, there has been a surge of activities in proposing “exactly solvable” quantum spin chains with surprising high amount of ground state entanglement–exponentially more than the critical systems that have $\log(n)$ von Neumann entropy. We discuss these models from first principles. For a spin chain of length $n$, we prove that the ground state entanglement entropy scales as $\sqrt(n)$ and in some cases even extensive (i.e., as $n$) despite the underlying Hamiltonian being: (1) Local (2) Having a unique ground state and (3) Translationally invariant in the bulk. These models have rich connections with combinatorics, random walks, Markov chains, and universality of Brownian excursions. Lastly, we develop techniques for proving the gap. As a consequence, the gap of Motzkin and Fredkin spin chains are proved to vanish as 1/n^c with c>2; this rules out the possibility of these models to be relativistic conformal field theories in the continuum limit. Time permitting we will discuss more recent developments in this direction and ‘generic’ aspects of local spin chains.
Current Developments In Mathematics 2018
Current Developments in Mathematics 2018 Conference.
Friday, Nov. 16, 2018 2:15 pm – 6:00 pm
Saturday, Nov. 17, 2018 9:00 am – 5:00 pm
Harvard University Science Center, Hall B
Visit the conference page here
Workshop on Additive Combinatorics, Oct. 2-6, 2017
The workshop on additive combinatorics will take place October 2-6, 2017 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
Additive combinatorics is a mathematical area bordering on number theory, discrete mathematics, harmonic analysis and ergodic theory. It has achieved a number of successes in pure mathematics in the last two decades in quite diverse directions, such as:
- The first sensible bounds for Szemerédi’s theorem on progressions (Gowers);
- Linear patterns in the primes (Green, Tao, Ziegler);
- Construction of expanding sets in groups and expander graphs (Bourgain, Gamburd);
- The Kakeya Problem in Euclidean harmonic analysis (Bourgain, Katz, Tao).
Ideas and techniques from additive combinatorics have also had an impact in theoretical computer science, for example
- Constructions of pseudorandom objects (eg. extractors and expanders);
- Constructions of extremal objects (eg. BCH codes);
- Property testing (eg. testing linearity);
- Algebraic algorithms (eg. matrix multiplication).
The main focus of this workshop will be to bring together researchers involved in additive combinatorics, with a particular inclination towards the links with theoretical computer science. Thus it is expected that a major focus will be additive combinatorics on the boolean cube (Z/2Z)^n , which is the object where the exchange of ideas between pure additive combinatorics and theoretical computer science is most fruitful. Another major focus will be the study of pseudorandom phenomena in additive combinatorics, which has been an important contributor to modern methods of generating provably good randomness through deterministic methods. Other likely topics of discussion include the status of major open problems (the polynomial Freiman-Ruzsa conjecture, inverse theorems for the Gowers norms with bounds, explicit correlation bounds against low degree polynomials) as well as the impact of new methods such as the introduction of algebraic techniques by Croot–Pach–Lev and Ellenberg–Gijswijt.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Confirmed participants include:
- Arnab Bhattacharyya (Indian Institute of Science)
- Thomas Bloom (University of Bristol)
- Jop Briët (Centrum Wiskunde & Informatica, Amsterdam)
- Mei-Chu Chang (University of California, Riverside)
- Noam Elkies (Harvard University)
- Asaf Ferber (MIT)
- Jacob Fox (Stanford University)
- Shafi Goldwasser (MIT)
- Elena Grigorescu (Purdue University)
- Hamed Hatami (McGill University)
- Pooya Hatami (Institute for Advanced Study)
- Kaave Hosseini (University of California, San Diego)
- Guy Kindler (Hebrew University of Jerusalem)
- Vsevolod Lev (University of Haifa at Oranim)
- Sean Prendiville (University of Manchester)
- Ronitt Rubinfeld (MIT)
- Will Sawin (ETH Zürich)
- Fernando Shao (Oxford University)
- Olof Sisask (KTH Royal Institute of Technology)
- Madhur Tulsiani (University of Chicago)
- Julia Wolf (University of Bristol)
- Emanuele Viola (Northeastern University)
- Yufei Zhao (MIT)
Co-organizers of this workshop include Ben Green, Swastik Kopparty, Ryan O’Donnell, Tamar Ziegler.
Monday, October 2
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Jacob Fox Tower-type bounds for Roth’s theorem with popular differences Abstract: A famous theorem of Roth states that for any $\alpha > 0$ and $n$ sufficiently large in terms of $\alpha$, any subset of $\{1, \dots, n\}$ with density $\alpha$ contains a 3-term arithmetic progression. Green developed an arithmetic regularity lemma and used it to prove that not only is there one arithmetic progression, but in fact there is some integer $d > 0$ for which the density of 3-term arithmetic progressions with common difference $d$ is at least roughly what is expected in a random set with density $\alpha$. That is, for every $\epsilon > 0$, there is some $n(\epsilon)$ such that for all $n > n(\epsilon)$ and any subset $A$ of $\{1, \dots, n\}$ with density $\alpha$, there is some integer $d > 0$ for which the number of 3-term arithmetic progressions in $A$ with common difference $d$ is at least $(\alpha^3-\epsilon)n$. We prove that $n(\epsilon)$ grows as an exponential tower of 2’s of height on the order of $\log(1/\epsilon)$. We show that the same is true in any abelian group of odd order $n$. These results are the first applications of regularity lemmas for which the tower-type bounds are shown to be necessary.
The first part of the talk by Jacob Fox includes an overview and discusses the upper bound. The second part of the talk by Yufei Zhao focuses on the lower bound construction and proof. These results are all joint work with Huy Tuan Pham.
10:20-11:00am Coffee Break 11:00-11:50am Yufei Zhao Tower-type bounds for Roth’s theorem with popular differences Abstract: Continuation of first talk by Jacob Fox. The first part of the talk by Jacob Fox includes an overview and discusses the upper bound. The second part of the talk by Yufei Zhao focuses on the lower bound construction and proof. These results are all joint work with Huy Tuan Pham.
12:00-1:30pm Lunch 1:30-2:20pm Jop Briët Locally decodable codes and arithmetic progressions in random settings Abstract: This talk is about a common feature of special types of error correcting codes, so-called locally decodable codes (LDCs), and two problems on arithmetic progressions in random settings, random differences in Szemerédi’s theorem and upper tails for arithmetic progressions in a random set in particular. It turns out that all three can be studied in terms of the Gaussian width of a set of vectors given by a collection of certain polynomials. Using a matrix version of the Khintchine inequality and a lemma that turns such polynomials into matrices, we give an alternative proof for the best-known lower bounds on LDCs and improved versions of prior results due to Frantzikinakis et al. and Bhattacharya et al. on arithmetic progressions in the aforementioned random settings.
Joint work with Sivakanth Gopi
2:20-3:00pm Coffee Break 3:00-3:50pm Fernando Shao Large deviations for arithmetic progressions
Abstract: We determine the asymptotics of the log-probability that the number of k-term arithmetic progressions in a random subset of integers exceeds its expectation by a constant factor. This is the arithmetic analog of subgraph counts in a random graph. I will highlight some open problems in additive combinatorics that we encountered in our work, namely concerning the “complexity” of the dual functions of AP-counts.
4:00-6:00pm Welcome Reception Tuesday, October 3
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Emanuele Viola Interleaved group products Authors: Timothy Gowers and Emanuele Viola
Abstract: Let G be the special linear group SL(2,q). We show that if (a1,a2) and (b1,b2) are sampled uniformly from large subsets A and B of G^2 then their interleaved product a1 b1 a2 b2 is nearly uniform over G. This extends a result of Gowers (2008) which corresponds to the independent case where A and B are product sets. We obtain a number of other results. For example, we show that if X is a probability distribution on G^m such that any two coordinates are uniform in G^2, then a pointwise product of s independent copies of X is nearly uniform in G^m, where s depends on m only. Similar statements can be made for other groups as well.
These results have applications in computer science, which is the area where they were first sought by Miles and Viola (2013).
10:20-11:00am Coffee Break 11:00-11:50am Vsevolod Lev On Isoperimetric Stability Abstract: We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-c)|S||A|$ with a real $c\in(0,1]$, then $|A|\ge4^{(1-1/d)c|S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible.
As a corollary, we derive an upper bound for the size of the largest independent subset of the set of popular differences of a finite subset of an abelian group. For groups of exponent $2$ and $3$, our bound translates into a sharp estimate for the additive dimension of the popular difference set.
We also prove, as an auxiliary result, the following estimate of possible independent interest: if $A\subseteq{\mathbb Z}^n$ is a finite, non-empty downset, then, denoting by $w(z)$ the number of non-zero components of the vector $z\in\mathbb{Z}^n$, we have $$ \frac1{|A|} \sum_{a\in A} w(a) \le \frac12\, \log_2 |A|. $$
12:00-1:30pm Lunch 1:30-2:20pm Elena Grigorescu NP-Hardness of Reed-Solomon Decoding and the Prouhet-Tarry-Escott Problem Abstract: I will discuss the complexity of decoding Reed-Solomon codes, and some results establishing NP-hardness for asymptotically smaller decoding radii than the maximum likelihood decoding radius. These results follow from the study of a generalization of the classical Subset Sum problem to higher moments, which may be of independent interest. I will further discuss a connection with the Prouhet-Tarry-Escott problem studied in Number Theory, which turns out to capture a main barrier in extending our techniques to smaller radii.
Joint work with Venkata Gandikota and Badih Ghazi.
2:20-3:00pm Coffee Break 3:00-3:50pm Sean Prendiville Partition regularity of certain non-linear Diophantine equations. Abstract: We survey some results in additive Ramsey theory which remain valid when variables are restricted to sparse sets of arithmetic interest, in particular the partition regularity of a class of non-linear Diophantine equations in many variables.
Wednesday, October 4
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Olof Sisask Bounds on capsets via properties of spectra Abstract: A capset in F_3^n is a subset A containing no three distinct elements x, y, z satisfying x+z=2y. Determining how large capsets can be has been a longstanding problem in additive combinatorics, particularly motivated by the corresponding question for subsets of {1,2,…,N}. While the problem in the former setting has seen spectacular progress recently through the polynomial method of Croot–Lev–Pach and Ellenberg–Gijswijt, such progress has not been forthcoming in the setting of the integers. Motivated by an attempt to make progress in this setting, we shall revisit the approach to bounding the sizes of capsets using Fourier analysis, and in particular the properties of large spectra. This will be a two part talk, in which many of the ideas will be outlined in the first talk, modulo the proof of a structural result for sets with large additive energy. This structural result will be discussed in the second talk, by Thomas Bloom, together with ideas on how one might hope to achieve Behrend-style bounds using this method.
