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.
Title: Symmetry in quantum field theory and quantum gravity 1
Abstract: In this talk I will give an overview of semi-recent work with Hirosi Ooguri arguing that three old conjectures about symmetry in quantum gravity are true in the AdS/CFT correspondence. These conjectures are 1) that there are no global symmetries in quantum gravity, 2) that dynamical objects transforming in all irreducible representations of any gauge symmetry must exist, and 3) all internal gauge symmetries must be compact. Along the way I will need to carefully define what we mean by gauge and global symmetries in quantum field theory and quantum gravity, which leads to interesting applications in various related fields. These definitions will be the focus of the first talk, while the second will apply them to AdS/CFT to prove conjectures 1-3).
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.
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
Abstract: The talk concerns recent work with Tamas Hausel in asking how SYZ mirror symmetry works for the moduli space of Higgs bundles. Focusing on C^*-invariant Lagrangian submanifolds, we use the notion of virtual multiplicity as a tool firstly to examine if the Lagrangian is closed, but also to open up new features involving finite-dimensional algebras which are deformations of cohomology algebras. Answering some of the questions raised requires revisiting basic constructions of stable bundles on curves.
Abstract: Deep learning is an exciting approach to modern artificial intelligence based on artificial neural networks. The goal of this talk is to provide a blueprint — using tools from physics — for theoretically analyzing deep neural networks of practical relevance. This task will encompass both understanding the statistics of initialized deep networks and determining the training dynamics of such an ensemble when learning from data.
In terms of their “microscopic” definition, deep neural networks are a flexible set of functions built out of many basic computational blocks called neurons, with many neurons in parallel organized into sequential layers. Borrowing from the effective theory framework, we will develop a perturbative 1/n expansion around the limit of an infinite number of neurons per layer and systematically integrate out the parameters of the network. We will explain how the network simplifies at large width and how the propagation of signals from layer to layer can be understood in terms of a Wilsonian renormalization group flow. This will make manifest that deep networks have a tuning problem, analogous to criticality, that needs to be solved in order to make them useful. Ultimately we will find a “macroscopic” description for wide and deep networks in terms of weakly-interacting statistical models, with the strength of the interactions between the neurons growing with depth-to-width aspect ratio of the network. Time permitting, we will explain how the interactions induce representation learning.
This talk is based on a book, The Principles of Deep Learning Theory, co-authored with Sho Yaida and based on research also in collaboration with Boris Hanin. It will be published next year by Cambridge University Press.
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.
Title: Symmetry in quantum field theory and quantum gravity 1
Abstract: In this talk I will give an overview of semi-recent work with Hirosi Ooguri arguing that three old conjectures about symmetry in quantum gravity are true in the AdS/CFT correspondence. These conjectures are 1) that there are no global symmetries in quantum gravity, 2) that dynamical objects transforming in all irreducible representations of any gauge symmetry must exist, and 3) all internal gauge symmetries must be compact. Along the way I will need to carefully define what we mean by gauge and global symmetries in quantum field theory and quantum gravity, which leads to interesting applications in various related fields. These definitions will be the focus of the first talk, while the second will apply them to AdS/CFT to prove conjectures 1-3).
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.
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
Abstract: The talk concerns recent work with Tamas Hausel in asking how SYZ mirror symmetry works for the moduli space of Higgs bundles. Focusing on C^*-invariant Lagrangian submanifolds, we use the notion of virtual multiplicity as a tool firstly to examine if the Lagrangian is closed, but also to open up new features involving finite-dimensional algebras which are deformations of cohomology algebras. Answering some of the questions raised requires revisiting basic constructions of stable bundles on curves.
Abstract: Deep learning is an exciting approach to modern artificial intelligence based on artificial neural networks. The goal of this talk is to provide a blueprint — using tools from physics — for theoretically analyzing deep neural networks of practical relevance. This task will encompass both understanding the statistics of initialized deep networks and determining the training dynamics of such an ensemble when learning from data.
In terms of their “microscopic” definition, deep neural networks are a flexible set of functions built out of many basic computational blocks called neurons, with many neurons in parallel organized into sequential layers. Borrowing from the effective theory framework, we will develop a perturbative 1/n expansion around the limit of an infinite number of neurons per layer and systematically integrate out the parameters of the network. We will explain how the network simplifies at large width and how the propagation of signals from layer to layer can be understood in terms of a Wilsonian renormalization group flow. This will make manifest that deep networks have a tuning problem, analogous to criticality, that needs to be solved in order to make them useful. Ultimately we will find a “macroscopic” description for wide and deep networks in terms of weakly-interacting statistical models, with the strength of the interactions between the neurons growing with depth-to-width aspect ratio of the network. Time permitting, we will explain how the interactions induce representation learning.