Joint work with Thomas Bloom.
10:20-11:00am Coffee Break 11:00-11:50am Thomas Bloom Bounds on capsets via properties of spectra This is a continuation of the previous talk by Olof Sisask.
12:00-1:30pm Lunch 1:30-2:20pm Hamed Hatami Polynomial method and graph bootstrap percolation Abstract: We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to determine the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel, and the latter provides an alternative and simpler proof for one of their main results. This is based on a joint work with Lianna Hambardzumyan and Yingjie Qian.
2:20-3:00pm Coffee Break 3:00-3:50pm Arnab Bhattacharyya Algorithmic Polynomial Decomposition Abstract: Fix a prime p. Given a positive integer k, a vector of positive integers D = (D_1, …, D_k) and a function G: F_p^k → F_p, we say a function P: F_p^n → F_p admits a (k, D, G)-decomposition if there exist polynomials P_1, …, P_k: F_p^n -> F_p with each deg(P_i) <= D_i such that for all x in F_p^n, P(x) = G(P_1(x), …, P_k(x)). For instance, an n-variate polynomial of total degree d factors nontrivially exactly when it has a (2, (d-1, d-1), prod)-decomposition where prod(a,b) = ab.
When show that for any fixed k, D, G, and fixed bound d, we can decide whether a given polynomial P(x_1, …, x_n) of degree d admits a (k,D,G)-decomposition and if so, find a witnessing decomposition, in poly(n) time. Our approach is based on higher-order Fourier analysis. We will also discuss improved analyses and algorithms for special classes of decompositions.
Joint work with Pooya Hatami, Chetan Gupta and Madhur Tulsiani.
Thursday, October 5
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Madhur Tulsiani Higher-order Fourier analysis and approximate decoding of Reed-Muller codes Abstract: Decomposition theorems proved by Gowers and Wolf provide an appropriate notion of “Fourier transform” for higher-order Fourier analysis. I will discuss some questions and techniques that arise from trying to develop polynomial time algorithms for computing these decompositions.
I will discuss constructive proofs of these decompositions based on boosting, which reduce the problem of computing these decompositions to a certain kind of approximate decoding problem for codes. I will also discuss some earlier and recent works on this decoding problem.
Based on joint works with Arnab Bhattacharyya, Eli Ben-Sasson, Pooya Hatami, Noga Ron-Zewi and Julia Wolf.
10:20-11:00am Coffee Break 11:00-11:50am Julia Wolf Stable arithmetic regularity The arithmetic regularity lemma in the finite-field model, proved by Green in 2005, states that given a subset A of a finite-dimensional vector space over a prime field, there exists a subspace H of bounded codimension such that A is Fourier-uniform with respect to almost all cosets of H. It is known that in general, the growth of the codimension of H is required to be of tower type depending on the degree of uniformity, and that one must allow for a small number of non-uniform cosets.
Our main result is that, under a natural model-theoretic assumption of stability, the tower-type bound and non-uniform cosets in the arithmetic regularity lemma are not necessary. Specifically, we prove an arithmetic regularity lemma for k-stable subsets in which the bound on the codimension of the subspace is a polynomial (depending on k) in the degree of uniformity, and in which there are no non-uniform cosets.
This is joint work with Caroline Terry.
12:00-1:30pm Lunch 1:30-2:20pm Will Sawin Constructions of Additive Matchings
Abstract: I will explain my work, with Robert Kleinberg and David Speyer, constructing large tri-colored sum-free sets in vector spaces over finite fields, and how it shows that some additive combinatorics problems over finite fields are harder than corresponding problems over the integers.
2:20-3:00pm Coffee Break 3:00-3:50pm Mei-Chu Chang Arithmetic progressions in multiplicative groups of finite fields Abstract: Let G be a multiplicative subgroup of the prime field F_p of size |G|> p^{1-\kappa} and r an arbitrarily fixed positive integer. Assuming \kappa=\kappa(r)>0 and p large enough, it is shown that any proportional subset A of G contains non-trivial arithmetic progressions of length r.
Friday, October 6
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:20am Asaf Ferber On a resilience version of the Littlewood-Offord problem Abstract: In this talk we consider a resilience version of the classical Littlewood-Offord problem. That is, consider the sum X=a_1x_1+…a_nx_n, where the a_i-s are non-zero reals and x_i-s are i.i.d. random variables with (x_1=1)= P(x_1=-1)=1/2. Motivated by some problems from random matrices, we consider the question: how many of the x_i-s can we typically allow an adversary to change without making X=0? We solve this problem up to a constant factor and present a few interesting open problems.
Joint with: Afonso Bandeira (NYU) and Matthew Kwan (ETH, Zurich).
10:20-11:00am Coffee Break 11:00-11:50am Kaave Hosseini Protocols for XOR functions and Entropy decrement Abstract: Let f:F_2^n –> {0,1} be a function and suppose the matrix M defined by M(x,y) = f(x+y) is partitioned into k monochromatic rectangles. We show that F_2^n can be partitioned into affine subspaces of co-dimension polylog(k) such that f is constant on each subspace. In other words, up to polynomial factors, deterministic communication complexity and parity decision tree complexity are equivalent.
This relies on a novel technique of entropy decrement combined with Sanders’ Bogolyubov-Ruzsa lemma.
Joint work with Hamed Hatami and Shachar Lovett
12:00-1:30pm Lunch 1:30-2:20pm Guy Kindler From the Grassmann graph to Two-to-Two games
Abstract: In this work we show a relation between the structure of the so called Grassmann graph over Z_2 and the Two-to-Two conjecture in computational complexity. Specifically, we present a structural conjecture concerning the Grassmann graph (together with an observation by Barak et. al., one can view this as a conjecture about the structure of non-expanding sets in that graph) which turns out to imply the Two-to-Two conjecture.
The latter conjecture its the lesser-known and weaker sibling of the Unique-Games conjecture [Khot02], which states that unique games (a.k.a. one-to-one games) are hard to approximate. Indeed, if the Grassmann-Graph conjecture its true, it would also rule out some attempts to refute the Unique-Games conjecture, as these attempts provide potentially efficient algorithms to solve unique games, that would actually also solve two-to-two games if they work at all.
These new connections between the structural properties of the Grassmann graph and complexity theoretic conjectures highlight the Grassmann graph as an interesting and worthy object of study. We may indicate some initial results towards analyzing its structure.
This is joint work with Irit Dinur, Subhash Khot, Dror Minzer, and Muli Safra.
Fluid turbulence and Singularities of the Euler/ Navier Stokes equations
The Workshop on Fluid turbulence and Singularities of the Euler/ Navier Stokes equations will take place on March 13-15, 2019. This is the first of two workshop organized by Michael Brenner, Shmuel Rubinstein, and Tom Hou. The second, Machine Learning for Multiscale Model Reduction, will take place on March 27-29, 2019. Both workshops will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Speakers:
- Claude Bardos, University of Paris
- Jiajie Chen, Caltech
- Peter Constantin, Princeton
- Diego Cordoba, ICMAT
- Tarek Elgindi, UCSD
- Susumu Goto, Osaka
- Alexander Kiselev, Duke University
- Alain Pumir, ENS Lyon
- Shmuel Rubinstein, Harvard SEAS
- Vladimir Sverak, University of Minnesota
- Edriss S. Titi, TAMU
- Vlad Vicol, Courant
- Sijue Wu, University of Michigan
- Andrej Zlatos, UCSD
Blockchain Conference
On January 24-25, 2019 the Center of Mathematical Sciences will be hosting a conference on distributed-ledger (blockchain) technology. The conference is intended to cover a broad range of topics, from abstract mathematical aspects (cryptography, game theory, graph theory, theoretical computer science) to concrete applications (in accounting, government, economics, finance, management, medicine). The talks will take place in Science Center, Hall D.
https://youtu.be/FyKCCutxMYo
Photos
Speakers:
- Joseph Abadi, Princeton University
- Benedikt Bunz, Stanford University
- Jake Cacciapaglia, Nebula Genomics/Harvard Medical School
- Eduardo Castello, Massachusetts Institute of Technology
- Alisa DiCaprio, R3
- Zhiguo He, University of Chicago
- Steven Kou, Boston University
- Anne Lafarre, Tilburg University
- Jacob Leshno, University of Chicago
- Bruce Schneier, Harvard Kennedy School
- David Schwartz, Ripple
- Elaine Shi, Cornell University/Thunder Research
- Hong Wan, NCSU
Workshop on Algebraic Methods in Combinatorics
The workshop on Algebraic Methods in Combinatorics will take place November 13-17, 2017 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
The main focus of the workshop is the application of algebraic method to study problems in combinatorics. In recent years there has been a large number of results in which the use of algebraic technique has resulted in significant improvements to long standing open problems. Such problems include the finite field Kakeya problem, the distinct distance problem of Erdos and, more recently, the cap-set problem. The workshop will include talks on all of the above mentioned problem as well as on recent development in related areas combining combinatorics and algebra.
Participation: The workshop is open to participation by all interested researchers, subject to capacity. Click here to register.
A list of lodging options convenient to the Center can also be found on our recommended lodgings page.
Confirmed participants include:
- Abdul Basit, Rutgers
- Boris Bukh, Carnegie Mellon University
- Pete L. Clark, University of Georgia
- David Conlon, University of Oxford
- Frank de Zeeuw, EPFL
- Thao Thi Thu Do, MIT
- Noam Elkies, Harvard University
- Jordan Ellenberg, University of Wisconsin
- Dion Gijswijt, Delft Institute of Technology
- Sivankanth Gopi, Princeton University
- Venkatesan Guruswami, Carnegie Mellon University
- Marina Iliopoulou, University of California, Berkeley
- Robert Kleinberg, Cornell University
- Michael Krivelevich, Tel Aviv University
- Vsevelod Lev, University of Haifa at Oranim
- László Miklós Lovász, UCLA
- Ben Lund, Rutgers
- Péter Pach, Budapest University of Technology and Economics
- János Pach, New York University
- Zuzana Patáková, Institute of Science and Technology Austria
- Orit Raz, Institute for Advanced Study
- Oliver Roche-Newton, Johannes Kepler University
- Misha Rudnev, University of Bristol
- Adam Sheffer, California Institute of Technology
- Amir Shpilka, Tel-Aviv University
- Noam Solomon, Harvard CMSA
- Jozsef Solymosi, University of British Columbia
- Benny Sudakov, ETH, Zurich
- Andrew Suk, University of California, San Diego
- Tibor Szabó, Freie Universität Berlin
- Chris Umans, California Institute of Technology
- Avi Wigderson, Princeton University
- Josh Zahl, University of British Columbia
Co-organizers of this workshop include Zeev Dvir, Larry Guth, and Shubhangi Saraf.