This talk is based on a book, The Principles of Deep Learning Theory, co-authored with Sho Yaida and based on research also in collaboration with Boris Hanin. It will be published next year by Cambridge University Press.
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.
Title: Symmetry in quantum field theory and quantum gravity 2
Abstract: In this talk I will give an overview of semi-recent work with Hirosi Ooguri arguing that three old conjectures about symmetry in quantum gravity are true in the AdS/CFT correspondence. These conjectures are 1) that there are no global symmetries in quantum gravity, 2) that dynamical objects transforming in all irreducible representations of any gauge symmetry must exist, and 3) all internal gauge symmetries must be compact. Along the way I will need to carefully define what we mean by gauge and global symmetries in quantum field theory and quantum gravity, which leads to interesting applications in various related fields. These definitions will be the focus of the first talk, while the second will apply them to AdS/CFT to prove conjectures 1-3).
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.
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
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.
Abstract: In this talk I will review our ongoing theoretical and experimental efforts toward deciphering the hydrodynamic behavior of confluent epithelia. The ability of epithelial cells to collectively flow lies at the heart of a myriad of processes that are instrumental for life, such as embryonic morphogenesis and wound healing, but also of life-threatening conditions, such as metastatic cancer. Understanding the physical origin of these mechanisms requires going beyond the current hydrodynamic theories of complex fluids and introducing a new theoretical framework, able to account for biomechanical activity as well as for scale-dependent liquid crystalline order.
Title: Polyhomogeneous expansions and Z/2-harmonic spinors branching along graphs
Abstract: In this talk, we will first reformulate the linearization of the moduli space of Z/2-harmonic spinorsv branching along a knot. This formula tells us that the kernel and cokernel of the linearization are isomorphic to the kernel and cokernel of the Dirac equation with a polyhomogeneous boundary condition. In the second part of this talk, I will describe the polyhomogenous expansions for the Z/2-harmonic spinors branching along graphs and formulate the Dirac equation with a suitable boundary condition that can describe the perturbation of graphs with some restrictions. This is joint work with Andriy Haydys and Rafe Mazzeo.
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.
Title: Symmetry in quantum field theory and quantum gravity 2
Abstract: In this talk I will give an overview of semi-recent work with Hirosi Ooguri arguing that three old conjectures about symmetry in quantum gravity are true in the AdS/CFT correspondence. These conjectures are 1) that there are no global symmetries in quantum gravity, 2) that dynamical objects transforming in all irreducible representations of any gauge symmetry must exist, and 3) all internal gauge symmetries must be compact. Along the way I will need to carefully define what we mean by gauge and global symmetries in quantum field theory and quantum gravity, which leads to interesting applications in various related fields. These definitions will be the focus of the first talk, while the second will apply them to AdS/CFT to prove conjectures 1-3).
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.
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
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.
Abstract: In this talk I will review our ongoing theoretical and experimental efforts toward deciphering the hydrodynamic behavior of confluent epithelia. The ability of epithelial cells to collectively flow lies at the heart of a myriad of processes that are instrumental for life, such as embryonic morphogenesis and wound healing, but also of life-threatening conditions, such as metastatic cancer. Understanding the physical origin of these mechanisms requires going beyond the current hydrodynamic theories of complex fluids and introducing a new theoretical framework, able to account for biomechanical activity as well as for scale-dependent liquid crystalline order.
Title: Polyhomogeneous expansions and Z/2-harmonic spinors branching along graphs
Abstract: In this talk, we will first reformulate the linearization of the moduli space of Z/2-harmonic spinorsv branching along a knot. This formula tells us that the kernel and cokernel of the linearization are isomorphic to the kernel and cokernel of the Dirac equation with a polyhomogeneous boundary condition. In the second part of this talk, I will describe the polyhomogenous expansions for the Z/2-harmonic spinors branching along graphs and formulate the Dirac equation with a suitable boundary condition that can describe the perturbation of graphs with some restrictions. This is joint work with Andriy Haydys and Rafe Mazzeo.
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.
Title: Black Holes, 2D Gravity, and Random Matrices
Abstract: I will discuss old and new connections between black hole physics, 2D quantum gravity, and random matrix theory. Black holes are believed to be very complicated, strongly interacting quantum mechanical systems, and certain aspects of their Hamiltonians should be well approximated by random matrix theory. The near-horizon effective dynamics of near-extremal black holes is two-dimensional, and many theories of 2D quantum gravity are known to have random matrix descriptions. All of these expectations were recently borne out in surprising detail with the solution of the Jackiw-Teitelboim (JT) model, but this result raises more questions than it answers. If time permits, I will discuss some extensions of these results and possible future directions.
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.
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 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.
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.
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 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.
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.
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 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.
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.
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
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.
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.
Abstract: Let M_n be drawn uniformly from all n by n symmetric matrices with entries in {-1,1}. In this talk I’ll consider the following basic question: what is the probability that M_n is singular? I’ll discuss recent joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe where we show that this probability is exponentially small. I hope to make the talk accessible to a fairly general audience.
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.
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 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.
Abstract: Let M_n be drawn uniformly from all n by n symmetric matrices with entries in {-1,1}. In this talk I’ll consider the following basic question: what is the probability that M_n is singular? I’ll discuss recent joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe where we show that this probability is exponentially small. I hope to make the talk accessible to a fairly general audience.
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.
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 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.
Abstract: I will describe some of my recent work on defects in supersymmetric field theories. The first part of my talk is focused on line defects in certain large classes of 4d N=2 theories and 3d N=2 theories. I will describe geometric methods to compute the ground states spectrum of the bulk-defect system, as well as implications on the construction of link invariants. In the second part I will talk about some perspectives of surface defects in 4d N=2 theories and related applications on the exact WKB method for ordinary differential equations. This talk is based on past joint work with A. Neitzke, various work in progress with D. Gaiotto, S. Jeong, A. Khan, G. Moore, as well as work by myself.
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.
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
Abstract: Transformer models yield impressive results on many NLP and sequence modeling tasks. Remarkably, Transformers can handle long sequences which allows them to produce long coherent outputs: full paragraphs produced by GPT-3 or well-structured images produced by DALL-E. These large language models are impressive but also very inefficient and costly, which limits their applications and accessibility. We postulate that having an explicit hierarchical architecture is the key to Transformers that efficiently handle long sequences. To verify this claim, we first study different ways to upsample and downsample activations in Transformers so as to make them hierarchical. We use the best performing upsampling and downsampling layers to create Hourglass – a hierarchical Transformer language model. Hourglass improves upon the Transformer baseline given the same amount of computation and can yield the same results as Transformers more efficiently. In particular, Hourglass sets new state-of-the-art for Transformer models on the ImageNet32 generation task and improves language modeling efficiency on the widely studied enwik8 benchmark.
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.
Abstract: I will describe some of my recent work on defects in supersymmetric field theories. The first part of my talk is focused on line defects in certain large classes of 4d N=2 theories and 3d N=2 theories. I will describe geometric methods to compute the ground states spectrum of the bulk-defect system, as well as implications on the construction of link invariants. In the second part I will talk about some perspectives of surface defects in 4d N=2 theories and related applications on the exact WKB method for ordinary differential equations. This talk is based on past joint work with A. Neitzke, various work in progress with D. Gaiotto, S. Jeong, A. Khan, G. Moore, as well as work by myself.
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.
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
Abstract: Transformer models yield impressive results on many NLP and sequence modeling tasks. Remarkably, Transformers can handle long sequences which allows them to produce long coherent outputs: full paragraphs produced by GPT-3 or well-structured images produced by DALL-E. These large language models are impressive but also very inefficient and costly, which limits their applications and accessibility. We postulate that having an explicit hierarchical architecture is the key to Transformers that efficiently handle long sequences. To verify this claim, we first study different ways to upsample and downsample activations in Transformers so as to make them hierarchical. We use the best performing upsampling and downsampling layers to create Hourglass – a hierarchical Transformer language model. Hourglass improves upon the Transformer baseline given the same amount of computation and can yield the same results as Transformers more efficiently. In particular, Hourglass sets new state-of-the-art for Transformer models on the ImageNet32 generation task and improves language modeling efficiency on the widely studied enwik8 benchmark.
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.