Click here for a list of registrants.
Monday, Nov. 13
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Jozsef Solymosi On the unit distance problem
Abstract: Erdos’ Unit Distances conjecture states that the maximum number of unit distances determined by n points in the plane is almost linear, it is O(n^{1+c}) where c goes to zero as n goes to infinity. In this talk I will survey the relevant results and propose some questions which would imply that the maximum number of unit distances is o(n^{4/3}).
10:30-11:00am Coffee Break 11:00-12:00pm Orit Raz Intersection of linear subspaces in R^d and instances of the PIT problem Abstract: In the talk I will tell about a new deterministic, strongly polynomial time algorithm which can be viewed in two ways. The first is as solving a derandomization problem, providing a deterministic algorithm to a new special case of the PIT (Polynomial Identity Testing) problem. The second is as computing the dimension of the span of a collection of flats in high dimensional space. The talk is based on a joint work with Avi Wigderson.
12:00-1:30pm Lunch 1:30-2:30pm Andrew Hoon Suk Ramsey numbers: combinatorial and geometric
Abstract: In this talk, I will discuss several results on determining the tower growth rate of Ramsey numbers arising in combinatorics and in geometry. These results are joint work with David Conlon, Jacob Fox, Dhruv Mubayi, Janos Pach, and Benny Sudakov.
2:30-3:00pm Coffee Break 3:00-4:00pm Josh Zahl Cutting curves into segments and incidence geometry
4:00-6:00pm Welcome Reception Tuesday, Nov. 14
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Péter Pál Pach Polynomials, rank and cap sets
Abstract: In this talk we will look at a new variant of the polynomial method which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like $\mathbb{Z}_4^n$ and $\mathbb{F}_q^n$ are exponentially small (compared to the size of the group). We will discuss lower and upper bounds for the size of the extremal subsets and mention further applications of the method.
10:30-11:00am Coffee Break 11:00-12:00pm Jordan Ellenberg The Degeneration Method
Abstract: In algebraic geometry, a very popular way to study (nice, innocent, nonsingular) varieties is to degenerate them to (weird-looking, badly singular, nonreduced) varieties (which are actually not even varieties but schemes.) I will talk about some results in combinatorics using this approach (joint with Daniel Erman) and some ideas for future applications of the method.
12:00-1:30pm Lunch 1:30-2:30pm Larry Guth The polynomial method in Fourier analysis Abstract: This will be a survey talk about how the polynomial method helps to understand problems in Fourier analysis. We will review some applications of the polynomial method to problems in combinatorial geometry. Then we’ll discuss some problems in Fourier analysis, explain the analogy with combinatorial problems, and discuss how to adapt the polynomial method to the Fourier analysis setting.
2:30-3:00pm
Coffee Break 3:00-4:00pm Open Problem Wednesday, Nov. 15
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Avi Wigderson The “rank method” in arithmetic complexity: Lower bounds and barriers to lower bounds
Abstract: Why is it so hard to find a hard function? No one has a clue! In despair, we turn to excuses called barriers. A barrier is a collection of lower bound techniques, encompassing as much as possible from those in use, together with a proof that these techniques cannot prove any lower bound better than the state-of-art (which is often pathetic, and always very far from what we expect for complexity of random functions).
In the setting of Boolean computation of Boolean functions (where P vs. NP is the central open problem), there are several famous barriers which provide satisfactory excuses, and point to directions in which techniques may be strengthened.
In the setting of Arithmetic computation of polynomials and tensors (where VP vs. VNP is the central open problem) we have no satisfactory barriers, despite some recent interesting attempts.
This talk will describe a new barrier for the Rank Method in arithmetic complexity, which encompass most lower bounds in this field. It also encompass most lower bounds on tensor rank in algebraic geometry (where the the rank method is called Flattening).
I will describe the rank method, explain how it is used to prove lower bounds, and then explain its limits via the new barrier result. As an example, it shows that while the best lower bound on the tensor rank of any explicit 3-dimensional tensor of side n (which is achieved by a rank method) is 2n, no rank method can prove a lower bound which exceeds 8n
(despite the fact that a random such tensor has rank quadratic in n).
No special background knowledge is assumed. The audience is expected to come up with new lower bounds, or else, with new excuses for their absence.
10:30-11:00am Coffee Break 11:00-12:00pm Venkat Guruswami Subspace evasion, list decoding, and dimension expanders
Abstract: A subspace design is a collection of subspaces of F^n (F = finite field) most of which are disjoint from every low-dimensional subspace of F^n. This notion was put forth in the context of algebraic list decoding where it enabled the construction of optimal redundancy list-decodable codes over small alphabets as well as for error-correction in the rank-metric. Explicit subspace designs with near-optimal parameters have been constructed over large fields based on polynomials with structured roots. (Over small fields, a construction via cyclotomic function fields with slightly worse parameters is known.) Both the analysis of the list decoding algorithm as well as the subspace designs crucially rely on the *polynomial method*.
Subspace designs have since enabled progress on linear-algebraic analogs of Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In particular, they yield an explicit construction of constant-degree dimension expanders over large fields. While constructions of such dimension expanders are known over any field, they are based on a reduction to a highly non-trivial form of vertex expanders called monotone expanders. In contrast, the subspace design approach is simpler and works entirely within the linear-algebraic realm. Further, in recent (ongoing) work, their combination with rank-metric codes yields dimension expanders with expansion proportional to the degree.
This talk will survey these developments revolving around subspace designs, their motivation, construction, analysis, and connections.
(Based on several joint works whose co-authors include Chaoping Xing, Swastik Kopparty, Michael Forbes, Nicolas Resch, and Chen Yuan.)
12:00-1:30pm Lunch 1:30-2:30pm David Conlon Finite reflection groups and graph norms
Abstract: For any given graph $H$, we may define a natural corresponding functional $\|.\|_H$. We then say that $H$ is norming if $\|.\|_H$ is a semi-norm. A similar notion $\|.\|_{r(H)}$ is defined by $\| f \|_{r(H)} := \| | f | \|_H$ and $H$ is said to be weakly norming if $\|.\|_{r(H)}$ is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. Using results from the theory of finite reflection groups, we identify a much larger class of weakly norming graphs. This result includes all previous examples of weakly norming graphs and adds many more. We also discuss several applications of our results. In particular, we define and compare a number of generalisations of Gowers’ octahedral norms and we prove some new instances of Sidorenko’s conjecture. Joint work with Joonkyung Lee.
2:30-3:00pm Coffee Break 3:00-4:00pm Laszlo Miklós Lovasz Removal lemmas for triangles and k-cycles.
Abstract: Let p be a fixed prime. A k-cycle in F_p^n is an ordered k-tuple of points that sum to zero; we also call a 3-cycle a triangle. Let N=p^n, (the size of F_p^n). Green proved an arithmetic removal lemma which says that for every k, epsilon>0 and prime p, there is a delta>0 such that if we have a collection of k sets in F_p^n, and the number of k-cycles in their cross product is at most a delta fraction of all possible k-cycles in F_p^n, then we can delete epsilon times N elements from the sets and remove all k-cycles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to our work, the best known bound for any k, due to Fox, showed that 1/delta can be taken to be an exponential tower of twos of height logarithmic in 1/epsilon (for a fixed k).
In this talk, we will discuss recent work on Green’s problem. For triangles, we prove an essentially tight bound for Green’s arithmetic triangle removal lemma in F_p^n, using the recent breakthroughs with the polynomial method. For k-cycles, we also prove a polynomial bound, however, the question of the optimal exponent is still open.
The triangle case is joint work with Jacob Fox, and the k-cycle case with Jacob Fox and Lisa Sauermann.
Thursday, Nov. 16
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Janos Pach Let’s talk about multiple crossings Abstract: Let k>1 be a fixed integer. It is conjectured that any graph on n vertices that can be drawn in the plane without k pairwise crossing edges has O(n) edges. Two edges of a hypergraph cross each other if neither of them contains the other, they have a nonempty intersection, and their union is not the whole vertex set. It is conjectured that any hypergraph on n vertices that contains no k pairwise crossing edges has at most O(n) edges. We discuss the relationship between the above conjectures and explain some partial answers, including a recent result of Kupavskii, Tomon, and the speaker, improving a 40 years old bound of Lomonosov.
10:30-11:00am Coffee Break 11:00-12:00pm Misha Rudnev Few products, many sums
Abstract: This is what I like calling “weak Erd\H os-Szemer\’edi conjecture”, still wide open over the reals and in positive characteristic. The talk will focus on some recent progress, largely based on the ideas of I. D. Shkredov over the past 5-6 years of how to use linear algebra to get the best out of the Szemer\’edi-Trotter theorem for its sum-product applications. One of the new results is strengthening (modulo the log term hidden in the $\lesssim$ symbol) the textbook Elekes inequality
$$
|A|^{10} \ll |A-A|^4|AA|^4
$$
to
$$|A|^{10}\lesssim |A-A|^3|AA|^5.$$
The other is the bound
$$E(H) \lesssim |H|^{2+\frac{9}{20}}$$ for additive energy of sufficiently small multiplicative subgroups in $\mathbb F_p$.
12:00-1:30pm Lunch 1:30-2:30pm Adam Sheffer Geometric Energies: Between Discrete Geometry and Additive Combinatorics
Abstract: We will discuss the rise of geometric variants of the concept of Additive energy. In recent years such variants are becoming more common in the study of Discrete Geometry problems. We will survey this development and then focus on a recent work with Cosmin Pohoata. This work studies geometric variants of additive higher moment energies, and uses those to derive new bounds for several problems in Discrete Geometry.