Title: The Noether Theorems in Geometry: Then and Now
Abstract: The 1918 Noether theorems were a product of the general search for energy and momentum conservation in Einstein’s newly formulated theory of general relativity. Although widely referred to as the connection between symmetry and conservation laws, the theorems themselves are often not understood properly and hence have not been as widely used as they might be. In the first part of the talk, I outline a brief history of the theorems, explain a bit of the language, translate the first theorem into coordinate invariant language and give a few examples. I will mention only briefly their importance in physics and integrable systems. In the second part of the talk, I describe why they are still relevant in geometric analysis: how they underlie standard techniques and why George Daskalopoulos and I came to be interested in them for our investigation into the best Lipschitz maps of Bill Thurston. Some applications to integrals on a domain a hyperbolic surface leave open possibilities for applications to integrals on domains which are locally symmetric spaces of higher dimension. The talk finishes with an example or two from the literature.
Beginning in Spring 2020, the CMSA will be hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars. Talks will take place throughout the semester. All talks will take place virtually. You must register to attend. Recordings will be posted to this page.
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.
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
Abstract: In 1977, Yau proved that a Kahler manifold with zero first Chern class admits a Ricci flat metric, which is uniquely determined by certain “moduli” data. These metrics have been very important in mathematics and in theoretical physics, but despite much subsequent work we have no analytical expressions for them. But significant progress has been made on computing numerical approximations. We give an introduction (not assuming knowledge of complex geometry) to these problems and describe these methods.
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.
Title: The Noether Theorems in Geometry: Then and Now
Abstract: The 1918 Noether theorems were a product of the general search for energy and momentum conservation in Einstein’s newly formulated theory of general relativity. Although widely referred to as the connection between symmetry and conservation laws, the theorems themselves are often not understood properly and hence have not been as widely used as they might be. In the first part of the talk, I outline a brief history of the theorems, explain a bit of the language, translate the first theorem into coordinate invariant language and give a few examples. I will mention only briefly their importance in physics and integrable systems. In the second part of the talk, I describe why they are still relevant in geometric analysis: how they underlie standard techniques and why George Daskalopoulos and I came to be interested in them for our investigation into the best Lipschitz maps of Bill Thurston. Some applications to integrals on a domain a hyperbolic surface leave open possibilities for applications to integrals on domains which are locally symmetric spaces of higher dimension. The talk finishes with an example or two from the literature.
Beginning in Spring 2020, the CMSA will be hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars. Talks will take place throughout the semester. All talks will take place virtually. You must register to attend. Recordings will be posted to this page.
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.
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
Abstract: In 1977, Yau proved that a Kahler manifold with zero first Chern class admits a Ricci flat metric, which is uniquely determined by certain “moduli” data. These metrics have been very important in mathematics and in theoretical physics, but despite much subsequent work we have no analytical expressions for them. But significant progress has been made on computing numerical approximations. We give an introduction (not assuming knowledge of complex geometry) to these problems and describe these methods.
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.
Title: On the solution space of the Ising perceptron model
Abstract: Consider the discrete cube $\{-1,1\}^N$ and a random collection of half spaces which includes each half space $H(x) := \{y \in \{-1,1\}^N: x \cdot y \geq \kappa \sqrt{N}\}$ for $x \in \{-1,1\}^N$ independently with probability $p$. The solution space is the intersection of these half spaces. In this talk, we will talk about its sharp threshold phenomenon, the frozen structure of the solution space, and the Gardner formula.
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.
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
Title: Gravitational anomaly of 3 + 1 dimensional Z2 toric code with fermionic charges and ferionic loop self-statistics
Abstract: Quasiparticle excitations in 3 + 1 dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in 3 + 1 dimensional topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics μ = ±1, that distinguishes two bosonic topological orders that both superficially resemble 3 + 1d Z2 gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic Z2 gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second has fermionic charges and fermionic loops (FcFl) and, as we argue, can only exist at the boundary of a non-trivial 4 + 1d invertible phase, stable without any symmetries i.e., it possesses a gravitational anomaly. We substantiate these claims by constructing an explicit exactly solvable 4 + 1d Walker–Wang model and computing the loop self-statistics in the fermionic Z2 gauge theory hosted at its boundary. We also show that the FcFl phase has the same gravitational anomaly as all-fermion quantum electrodynamics. Our results are in agreement with the recent classification of nondegenerate braided fusion 2- categories, and with the cobordism prediction of a non-trivial Z2-classified 4+1d invertible phase with action S = (1/2) w2 w3.
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.
Title: On the solution space of the Ising perceptron model
Abstract: Consider the discrete cube $\{-1,1\}^N$ and a random collection of half spaces which includes each half space $H(x) := \{y \in \{-1,1\}^N: x \cdot y \geq \kappa \sqrt{N}\}$ for $x \in \{-1,1\}^N$ independently with probability $p$. The solution space is the intersection of these half spaces. In this talk, we will talk about its sharp threshold phenomenon, the frozen structure of the solution space, and the Gardner formula.
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.
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
Title: Gravitational anomaly of 3 + 1 dimensional Z2 toric code with fermionic charges and ferionic loop self-statistics
Abstract: Quasiparticle excitations in 3 + 1 dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in 3 + 1 dimensional topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics μ = ±1, that distinguishes two bosonic topological orders that both superficially resemble 3 + 1d Z2 gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic Z2 gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second has fermionic charges and fermionic loops (FcFl) and, as we argue, can only exist at the boundary of a non-trivial 4 + 1d invertible phase, stable without any symmetries i.e., it possesses a gravitational anomaly. We substantiate these claims by constructing an explicit exactly solvable 4 + 1d Walker–Wang model and computing the loop self-statistics in the fermionic Z2 gauge theory hosted at its boundary. We also show that the FcFl phase has the same gravitational anomaly as all-fermion quantum electrodynamics. Our results are in agreement with the recent classification of nondegenerate braided fusion 2- categories, and with the cobordism prediction of a non-trivial Z2-classified 4+1d invertible phase with action S = (1/2) w2 w3.
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.
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.
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 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.
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.
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 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.
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.
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 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.
Abstract: A long-standing problem in random graph theory has been to determine asymptotically the length of a longest induced path in sparse random graphs. Independent work of Luczak and Suen from the 90s showed the existence of an induced path of roughly half the optimal size, which seems to be a barrier for certain natural approaches. Recently, in joint work with Draganic and Krivelevich, we solved this problem. In the talk, I will discuss the history of the problem and give an overview of the proof.
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.
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 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.
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.
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
Abstract: A central challenge in quantum many-body physics is to estimate the properties of natural quantum states, such as the quantum ground states and Gibbs states. Quantum Hamiltonian complexity offers a computational perspective on this challenge and classifies these natural quantum states using the language of quantum complexity classes. This talk will provide a gentle introduction to the field and highlight its success in pinning down the hardness of a wide variety of quantum states. In particular, we will consider the gapped ground states and Gibbs states on low dimensional lattices, which are believed to exhibit ‘low complexity’ due to the widely studied area law behaviour. Here, we will see the crucial role of complexity-theoretic methods in progress on the ‘area law conjecture’ and in the development of efficient algorithms to classically simulate quantum many-body systems.
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.
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.
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
Abstract: A central challenge in quantum many-body physics is to estimate the properties of natural quantum states, such as the quantum ground states and Gibbs states. Quantum Hamiltonian complexity offers a computational perspective on this challenge and classifies these natural quantum states using the language of quantum complexity classes. This talk will provide a gentle introduction to the field and highlight its success in pinning down the hardness of a wide variety of quantum states. In particular, we will consider the gapped ground states and Gibbs states on low dimensional lattices, which are believed to exhibit ‘low complexity’ due to the widely studied area law behaviour. Here, we will see the crucial role of complexity-theoretic methods in progress on the ‘area law conjecture’ and in the development of efficient algorithms to classically simulate quantum many-body systems.
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.
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.
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
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.
Title: Quadratic reciprocity from a family of adelic conformal field theories
Abstract: We consider a deformation of the 2d free scalar field action by raising the Laplacian to a positive real power. It turns out that the resulting non-local generalized free action is invariant under two commuting actions of the global conformal symmetry algebra, although it’s no longer invariant under the local conformal symmetry algebra. Furthermore, there is an adelic version of this family of global conformal field theories, parametrized by the choice of a number field, together with a Hecke character. Tate’s thesis plays an important role here in calculating Green’s functions of these theories, and in ensuring the adelic compatibility of these theories. In particular, the local L-factors contribute to prefactors of these Green’s functions. We shall try to see quadratic reciprocity from this context, as a consequence of an adelic version of holomorphic factorization of these theories. This is work in progress with B. Stoica and X. Zhong.
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.
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.
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
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.