2:30-3:00pm Coffee Break 3:00-4:00pm Boris Bukh Ranks of matrices with few distinct entries
Abstract: Many applications of linear algebra method to combinatorics rely on the bounds on ranks of matrices with few distinct entries and constant diagonal. In this talk, I will explain some of these application. I will also present a classification of sets L for which no low-rank matrix with entries in L exists.
Friday, Nov. 17
Time Speaker Title/Abstract 9:00-9:30am Breakfast 9:30-10:30am Benny Sudakov Submodular minimization and set-systems with restricted intersections
Abstract: Submodular function minimization is a fundamental and efficiently solvable problem class in combinatorial optimization with a multitude of applications in various fields. Surprisingly, there is only very little known about constraint types under which it remains efficiently solvable. The arguably most relevant non-trivial constraint class for which polynomial algorithms are known are parity constraints, i.e., optimizing submodular function only over sets of odd (or even) cardinality. Parity constraints capture classical combinatorial optimization problems like the odd-cut problem, and they are a key tool in a recent technique to efficiently solve integer programs with a constraint matrix whose subdeter-minants are bounded by two in absolute value.
We show that efficient submodular function minimization is possible even for a significantly larger class than parity constraints, i.e., over all sets (of any given lattice) of cardinality r mod m, as long as m is a constant prime power. To obtain our results, we combine tools from Combinatorial Optimization, Combinatorics, and Number Theory. In particular, we establish an interesting connection between the correctness of a natural algorithm, and the non-existence of set systems with specific intersection properties.
Joint work with M. Nagele and R. Zenklusen
10:30-11:00am Coffee Break 11:00-12:00pm Robert Kleinberg Explicit sum-of-squares lower bounds via the polynomial method
Abstract: The sum-of-squares (a.k.a. Positivstellensatz) proof system is a powerful method for refuting systems of multivariate polynomial inequalities, i.e. proving that they have no solutions. These refutations themselves involve sum-of-squares (sos) polynomials, and while any unsatisfiable system of inequalities has a sum-of-squares refutation, the sos polynomials involved might have arbitrarily high degree. However, if a system admits a refutation where all polynomials involved have degree at most d, then the refutation can be found by an algorithm with running time polynomial in N^d, where N is the combined number of variables and inequalities in the system.
Low-degree sum-of-squares refutations appear throughout mathematics. For example, the above proof search algorithm captures as a special case many a priori unrelated algorithms from theoretical computer science; one example is Goemans and Williamson’s algorithm to approximate the maximum cut in a graph. Specialized to extremal graph theory, they become equivalent to flag algebras. They have also seen practical use in robotics and optimal control.
Therefore, it is of interest to identify “hard” systems of low-degree polynomial inequalities that have no solutions but also have no low-degree sum-of-squares refutations. Until recently, the only known examples were either not explicit (i.e., known to exist by non-constructive means such as the probabilistic method) or not robust (i.e., a system is constructed which is not refutable by degree d sos polynomials, but becomes refutable when perturbed by an amount tending to zero with d). We present a new family of instances derived from the cap-set problem, and we show a super-constant lower bound on the degree of its sum-of-squares refutations. Our instances are both explicit and robust.
This is joint work with Sam Hopkins.
12:00-1:30pm Lunch Naturalness and muon anomalous magnetic moment
Title: Naturalness and muon anomalous magnetic moment
Abstract: We study a model for explaining the apparent deviation of the muon anomalous magnetic moment, (g-2), from the Standard Model expectation. There are no new scalars and hence no new hierarchy puzzles beyond those associated with the Standard model Higgs; the only new particles that are relevant for (g-2) are vector-like singlet and doublet leptons. Interestingly, this simple model provides a calculable example violating the Wilsonian notion of naturalness: despite the absence of any symmetries prohibiting its generation, the coefficient of the naively leading dimension-six operator for (g−2) vanishes at one-loop. While effective field theorists interpret this either as a surprising UV cancellation of power divergences, or as a delicate cancellation between matching UV and calculable IR corrections to (g−2) from parametrically separated scales, there is a simple explanation in the full theory: the loop integrand is a total derivative of a function vanishing in both the deep UV and IR. The leading contribution to (g−2) arises from dimension-eight operators, and thus the required masses of new fermions are lower than naively expected, with a sizable portion of parameter space already covered by direct searches at the LHC. All of the the viable parameter can be probed by the LHC and planned future colliders.
Exotic quantum matter: From lattice gauge theory to hyperbolic lattices
Title: Exotic quantum matter: From lattice gauge theory to hyperbolic lattices
Abstract: This talk, in two parts, will discuss two (unrelated) instances of exotic quantum matter. In the first part, I will discuss quantum critical points describing possible transitions out of the Dirac spin liquid, towards either symmetry-breaking phases or topologically ordered spin liquids. I will also comment on the role of instanton zero modes for symmetry breaking in parton gauge theories. In the second part, I will propose an extension of Bloch band theory to hyperbolic lattices, such as those recently realized in circuit QED experiments, based on ideas from algebraic geometry and Riemann surface theory.
Cornering the universal shape of fluctuations and entanglement
Title: Cornering the universal shape of fluctuations and entanglement
Abstract: Understanding the fluctuations of observables is one of the main goals in physics. We investigate such fluctuations when a subregion of the full system can be observed, focusing on geometries with corners. We report that the dependence on the opening angle is super-universal: up to a numerical prefactor, this function does not depend on anything, provided the system under study is uniform, isotropic, and correlations do not decay too slowly. The prefactor contains important physical information: we show in particular that it gives access to the long-wavelength limit of the structure factor. We illustrate our findings with several examples: classical fluids, fractional quantum Hall (FQH) states, scale invariant quantum critical theories, and metals. Finally, we discuss connections with the entanglement entropy, including new results for Laughlin FQH states.
Ref: arXiv:2102.06223
Quantum gravity from quantum matter
Title: Quantum gravity from quantum matter
Abstract: We present a model of quantum gravity in which dimension, topology and geometry of spacetime are collective dynamical variables that describe the pattern of entanglement of underlying quantum matter. As spacetimes with arbitrary dimensions can emerge, the gauge symmetry is generalized to a group that includes diffeomorphisms in general dimensions. The gauge symmetry obeys a first-class constraint operator algebra, and is reduced to a generalized hypersurface deformation algebra in states that exhibit classical spacetimes. In the semi-classical limit, we find a saddle-point solution that describes a series of (3+1)-dimensional de Sitter-like spacetimes with the Lorentzian signature bridged by Euclidean spaces in between.
9/23/2021 Interdisciplinary Science Seminar
Title: The number of n-queens configurations
Abstract: The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. The problem has a storied history and was studied by such eminent mathematicians as Gauss and Polya. The problem has also found applications in fields such as algorithm design and circuit development.
Despite much study, until recently very little was known regarding the asymptotics of Q(n). We apply modern methods from probabilistic combinatorics to reduce understanding Q(n) to the study of a particular infinite-dimensional convex optimization problem. The chief implication is that (in an appropriate sense) for a~1.94, Q(n) is approximately (ne^(-a))^n. Furthermore, our methods allow us to study the typical “shape” of n-queens configurations.
10/7/2021 Interdisciplinary Science Seminar
Title: SiRNA Targeting TCRb: A Proposed Therapy for the Treatment of Autoimmunity
Abstract: As of 2018, the United States National Institutes of Health estimate that over half a billion people worldwide are affected by autoimmune disorders. Though these conditions are prevalent, treatment options remain relatively poor, relying primarily on various forms of immunosuppression which carry potentially severe side effects and often lose effectiveness over time. Given this, new forms of therapy are needed. To this end, we have developed methods for the creation of small-interfering RNA (siRNA) for hypervariable regions of the T-cell receptor β-chain gene (TCRb) as a highly targeted, novel means of therapy for the treatment of autoimmune disorders.
This talk will review the general mechanism by which autoimmune diseases occur and discuss the pros and cons of conventional pharmaceutical therapies as they pertain to autoimmune disease treatment. I will then examine the rational and design methodology for the proposed siRNA therapy and how it contrasts with contemporary methods for the treatment of these conditions. Additionally, the talk will compare the efficacy of multiple design strategies for such molecules by comparison over several metrics and discuss how this will be guiding future research.
10/14/2021 Interdisciplinary Science Seminar
Title: D3C: Reducing the Price of Anarchy in Multi-Agent Learning
Abstract: In multi-agent systems the complex interaction of fixed incentives can lead agents to outcomes that are poor (inefficient) not only for the group but also for each individual agent. Price of anarchy is a technical game theoretic definition introduced to quantify the inefficiency arising in these scenarios– it compares the welfare that can be achieved through perfect coordination against that achieved by self-interested agents at a Nash equilibrium. We derive a differentiable upper bound on a price of anarchy that agents can cheaply estimate during learning. Equipped with this estimator agents can adjust their incentives in a way that improves the efficiency incurred at a Nash equilibrium. Agents adjust their incentives by learning to mix their reward (equiv. negative loss) with that of other agents by following the gradient of our derived upper bound. We refer to this approach as D3C. In the case where agent incentives are differentiable D3C resembles the celebrated Win-Stay Lose-Shift strategy from behavioral game theory thereby establishing a connection between the global goal of maximum welfare and an established agent-centric learning rule. In the non-differentiable setting as is common in multiagent reinforcement learning we show the upper bound can be reduced via evolutionary strategies until a compromise is reached in a distributed fashion. We demonstrate that D3C improves outcomes for each agent and the group as a whole on several social dilemmas including a traffic network exhibiting Braess’s paradox a prisoner’s dilemma and several reinforcement learning domains.
10/21/2021 Interdisciplinary Science Seminar
Title: Mathematical resolution of the Liouville conformal field theory.
Abstract: The Liouville conformal field theory is a well-known beautiful quantum field theory in physics describing random surfaces. Only recently a mathematical approach based on a well-defined path integral to this theory has been proposed using probability by David, Kupiainen, Rhodes, Vargas.
Many works since the ’80s in theoretical physics (starting with Belavin-Polyakov-Zamolodchikov) tell us that conformal field theories in dimension 2 are in general « Integrable », the correlations functions are solutions of PDEs and can in principle be computed explicitely by using algebraic tools (vertex operator algebras, representations of Virasoro algebras, the theory of conformal blocks). However, for Liouville Theory this was not done at the mathematical level by algebraic methods.