Title: Quadratic reciprocity from a family of adelic conformal field theories
Abstract: We consider a deformation of the 2d free scalar field action by raising the Laplacian to a positive real power. It turns out that the resulting non-local generalized free action is invariant under two commuting actions of the global conformal symmetry algebra, although it’s no longer invariant under the local conformal symmetry algebra. Furthermore, there is an adelic version of this family of global conformal field theories, parametrized by the choice of a number field, together with a Hecke character. Tate’s thesis plays an important role here in calculating Green’s functions of these theories, and in ensuring the adelic compatibility of these theories. In particular, the local L-factors contribute to prefactors of these Green’s functions. We shall try to see quadratic reciprocity from this context, as a consequence of an adelic version of holomorphic factorization of these theories. This is work in progress with B. Stoica and X. Zhong.
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.
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.
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 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.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
During the 2021–22 academic year, the CMSA will be hosting a Colloquium, organized by Du Pei, Changji Xu, and Michael Simkin. It will take place on Wednesdays at 9:30am – 10:30am (Boston time). The meetings will take place virtually on Zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. The schedule below will be updated as talks are confirmed.
Abstract: In 1975, Stephen Hawking showed that black holes radiate away in a manner that violates quantum theory. Starting in 1997, it was observed that black holes in string theory did not have the form expected from general relativity: in place of “empty space will all the mass at the center,” one finds a “fuzzball” where the mass is distributed throughout the interior of the horizon. This resolves the paradox, but opposition to this resolution came from groups who sought to extrapolate some ideas in holography. In 2009 it was shown, using some theorems from quantum information theory, that these extrapolations were incorrect, and the fuzzball structure was essential for resolving the puzzle. Opposition continued along different lines, with a postulate that information would leak out through wormholes. Recently, it was shown that this wormhole idea had some basic flaws, leaving the fuzzball paradigm as the natural resolution of Hawking’s puzzle.
Abstract: Consider an agency holding a large database of sensitive personal information—say, medical records, census survey answers, web searches, or genetic data. The agency would like to discover and publicly release global characteristics of the data while protecting the privacy of individuals’ records.
I will discuss recent (and not-so-recent) results on this problem with a focus on the release of statistical models. I will first explain some of the fundamental limitations on the release of machine learning models—specifically, why such models must sometimes memorize training data points nearly completely. On the more positive side, I will present differential privacy, a rigorous definition of privacy in statistical databases that is now widely studied, and increasingly used to analyze and design deployed systems. I will explain some of the challenges of sound statistical inference based on differentially private statistics, and lay out directions for future investigation.
2/8/2022
Wenbin Yan (Tsinghua University) (special time: 9:30 pm ET)
Abstract: We introduce and study tetrahedron instantons. Physically they capture instantons on $\mathbb{C}^{3}$ in the presence of the most general intersecting codimension-two supersymmetric defects. In this talk, we will review instanton moduli spaces, explain the construction, moduli space and partition functions of tetrahedron instantons. We will also point out possible relations with M-theory index which could be a generalization of Gupakuma-Vafa theory.
Abstract: In 1960’s, Narasimhan and Seshadri discovered the equivalence between irreducible unitary flat bundles and stable bundles of degree $0$ on compact Riemann surfaces. In 1980’s, Donaldson, Uhlenbeck and Yau generalized it to the equivalence between irreducible Hermitian-Einstein bundles and stable bundles on smooth projective varieties. This is a surprising bridge connecting differential geometry and algebraic geometry. Since then, many interesting generalizations have been studied.
In this talk, we would like to review a stream in the study of such correspondences for Higgs bundles, integrable connections, $D$-modules and periodic monopoles.
Abstract: In the AdS/CFT correspondence, the holographic entropy cone, which identifies von Neumann entropies of CFT regions that are consistent with a semiclassical bulk dual, is currently known only up to n=5 regions. I explain that average entropies of p-partite subsystems can be checked for consistency with a semiclassical bulk dual far more easily, for an arbitrary number of regions n. This analysis defines the “Holographic Cone of Average Entropies” (HCAE). I conjecture the exact form of HCAE, and find that it has the following properties: (1) HCAE is the simplest it could be, namely it is a simplicial cone. (2) Its extremal rays represent stages of thermalization (black hole formation). (3) In a time-reversed picture, the extremal rays of HCAE represent stages of unitary black hole evaporation, as stipulated by the island solution of the black hole information paradox. (4) HCAE is bound by a novel, infinite family of holographic entropy inequalities. (5) HCAE is the simplest it could be also in its dependence on the number of regions n, namely its bounding inequalities are n-independent. (6) In a precise sense I describe, the bounding inequalities of HCAE unify (almost) all previously discovered holographic inequalities and strongly constrain future inequalities yet to be discovered. I also sketch an interpretation of HCAE in terms of error correction and the holographic Renormalization Group. The big lesson that HCAE seems to be teaching us is about the universality of black hole physics.