I’ll explain how to combine probabilistic, analytic and geometric tools to give explicit (although complicated) expressions for all the correlation functions on all Riemann surfaces in terms of certain holomorphic functions of the moduli parameters called conformal blocks, and of the structure constant (3-point function on the sphere). This gives a concrete mathematical proof of the so-called conformal bootstrap and of Segal’s gluing axioms for this CFT. The idea is to break the path integral on a closed surface into path integrals on pairs of pants and reduce all correlation functions to the 3-point correlation function on the Riemann sphere $S^2$. This amounts in particular to prove a spectral resolution of a certain operator acting on $L^2(H^{-s}(S^1))$ where $H^{-s}(S^1)$ is the Sobolev space of order -s<0 equipped with a Gaussian measure, which is viewed as the space of fields, and to construct a certain representation of the Virasoro algebra into unbounded operators acting on this Hilbert space.
This is joint work with A. Kupiainen, R. Rhodes and V. Vargas.
More Exact Results in Gauge Theories: Confinement and Chiral Symmetry Breaking
Title: More Exact Results in Gauge Theories: Confinement and Chiral Symmetry Breaking
Abstract: In this follow-up to Hitoshi Murayama’s talk “Some Exact Results in QCD-like and Chiral Gauge Theories”, I present a detailed analysis of the phases of $SO(N_c)$ gauge theory.
Starting with supersymmetric $SO(N_c)$ with $N_F$ flavors, we extrapolate to the non-supersymmetric limit using anomaly-mediated supersymmetry breaking (AMSB). Interestingly, the abelian Coulomb and free magnetic phases do not survive supersymmetry breaking and collapse to a confining phase. This provided one of the first demonstrations of true confinement with chiral symmetry breaking in a non-SUSY theory.ARCH: Know What Your Machine Doesn’t Know
Speaker: Jie Yang, Delft University of Technology
Title: ARCH: Know What Your Machine Doesn’t Know
Abstract: Despite their impressive performance, machine learning systems remain prohibitively unreliable in safety-, trust-, and ethically sensitive domains. Recent discussions in different sub-fields of AI have reached the consensus of knowledge need in machine learning; few discussions have touched upon the diagnosis of what knowledge is needed. In this talk, I will present our ongoing work on ARCH, a knowledge-driven, human-centered, and reasoning-based tool, for diagnosing the unknowns of a machine learning system. ARCH leverages human intelligence to create domain knowledge required for a given task and to describe the internal behavior of a machine learning system; it infers the missing or incorrect knowledge of the system with the built-in probabilistic, abductive reasoning engine. ARCH is a generic tool that can be applied to machine learning in different contexts. In the talk, I will present several applications in which ARCH is currently being developed and tested, including health, finance, and smart buildings.
Three-particle mechanism for pairing and superconductivity
Title: Three-particle mechanism for pairing and superconductivity
Abstract: I will present a new mechanism and an exact theory of electron pairing due to repulsive interaction in doped insulators. When the kinetic energy is small, the dynamics of adjacent electrons on the lattice is strongly correlated. By developing a controlled kinetic energy expansion, I will show that two doped charges can attract and form a bound state, despite and because of the underlying repulsion. This attraction by repulsion is enabled by the virtual excitation of a third electron in the filled band. This three-particle pairing mechanism leads to a variety of novel phenomena at finite doping, including spin-triplet superconductivity, pair density wave, BCS-BEC crossover and Feshbach resonance involving “trimers”. Possible realizations in moire materials, ZrNCl and WTe2 will be discussed.
[1] V. Crepel and L. Fu, Science Advances 7, eabh2233 (2021)
[2] V. Crepel and L. Fu, arXiv:2103.12060
[3] K. Slagle and L. Fu, Phys. Rev. B 102, 235423 (2020)The Hilbert Space of large N Chern-Simons matter theories
Title: The Hilbert Space of large N Chern-Simons matter theories
Abstract: We demonstrate that all known formulae for the thermal partition function for large N Chern Simons matter theory admit a simple Hilbert Space interpretation. In each case this quantity equals the partition function of an associated ungauged large $N$ matter theory with a particular local Lagrangian with one additional element: the Fock Space of this associated theory is projected down to the subspace of its WZW singlets. This projection, in particular, implies the previously encountered `Bosonic Exclusion Principle’, namely that no single particle state can be occupied by more than $k_B$ particles ($k_B$ is the Chern Simons level). Unlike its Gauss Law counterpart, the WZW constraint does not trivialize in the large volume limit. However thermodynamics does simplify in this limit; the final partition function reduces to a product of partition functions associated with each single particle state. These individual single particle state partition functions are a one parameter generalizations of their free boson and free fermion counterparts, and reduce to the later at extreme values of the ‘t Hooft coupling. At generic values of the rank and the level the occupation statistics of each energy level is given by a $q$ deformation of the usual free formulae of Bose and Fermi statistics.
Strong Coupling Theory of Magic-Angle Graphene: A Pedagogical Introduction
Title: Strong Coupling Theory of Magic-Angle Graphene: A Pedagogical Introduction
Abstract: In this talk, I will review a recently developed strong coupling theory of magic-angle twisted bilayer graphene. An advantage of this approach is that a single formulation can capture both the insulating and superconducting states, and with a few simplifying assumptions, can be treated analytically. I begin by reviewing the electronic structure of magic angle graphene’s flat bands, in a limit that exposes their peculiar band topology and geometry. I will show how similarities between the flat bands and the lowest Landau level can provide valuable insights into the effect of interactions and form the basis for an analytic treatment of the problem. At integer fillings, this approach points to flavor ordered insulators, which can be captured by a sigma-model in its ordered phase. Remarkably, topological textures of the sigma model carry electric charge which enables the same theory to describe the doped phases away from integer filling. I will show how this approach can lead to superconductivity on disordering the sigma model, and estimate the Tc for the superconductor. I will highlight the important role played by an effective super-exchange coupling both in pairing and in setting the effective mass of Cooper pairs. At the end, I will show how this theory provides criteria to predict which multilayer graphene stacks are expected to superconduct including the recently discovered alternating twist trilayer platform.
UV/IR and Effective Field Theory
Speaker: Nima Arkani-Hamed (IAS Princeton)
Title: UV/IR and Effective Field Theory
Tropical disk counts
Abstract: (joint with S. Venugopalan) I will describe version of the Fukaya algebra that appears in a tropical degeneration with the Lagrangian being one of the “tropical fibers”. An example is the count of “twenty-one disks in the cubic surface” (suggested by Sheridan) which is an open analog of the twenty-seven lines. As an application, I will explain why the Floer cohomology of such tropical fibers is well-defined; this is a generalization fo a result of Fukaya-Oh-Ohta-Ono for toric varieties.
RANDOM MATRIX PROGRAM
arge random matrices provide some of the simplest models for large, strongly correlated quantum systems. The statistics of the energy levels of ensembles of such systems are expected to exhibit universality, in the sense that they depend only on the symmetry class of the system. Recent advances have enabled a rigorous understanding of universality in the case of orthogonal, Hermitian, or symplectic matrices with independent entries, resolving a conjecture of Wigner-Dyson-Mehta dating back 50 years. These new developments have exploited techniques from a wide range of mathematical areas in addition to probability, including combinatorics, partial differential equations, and hydrodynamic limits. It is hoped that these new techniques will be useful in the analysis of universal behaviour in matrix ensembles with more complicated structure such as random regular graph models, or 2D matrix ensembles, as well as more physically relevant systems such as band matrices and random Schroedinger-type Hamiltonians. For some of these models, results in the direction of universality have already been obtained.
Here is a partial list of the mathematicians who are participating in this program
TOPOLOGICAL ASPECTS OF CONDENSED MATTER
During Academic year 2018-19, the CMSA will be hosting a Program on Topological Aspects of Condensed Matter. New ideas rooted in topology have recently had a big impact on condensed matter physics, and have highlighted new connections with high energy physics, mathematics and quantum information theory. Additionally, these ideas have found applications in the design of photonic systems and of materials with novel mechanical properties. The aim of this program will be to deepen these connections by foster discussion and seeding new collaborations within and across disciplines.
As part of the Program, the CMSA will be hosting two workshops:
- Workshop on Topology and Quantum Phases of Matter (August 27-28, 2018)
- Workshop on Topological Aspects of Condensed Matter (Septmeber 10-11, 2019)
.
Additionally, a weekly Topology Seminar will be held on Mondays from 10:00-11:30pm in CMSA room G10.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special programName Tentative Visiting Dates 11/12/2018-11/16/2018 Maissam Barkeshli 4/22/2019 – 4/26/2019 Xie Chen 4/15-17/2019 4/19-21/2019 4/24-30/2019 1/7/2019-1/11/2019 8/15/2018-8/30/2018 & 5/9/2019-5/19/2019 10/14/2018-10/27/2018 Anton Kapustin 8/26/2018-8/30/2018 & 3/28/2019-4/5/2019 3/11/2019-3/15/2019 Yuan-Ming Lu 4/29/2019-6/01/2019 4/2/2019- 4/19/2019 4/22/2019-5/22/2019 Chong Wang 10/22/2018-11/16/2018 4/1/2019-4/16/2019 Cenke Xu 8/26/2018-10/1/2018 4/1/2019-4/19/2019 5/1/2019-5/10/2019 Mathematical Biology
During Academic year 2018-19, the CMSA will be hosting a Program on Mathematical Biology.
Just over a century ago, the biologist, mathematician and philologist D’Arcy Thompson wrote “On growth and form”. The book was a visionary synthesis of the geometric biology of form at the time. It also served as a call for mathematical and physical approaches to understanding the evolution and development of shape.
In the century since its publication, we have seen a revolution in biology following the discovery of the genetic code, which has uncovered the molecular and cellular basis for life, combined with the ability to probe the chemical, structural, and dynamical nature of molecules, cells, tissues and organs across scales. In parallel, we have seen a blossoming of our understanding of spatiotemporal patterning in physical systems, and a gradual unveiling of the complexity of physical form. And in mathematics and computation, there has been a revolution in terms of posing and solving problems at the intersection of computational geometry, statistics and inference. So, how far are we from realizing a descriptive, predictive and controllable theory of biological shape?
In Fall 2018, CMSA will focus on a program that aims at recent mathematical advances in describing shape using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems
The CMSA will be hosting three workshops as part of this program. The Workshop on Morphometrics, Morphogenesis and Mathematics will take place on October 22-26.
A workshop on Morphogenesis: Geometry and Physics will take place on December 3-6, 2018.
A workshop on Invariance and Geometry in Sensation, Action and Cognition will take place on April 15-17, 2019.
SPACETIME AND QUANTUM MECHANICS, TOTAL POSITIVITY AND MOTIVES
Recent developments have poised this area to make serious advances in 2019, and we feel that bringing together many of the relevant experts for an intensive semester of discussions and collaboration will trigger some great things to happen. To this end, the organizers will host a small workshop during fall 2019, with between 20-30 participants. They will also invite 10-20 longer-term visitors throughout the semester. Additionally, there will be a seminar held weekly on Thursdays at 2:30pm in CMSA G10.
Organizers:
- Nima Arkani-Hamed (IAS)
- Lauren Williams (Harvard)
- Alexander Postnikov (MIT)
- Thomas Lam (Michigan)
.
Workshops:
Spacetime and Quantum Mechanics Workshop, October 28-30, 2019Here is a partial list of the mathematicians and physicists who have indicated that they will attend part or all of this special program as a visitor:
- Paolo Benincasa, 11/17/2019 – 11/29/2019
- Jacob Bourjaily, 9/1/2019 – 12/15/2019
- Francis Brown, 9/15/2019 – 9/20/2019
- Simon Caron-Huot, 9/30/2019 – 10/04/2019
- Lance Dixon, 9/9/2019 – 9/20/2019
- Charles Doran, 10/19/2019 – 11/1/2019
- James Drummond, 10/14/2019 – 10/18/2019
- Nick Early, 11/18/2019 – 11/22/2019
- Livia Ferro, 10/27/2019 – 11/9/2019
- Sergey Fomin, 10/6/2019 – 10/16/2019
- Sebastian Franco, 10/9/2019 – 10/19/2019
- Hadleigh Frost, 9/15/2019 – 12/20/2019
- Michael Green, 10/05/2019 – 10/13/2019
- Alexander Goncharov, 12/05/2019 – 12/20/2019
- Song He, 9/29/2019 – 11/10/2019
- Xuhua He, 10/30/2019-11/03/2019.
- Enrico Herrmann, 10/27/2019 – 11/9/2019
- Yutin Huang, 9/30/2019 – 10/12/2019
- Steven Karp, 10/11/2019 – 11/03/2019
- Tomasz Lukowski, 10/27/2019 – 11/11/2019
- Andrew McLeod, 10/6/2019 – 10/19/2019 & 11/3/2019 – 11/16/2019
- Sebastian Mizera, 10/28/2019 – 11/1/2019
- Erik Panzer, 9/15/2019 – 9/25/2019
- Matteo Parisi, 10/26/2019 – 11/10/2019
- Julio Parra-Martinez, 10/10/2019 – 05/12/2019
- Pierpaolo Mastrolia, 11/8/2019 – 11/16/2019
- Pasha Pylyavskyy, 9/8/2019 – 9/22/2019 & 10/14/2019 – 11/1/2019
- Junjie Rao, 10/25/2019 – 11/04/2019
- Giulio Salvatori, 9/3/2019 – 12/15/2019
- Michael Shapiro, 10/27/2019 – 11/2/2019
- David Speyer, 10/14/2019 – 10/18/2019
- Hugh Thomas, 10/27/2019 – 11/22/2019
- Jaroslav Trnka, 9/30/2019 – 10/04/2019, 10/28/2019 – 11/01/2019, 11/18/2019 – 11/22/2019
- Cristian Vergu, 11/10/2019 – 11/30/2019
- Matthias Volk, 10/14/2019 – 10/25/2019
- Matthew von Hippel, 11/11/2019 – 11/22/2019
- Pierre Vanhove, 10/22/2019 – 10/31/2019
- Matthias Wilhelm, 10/14/2019 – 10/25/2019
THE SIMONS COLLABORATION IN HOMOLOGICAL MIRROR SYMMETRY
The Simons Collaboration program in Homological Mirror Symmetry at Harvard CMSA and Brandeis University is part of the bigger Simons collaboration program on Homological mirror symmetry (https://schms.math.berkeley.edu) which brings to CMSA experts on algebraic geometry, Symplectic geometry, Arithmetic geometry, Quantum topology and mathematical aspects of high energy physics, specially string theory with the goal of proving the homological mirror symmetry conjecture (HMS) in full generality and explore its applications. Mirror symmetry, which emerged in the late 1980s as an unexpected physical duality between quantum field theories, has been a major source of progress in mathematics. At the 1994 ICM, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves of a smooth projective variety is equivalent to the Fukaya category of a mirror symplectic manifold (or Landau-Ginzburg model). We are happy to announce that the Simons Foundation has agreed to renew funding for the HMS collaboration program for three additional years.
A brief induction of the Brandeis-Harvard CMSA HMS/SYZ research agenda and team members are as follow:
Directors:
Shing-Tung Yau (Harvard University)Born in Canton, China, in 1949, S.-T. Yau grew up in Hong Kong, and studied in the Chinese University of Hong Kong from 1966 to 1969. He did his PhD at UC Berkeley from 1969 to 1971, as a student of S.S. Chern. He spent a year as a postdoc at the Institute for Advanced Study in Princeton, and a year as assistant professor at SUNY at Stony Brook. He joined the faculty at Stanford in 1973. On a Sloan Fellowship, he spent a semester at the Courant Institute in 1975. He visited UCLA the following year, and was offered a professorship at UC Berkeley in 1977. He was there for a year, before returning to Stanford. He was a plenary speaker at the 1978 ICM in Helsinki. The following year, he became a faculty member at the IAS in Princeton. He moved to UCSD in 1984. Yau came to Harvard in 1987, and was appointed the Higgins Professor of Mathematics in 1997. He has been at Harvard ever since. Yau has received numerous prestigious awards and honors throughout his career. He was named a California Scientist of the Year in 1979. In 1981, he received a Oswald Veblen Prize in Geometry and a John J. Carty Award for the Advancement of Science, and was elected a member of the US National Academy of Sciences. In 1982, he received a Fields Medal for “his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex MongeAmpre equations”. He was named Science Digest, America’s 100 Brightest Scientists under 40, in 1984. In 1991, he received a Humboldt Research Award from the Alexander von Humboldt Foundation in Germany. He was awarded a Crafoord Prize in 1994, a US National Medal of Science in 1997, and a China International Scientific and Technological Cooperation Award, for “his outstanding contribution to PRC in aspects of making progress in sciences and technology, training researchers” in 2003. In 2010, he received a Wolf Prize in Mathematics, for “his work in geometric analysis and mathematical physics”. Yau has also received a number of research fellowships, which include a Sloan Fellowship in 1975-1976, a Guggenheim Fellowship in 1982, and a MacArthur Fellowship in 1984-1985. Yau’s research interests include differential and algebraic geometry, topology, and mathematical physics. As a graduate student, he started to work on geometry of manifolds with negative curvature. He later became interested in developing the subject of geometric analysis, and applying the theory of nonlinear partial differential equations to solve problems in geometry, topology, and physics. His work in this direction include constructions of minimal submanifolds, harmonic maps, and canonical metrics on manifolds. The most notable, and probably the most influential of this, was his solution of the Calabi conjecture on Ricci flat metrics, and the existence of Kahler-Einstein metrics. He has also succeeded in applying his theory to solve a number of outstanding conjectures in algebraic geometry, including Chern number inequalities, and the rigidity of complex structures of complex projective spaces. Yau’s solution to the Calabi conjecture has been remarkably influential in mathematical physics over the last 30 years, through the creation of the theory of Calabi-Yau manifolds, a theory central to mirror symmetry. He and a team of outstanding mathematicians trained by him, have developed many important tools and concepts in CY geometry and mirror symmetry, which have led to significant progress in deformation theory, and on outstanding problems in enumerative geometry. Lian, Yau and his postdocs have developed a systematic approach to study and compute period integrals of CY and general type manifolds. Lian, Liu and Yau (independently by Givental) gave a proof of the counting formula of Candelas et al for worldsheet instantons on the quintic threefold. In the course of understanding mirror symmetry, Strominger, Yau, and Zaslow proposed a new geometric construction of mirror symmetry, now known as the SYZ construction. This has inspired a rapid development in CY geometry over the last two decades. In addition to CY geometry and mirror symmetry, Yau has done influential work on nonlinear partial differential equations, generalized geometry, Kahler geometry, and general relativity. His proof of positive mass conjecture is a widely regarded as a cornerstone in the classical theory of general relativity. In addition to publishing well over 350 research papers, Yau has trained more than 60 PhD students in a broad range of fields, and mentored dozens of postdoctoral fellows over the last 40 years.