Abstract: We consider SL_n-local systems on graphs on surfaces and show how the associated Kasteleyn matrix can be used to compute probabilities of various topological events involving the overlay of n independent dimer covers (or “n-webs”).
This is joint work with Dan Douglas and Haolin Shi.
Abstract: The low dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. We analyze the cases where the training data are distributed in a linear subspace of Rd. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network’s depth and noise in the codimension of the data manifold.
Abstract: Over the last century, ecologists, statisticians, physicists, financial quants, and other scientists discovered that, in many examples, the sample variance approximates a power of the sample mean of each of a set of samples of nonnegative quantities. This power-law relationship of variance to mean is known as a power variance function in statistics, as Taylor’s law in ecology, and as fluctuation scaling in physics and financial mathematics. This survey talk will emphasize ideas, motivations, recent theoretical results, and applications rather than detailed proofs. Many models intended to explain Taylor’s law assume the probability distribution underlying each sample has finite mean and variance. Recently, colleagues and I generalized Taylor’s law to samples from probability distributions with infinite mean or infinite variance and higher moments. For such heavy-tailed distributions, we extended Taylor’s law to higher moments than the mean and variance and to upper and lower semivariances (measures of upside and downside portfolio risk). In unpublished work, we suggest that U.S. COVID-19 cases and deaths illustrate Taylor’s law arising from a distribution with finite mean and infinite variance. This model has practical implications. Collaborators in this work are Mark Brown, Richard A. Davis, Victor de la Peña, Gennady Samorodnitsky, Chuan-Fa Tang, and Sheung Chi Phillip Yam.
Abstract: In this talk I first review some of the many appearances of localized degrees of freedom — edge modes — in a variety of physical systems. Edge modes are implicated for example in quantum entanglement and in various topological and holographic dualities. I then review recent work in which it has been realized that a careful treatment of such modes, paying attention to relevant symmetries, is required in order to properly understand such basic physical quantities as Noether charges. From many points of view, it is conjectured that this physics may be pointing at basic properties of quantum spacetimes and gravity.
Abstract: In the last three decades, the problem of consciousness – how and why physical systems such as the brain have conscious experiences – has received increasing attention among neuroscientists, psychologists, and philosophers. Recently, a decidedly mathematical perspective has emerged as well, which is now called Mathematical Consciousness Science. In this talk, I will give an introduction and overview of Mathematical Consciousness Science for mathematicians, including a bottom-up introduction to the problem of consciousness and how it is amenable to mathematical tools and methods.
Abstract: The standard definition of symmetries of a structure given on a set S (in the sense of Bourbaki) is the group of bijective maps S to S, compatible with this structure. But in fact, symmetries of various structures related to storing and transmitting information (such as information spaces) are naturally embodied in various classes of loops such as Moufang loops, – nonassociative analogs of groups.
The idea of symmetry as a group is closely related to classical physics, in a very definite sense, going back at least to Archimedes. When quantum physics started to replace classical, it turned out that classical symmetries must also be replaced by their quantum versions, e.g. quantum groups.
In this talk we explain how to define and study quantum versions of symmetries, relevant to information theory and other contexts
Abstract: Suppose we transmit n bits on a noisy channel that deletes some fraction of the bits arbitrarily. What’s the supremum p* of deletion fractions that can be corrected with a binary code of non-vanishing rate? Evidently p* is at most 1/2 as the adversary can delete all occurrences of the minority bit. It was unknown whether this simple upper bound could be improved, or one could in fact correct deletion fractions approaching 1/2.
We show that there exist absolute constants A and delta > 0 such that any subset of n-bit strings of size exp((log n)^A) must contain two strings with a common subsequence of length (1/2+delta)n. This immediately implies that the zero-rate threshold p* of worst-case bit deletions is bounded away from 1/2.
Our techniques include string regularity arguments and a structural lemma that classifies bit-strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence.