Professor Bong Lian (Brandeis University)Born in Malaysia in 1962, Bong Lian completed his PhD in physics at Yale University under the direction of G. Zuckerman in 1991. He joined the permanent faculty at Brandeis University in 1995, and has remained there since. Between 1995 and 2013, he had had visiting research positions at numerous places, including the National University of Taiwan, Harvard University, and Tsinghua University. Lian received a J.S. Guggenheim Fellowship in 2003. He was awarded a Chern Prize at the ICCM in Taipei in 2013, for his “influential and fundamental contributions in mathematical physics, in particular in the theory of vertex algebras and mirror symmetry.” He has also been co-Director, since 2014, of the Tsinghua Mathcamp, a summer outreach program launched by him and Yau for mathematically talented teenagers in China. Since 2008, Lian has been the President of the International Science Foundation of Cambridge, a non-profit whose stated mission is “to provide financial and logistical support to scholars and universities, to promote basic research and education in mathematical sciences, especially in the Far East.” Over the last 20 years, he has mentored a number of postdocs and PhD students. His research has been supported by an NSF Focused Research Grant since 2009. Published in well over 60 papers over 25 years, Lian’s mathematical work lies in the interface between representation theory, Calabi-Yau geometry, and string theory. Beginning in the late 80’s, Lian, jointly with Zuckerman, developed the theory of semi-infinite cohomology and applied it to problems in string theory. In 1994, he constructed a new invariant (now known as the Lian- Zuckerman algebra) of a topological vertex algebra, and conjectured the first example of a G algebra in vertex algebra theory. The invariant has later inspired a new construction of quantum groups by I. Frenkel and A. Zeitlin, as semi-infinite cohomology of braided vertex algebras, and led to a more recent discovery of new relationships between Courant algebroids, A-algebras, operads, and deformation theory of BV algebras. In 2010, he and his students Linshaw and Song developed important applications of vertex algebras in equivariant topology. Lian’s work in CY geometry and mirror symmetry began in early 90’s. Using a characteristic p version of higher order Schwarzian equations, Lian and Yau gave an elementary proof that the instanton formula of Candelas et al implies Clemens’s divisibility conjecture for the quintic threefold, for infinitely many degrees. In 1996, Lian (jointly with Hosono and Yau) answered the so-called Large Complex Structure Limit problem in the affirmative in many important cases. Around the same year, they announced their hyperplane conjecture, which gives a general formula for period integrals for a large class of CY manifolds, extending the formula of Candelas et al. Soon after, Lian, Liu and Yau (independently by Givental) gave a proof of the counting formula. In 2003, inspired by mirror symmetry, Lian (jointly with Hosono, Oguiso and Yau) discovered an explicit counting formula for Fourier-Mukai partners, and settled an old problem of Shioda on abelian and K3 surfaces. Between 2009 and 2014, Lian (jointly with Bloch, Chen, Huang, Song, Srinivas, Yau, and Zhu) developed an entirely new approach to study the so-called Riemann-Hilbert problem for period integrals of CY manifolds, and extended it to general type manifolds. The approach leads to an explicit description of differential systems for period integrals with many applications. In particular, he answered an old question in physics on the completeness of Picard-Fuchs systems, and constructed new differential zeros of hypergeometric functions.
Denis Auroux (Harvard University)Denis Auroux’s research concerns symplectic geometry and its applications to mirror symmetry. While his early work primarily concerned the topology of symplectic 4-manifolds, over the past decade Auroux has obtained pioneering results on homological mirror symmetry outside of the Calabi-Yau setting (for Fano varieties, open Riemann surfaces, etc.), and developed an extension of the SYZ approach to non-Calabi-Yau spaces.After obtaining his PhD in 1999 from Ecole Polytechnique (France), Auroux was employed as Chargé de Recherche at CNRS and CLE Moore Instructor at MIT, before joining the faculty at MIT in 2002 (as Assistant Professor from 2002 to 2004, and as Associate Professor from 2004 to 2009, with tenure starting in 2006). He then moved to UC Berkeley as a Full Professor in 2009.
Auroux has published over 30 peer-reviewed articles, including several in top journals, and given 260 invited presentations about his work. He received an Alfred P. Sloan Research Fellowship in 2005, was an invited speaker at the 2010 International Congress of Mathematicians, and in 2014 he was one of the two inaugural recipients of the Poincaré Chair at IHP. He has supervised 10 PhD dissertations, won teaching awards at MIT and Berkeley, and participated in the organization of over 20 workshops and conferences in symplectic geometry and mirror symmetry.
Senior Personnel:Artan Sheshmani (Harvard CMSA)
Artan Sheshmani’s research is focused on enumerative algebraic geometry and mathematical aspects of string theory. He is interested in applying techniques in algebraic geometry, such as, intersection theory, derived category theory, and derived algebraic geometry to construct and compute the deformation invariants of algebraic varieties, in particular Gromov-Witten (GW) or Donaldson-Thomas (DT) invariants. In the past Professor Sheshmani has worked on proving modularity property of certain DT invariants of K3-fibered threefolds (as well as their closely related Pandharipande-Thomas (PT) invariants), local surface threefolds, and general complete intersection Calabi-Yau threefolds. The modularity of DT/PT invariants in this context is predicted in a famous conjecture of string theory called S-duality modularity conjecture, and his joint work has provided the proof to some cases of it, using degenerations, virtual localizations, as well as wallcrossing techniques. Recently, Sheshmani has focused on proving a series of dualities relating the various enumerative invariants over threefolds, notably the GW invariants and invariants that arise in topological gauge theory. In particular in his joint work with Gholampour, Gukov, Liu, Yau he studied DT gauge theory and its reductions to D=4 and D=2 which are equivalent to local theory of surfaces in Calabi-Yau threefolds. Moreover, in a recent joint work with Yau and Diaconescu, he has studied the construction and computation of DT invariants of Calabi-Yau fourfolds via a suitable derived categorical reduction of the theory to the DT theory of threefolds. Currently Sheshmani is interested in a wide range of problems in enumerative geometry of CY varieties in dimensions 3,4,5.
Artan has received his PhD and Master’s degrees in pure mathematics under Sheldon Katz and Thomas Nevins from the University of Illinois at Urbana Champaign (USA) in 2011 and 2008 respectively. He holds a Master’s degree in Solid Mechanics (2004) and two Bachelor’s degrees, in Mechanical Engineering and Civil Engineering from the Sharif University of Technology, Tehran, Iran. Artan has been a tenured Associate Professor of Mathematics with joint affiliation at Harvard CMSA and center for Quantum Geometry of Moduli Spaces (QGM), since 2016. Before that he has held visiting Associate Professor and visiting Assistant Professor positions at MIT.
An Huang (Brandeis University)
The research of An Huang since 2011 has been focused on the interplay between algebraic geometry, the theory of special functions and mirror symmetry. With S. Bloch, B. Lian, V. Srinivas, S.-T. Yau, X. Zhu, he has developed the theory of tautological systems, and has applied it to settle several important problems concerning period integrals in relation to mirror symmetry. With B. Lian and X. Zhu, he has given a precise geometric interpretation of all solutions to GKZ systems associated to Calabi-Yau hypersurfaces in smooth Fano toric varieties. With B. Lian, S.-T. Yau, and C.-L. Yu, he has proved a conjecture of Vlasenko concerning an explicit formula for unit roots of the zeta functions of hypersurfaces, and has further related these roots to p-adic interpolations of complex period integrals. Beginning in 2018, with B. Stoica and S.-T. Yau, he has initiated the study of p-adic strings in curved spacetime, and showed that general relativity is a consequence of the self-consistency of quantum p-adic strings. One of the goals of this study is to understand p-adic A and B models.
An Huang received his PhD in Mathematics from the University of California at Berkeley in 2011. He was a postdoctoral fellow at the Harvard University Mathematics Department, and joined Brandeis University as an Assistant Professor in Mathematics in 2016.
Siu Cheong Lau (Boston University)The research interest of Siu Cheong Lau lies in SYZ mirror symmetry, symplectic and algebraic geometry. His thesis work has successfully constructed the SYZ mirrors for all toric Calabi-Yau manifolds based on quantum corrections by open Gromov-Witten invariants and their wall-crossing phenomenon. In collaboration with N.C. Leung, H.H. Tseng and K. Chan, he derived explicit formulas for the open Gromov-Witten invariants for semi-Fano toric manifolds which have an obstructed moduli theory. It has a beautiful relation with mirror maps and Seidel representations. Recently he works on a local-to-global approach to SYZ mirror symmetry. In joint works with C.H. Cho and H. Hong, he developed a noncommutative local mirror construction for immersed Lagrangians, and a natural gluing method to construct global mirrors. The construction has been realized in various types of geometries including orbifolds, focus-focus singularities and pair-of-pants decompositions of Riemann surfaces.
Siu-Cheong Lau has received the Doctoral Thesis Gold Award (2012) and the Best Paper Silver Award (2017) at the International Congress of Chinese Mathematicians. He was awarded the Simons Collaboration Grant in 2018. He received a Certificate of Teaching Excellence from Harvard University in 2014.
Affiliates:
- Netanel Rubin-Blaier (Cambridge)
- Kwokwai Chan (Chinese University of Hong Kong)
- Mandy Cheung (Harvard University, BP)
- Chuck Doran (University of Alberta)
- Honsol Hong (Yonsei University)
- Shinobu Hosono (Gakushuin University, Japan)
- Conan Leung (Chinese University of Hong Kong)
- Yu-shen Lin (Boston University)
- Hossein Movassati (IMPA Brazil)
- Arnav Tripathhy (Harvard University, BP)
Postdocs:
- Dennis Borisov
- Tsung-Ju Lee
- Dingxin Zhang
- Jingyu Zhao
- Yang Zhou
Jobs:
Postdoctoral Fellowship in Algebraic Geometry
Postdoctoral Fellowship in Mathematical Sciences
To learn about previous programming as part of the Simons Collaboration, click here.
- 0111/01/2021
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Bubble instability of mIIA on AdS_4 x S^6
Abstract: Recently, a set of non-supersymmetric AdS_4 vacua of massive type IIA string theory has been constructed. These vacua are perturbatively stable with respect to the full KK spectrum of type mIIA supergravity and furthermore, they are stable against a variety of non-perturbative decay channels. Hence, at this point, they represent a serious challenge to the AdS swampland conjecture. In my talk, I will review in detail the construction of these vacua as well as introduce a new decay channel, ultimately sealing their fate as being unstable.
- 0211/02/2021
Counting invariant curves on a Calabi-Yau threefold with an involution
Abstract: Gopakumar-Vafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a Calabi-Yau threefold. In this talk, I will give a general overview of two of the sheaf-theoretic approaches to defining these invariants: via stable pairs a la Pandharipande-Thomas (PT) and via perverse sheaves a la Maulik-Toda (MT). I will then outline a parallel theory of Gopakumar-Vafa invariants for a Calabi-Yau threefold X with an involution. They are integers n_beta(g,h) which give a virtual count of curves of genus g in the class beta which are invariant under the involution and whose quotient by the involution has genus h. I will give two definitions of n_beta(g,h) which are conjectured to be equivalent, one in terms of a version of PT theory, and one in terms of a version of MT theory. These invariants can be computed and the conjecture proved in the case where X=SxC where S is an Abelian or K3 surface with a symplectic involution. In these cases, the invariants are given by formulas expressed with Jacobi modular forms. In the case where S is an Abelian surface, the specialization of n_beta(g,h) to h=0 recovers the count of hyperelliptic curves on Abelian surfaces first computed by B-Oberdieck-Pandharipande-Yin. This is joint work with Stephen Pietromonaco.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero
Abstract: We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields. This is a joint work with J. Cao and R. Villaflor. (arXiv:2109.00587)
- 0311/03/2021
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Non-Invertible Duality Defects in 3+1 Dimensions
Speaker: Clay Cordova (U Chicago)
Title: Non-Invertible Duality Defects in 3+1 Dimensions
Abstract: For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions and determine the fusion rule. From modular invariance and a direct analysis of one-form symmetry-protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. By further assuming time-reversal symmetry, we find that the presence of certain duality defects implies that the low-energy phase has to be gapless unless the one-form symmetry is spontaneously broken. We give an explicit realization of this duality defect in the free Maxwell theory where the duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides.
When Computer Algebra Meets Satisfiability: A New Approach to Combinatorial Mathematics
Abstract: Solvers for the Boolean satisfiability (SAT) problem have been increasingly used to resolve problems in mathematics due to their excellent search algorithms. This talk will describe a new method for mathematical search that couples SAT solvers with computer algebra systems (CAS), thereby combining the expressiveness of CASs with the search power of SAT solvers. This paradigm has led to a number of results on long-standing mathematical questions such as the first computer-verifiable resolution of Lam’s problem and the discovery of a new infinite class of Williamson matrices.
- 0411/04/2021
Fusion Category Symmetries in Quantum Field Theory
Speaker: Yifan Wang (NYU)
Title: Fusion Category Symmetries in Quantum Field Theory
Abstract: Topological defects provide a modern perspective on symmetries in quantum field theory. They generalize the familiar inverti
ble symmetries described by groups to non-invertible symmetries described by fusion categories. Such generalized symmetries are ubiquitous in quantum field theory and provide new constraints on renormalization group flows and the IR phase diagram. In this talk I’ll review some recent progress in identifying and understanding fusion category symmetries in 1+1d conformal field theories. Time permitting, I’ll also comment on higher dimensional generalizations.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The stability of charged black holes
Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
11/4/21 CMSA Interdisciplinary Science Seminar
Title: Exploring Invertibility in Image Processing and Restoration
Abstract: Today’s smartphones have enabled numerous stunning visual effects from denoising to beautification, and we can share high-quality JPEG images easily on the internet, but it is still valuable for photographers and researchers to keep the original raw camera data for further post-processing (e.g., retouching) and analysis. However, the huge size of raw data hinders its popularity in practice, so can we almost perfectly restore the raw data from a compressed RGB image and thus avoid storing any raw data? This question leads us to design an invertible image signal processing pipeline. Then we further explore invertibility in other image processing and restoration tasks, including image compression, reversible image conversion (e.g., image-to-video conversion), and embedding novel views in a single JPEG image. We demonstrate that customized invertible neural networks are highly effective in these inherently non-invertible tasks.
- 0511/05/2021
The Greene-Plesser Construction Revisited
Member Seminar
Speaker: Chuck Doran
Title: The Greene-Plesser Construction Revisited
Abstract: The first known construction of mirror pairs of Calabi-Yau manifolds was the Greene-Plesser “quotient and resolve” procedure which applies to pencils of hypersurfaces in projective space. We’ll review this approach, uncover the hints it gives for some more general mirror constructions, and describe a brand-new variant that applies to pencils of hypersurfaces in Grassmannians. This last is joint work with Tom Coates and Elana Kalashnikov (arXiv:2110.0727).
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 0611/06/2021
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 3110/31/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 0111/01/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Bubble instability of mIIA on AdS_4 x S^6
Abstract: Recently, a set of non-supersymmetric AdS_4 vacua of massive type IIA string theory has been constructed. These vacua are perturbatively stable with respect to the full KK spectrum of type mIIA supergravity and furthermore, they are stable against a variety of non-perturbative decay channels. Hence, at this point, they represent a serious challenge to the AdS swampland conjecture. In my talk, I will review in detail the construction of these vacua as well as introduce a new decay channel, ultimately sealing their fate as being unstable.
- 0211/02/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Counting invariant curves on a Calabi-Yau threefold with an involution
Abstract: Gopakumar-Vafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a Calabi-Yau threefold. In this talk, I will give a general overview of two of the sheaf-theoretic approaches to defining these invariants: via stable pairs a la Pandharipande-Thomas (PT) and via perverse sheaves a la Maulik-Toda (MT). I will then outline a parallel theory of Gopakumar-Vafa invariants for a Calabi-Yau threefold X with an involution. They are integers n_beta(g,h) which give a virtual count of curves of genus g in the class beta which are invariant under the involution and whose quotient by the involution has genus h. I will give two definitions of n_beta(g,h) which are conjectured to be equivalent, one in terms of a version of PT theory, and one in terms of a version of MT theory. These invariants can be computed and the conjecture proved in the case where X=SxC where S is an Abelian or K3 surface with a symplectic involution. In these cases, the invariants are given by formulas expressed with Jacobi modular forms. In the case where S is an Abelian surface, the specialization of n_beta(g,h) to h=0 recovers the count of hyperelliptic curves on Abelian surfaces first computed by B-Oberdieck-Pandharipande-Yin. This is joint work with Stephen Pietromonaco.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero
Abstract: We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields. This is a joint work with J. Cao and R. Villaflor. (arXiv:2109.00587)
- 0311/03/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
Non-Invertible Duality Defects in 3+1 Dimensions
Speaker: Clay Cordova (U Chicago)
Title: Non-Invertible Duality Defects in 3+1 Dimensions
Abstract: For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions and determine the fusion rule. From modular invariance and a direct analysis of one-form symmetry-protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. By further assuming time-reversal symmetry, we find that the presence of certain duality defects implies that the low-energy phase has to be gapless unless the one-form symmetry is spontaneously broken. We give an explicit realization of this duality defect in the free Maxwell theory where the duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides.
When Computer Algebra Meets Satisfiability: A New Approach to Combinatorial Mathematics
Abstract: Solvers for the Boolean satisfiability (SAT) problem have been increasingly used to resolve problems in mathematics due to their excellent search algorithms. This talk will describe a new method for mathematical search that couples SAT solvers with computer algebra systems (CAS), thereby combining the expressiveness of CASs with the search power of SAT solvers. This paradigm has led to a number of results on long-standing mathematical questions such as the first computer-verifiable resolution of Lam’s problem and the discovery of a new infinite class of Williamson matrices.
- 0411/04/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Fusion Category Symmetries in Quantum Field Theory
Speaker: Yifan Wang (NYU)
Title: Fusion Category Symmetries in Quantum Field Theory
Abstract: Topological defects provide a modern perspective on symmetries in quantum field theory. They generalize the familiar inverti
ble symmetries described by groups to non-invertible symmetries described by fusion categories. Such generalized symmetries are ubiquitous in quantum field theory and provide new constraints on renormalization group flows and the IR phase diagram. In this talk I’ll review some recent progress in identifying and understanding fusion category symmetries in 1+1d conformal field theories. Time permitting, I’ll also comment on higher dimensional generalizations.
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
The stability of charged black holes
Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
11/4/21 CMSA Interdisciplinary Science Seminar
Title: Exploring Invertibility in Image Processing and Restoration
Abstract: Today’s smartphones have enabled numerous stunning visual effects from denoising to beautification, and we can share high-quality JPEG images easily on the internet, but it is still valuable for photographers and researchers to keep the original raw camera data for further post-processing (e.g., retouching) and analysis. However, the huge size of raw data hinders its popularity in practice, so can we almost perfectly restore the raw data from a compressed RGB image and thus avoid storing any raw data? This question leads us to design an invertible image signal processing pipeline. Then we further explore invertibility in other image processing and restoration tasks, including image compression, reversible image conversion (e.g., image-to-video conversion), and embedding novel views in a single JPEG image. We demonstrate that customized invertible neural networks are highly effective in these inherently non-invertible tasks.
- 0511/05/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
The Greene-Plesser Construction Revisited
Member Seminar
Speaker: Chuck Doran
Title: The Greene-Plesser Construction Revisited
Abstract: The first known construction of mirror pairs of Calabi-Yau manifolds was the Greene-Plesser “quotient and resolve” procedure which applies to pencils of hypersurfaces in projective space. We’ll review this approach, uncover the hints it gives for some more general mirror constructions, and describe a brand-new variant that applies to pencils of hypersurfaces in Grassmannians. This last is joint work with Tom Coates and Elana Kalashnikov (arXiv:2110.0727).
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374
- 0611/06/2021
Swampland Program
Please visit the Swampland Initiative for current events.
The Harvard Swampland Initiative is an immersive program aiming to bring together leading experts with the goal of exploring the boundaries of the quantum gravity landscape. Through workshops, seminars, and collaborative research, participants collectively navigate the Swampland, advancing our comprehension of the fundamental principles of quantum gravity.
During the 2021-2022 academic year, the CMSA hosted a program on the so-called “Swampland.”
The Swampland program aims to determine which low-energy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last few years.
The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics.
This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward graduate students.
Seminars
Swampland Seminar Series & Group Meetings
Program Visitors
- Pieter Bomans, Princeton, 10/30/21 – 11/02/21
- Irene Valenzuela, Instituto de Física Teórica, 02/14/22 – 02/21/22
- Mariana Grana, CEA/Saclay, 03/21/22 – 03/25/22
- Hector Parra De Freitas, IPHT Saclay, 03/21/22 – 04/01/22
- Timo Weigand, 03/21/22 – 03/28/22
- Gary Shiu, University of Wisconsin-Madison, 04/03/22 – 04/10/22
- Thomas van Riet, Leuven University, 04/03/22 – 04/09/22
- Lars Aalsma, University of Wisconsin-Madison, 04/11/22 – 04/15/22
- Sergio Cecotti, 05/08/22 – 05/21/22
- Tom Rudelius, 05/09/22 – 05/13/22
https://sites.harvard.edu/swampland-initiative/
Math Science Lectures in Honor of Raoul Bott: Michael Freedman
On October 4th and October 5th, 2021, Harvard CMSA hosted the annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker was Michael Freedman (Microsoft). The lectures took place on Zoom.This will be the third annual lecture series held in honor of Raoul Bott.
Lecture 1
October 4th, 11:00am (Boston time)Title: The Universe from a single Particle Abstract: I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709.
Lecture 2
October 5th, 11:00am (Boston time)Title: Controlled Mather Thurston Theorems. Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374