Abstract: Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, a precursor of general relativity characterized by a dimensionless coupling constant (the “Foeppl-von Karman number”) that can easily reach vK = 10^7 in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics, as exemplified by experiments that twist and bend individual atomically-thin free-standing graphene sheets (with vK = 10^13!) A crumpling transition out of the flat phase for thermalized elastic membranes has been predicted when kT is large compared to the microscopic bending stiffness, which could have interesting consequences for Dirac cones of electrons embedded in graphene. It may be possible to lower the crumpling temperature for graphene to more readily accessible range by inserting a regular lattice of laser-cut perforations, an expectation an confirmed by extensive molecular dynamics simulations. We then move on to analyze the physics of sheets mutilated with puckers and stitches. Puckers and stitches lead to Ising-like phase transitions riding on a background of flexural phonons, as well as an anomalous coefficient of thermal expansion. Finally, we argue that thin membranes with a background curvature lead to thermalized spherical shells that must collapse beyond a critical size at room temperature, even in the absence of an external pressure.
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.
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.
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.
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.
Abstract: I will give a survey of the program of categorification for quantum groups, some of its recent development and applications to representation theory.
10/27/2021
Karim Adiprasito, Hebrew University and University of Copenhagen
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.
Abstract: We will explain how to model the Hitchin integrable system on a certain Lagrangian upward flow as the spectrum of equivariant cohomology of a Grassmannian.
Abstract: Many combinatorial objects can be thought of as a hypergraph decomposition, i.e. a partition of (the edge set of) one hypergraph into (the edge sets of) copies of some other hypergraphs. For example, a Steiner Triple System is equivalent to a decomposition of a complete graph into triangles. In general, Steiner Systems are equivalent to decompositions of complete uniform hypergraphs into other complete uniform hypergraphs (of some specified sizes). The Existence Conjecture for Combinatorial Designs, which I proved in 2014, states that, bar finitely many exceptions, such decompositions exist whenever the necessary ‘divisibility conditions’ hold. I also obtained a generalisation to the quasirandom setting, which implies an approximate formula for the number of designs; in particular, this resolved Wilson’s Conjecture on the number of Steiner Triple Systems. A more general result that I proved in 2018 on decomposing lattice-valued vectors indexed by labelled complexes provides many further existence and counting results for a wide range of combinatorial objects, such as resolvable designs (the generalised form of Kirkman’s Schoolgirl Problem), whist tournaments or generalised Sudoku squares. In this talk, I plan to review this background and then describe some more recent and ongoing applications of these results and developments of the ideas behind them.
Abstract: Enumerative geometry is a venerable subfield of Mathematics, with roots dating back to Greek Antiquity and a present inextricably linked with developments in other domains. Since the early 90s, in particular, the interaction with String Theory has sent shockwaves through the subject, giving both unexpected new perspectives and a remarkably powerful, physics-motivated toolkit to tackle several traditionally hard questions in the field. I will survey some recent developments in this vein for the case of enumerative invariants associated to a pair (X, D), with X a complex algebraic surface and D a singular anticanonical divisor in it. I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to (X, D), including the log Gromov-Witten invariants of the pair, the Gromov-Witten invariants of an associated higher dimensional Calabi-Yau variety, the open Gromov-Witten invariants of certain special Lagrangians in toric Calabi–Yau threefolds, the Donaldson–Thomas theory of a class of symmetric quivers, and certain open and closed Gopakumar-Vafa-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.
Abstract: For a simple and simply connected complex group G, I will discuss some elements of the proof of the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over families of smooth projective curves with marked points. Under the isomorphism with the bundle of conformal blocks, this connection is equivalent to the one constructed by conformal field theory. This is joint work with Indranil Biswas and Swarnava Mukhopadhyay.
Abstract: Tree decompositions are a powerful tool in both structural graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph.
Tree decompositions have traditionally been used in the context of forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.
Abstract: I will present a construction of the object in the title which, applied to the classical Toda system, controls the theory of categorical representations of compact Lie groups, along with applications (some conjectural, some rigorous) to gauged Gromov-Witten theory. Time permitting, we will review applications to Coulomb branches and the categorified Weyl character formula.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
During the 2021–22 academic year, the CMSA will be hosting a Colloquium, organized by Du Pei, Changji Xu, and Michael Simkin. It will take place on Wednesdays at 9:30am – 10:30am (Boston time). The meetings will take place virtually on Zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. The schedule below will be updated as talks are confirmed.
Abstract: In 1975, Stephen Hawking showed that black holes radiate away in a manner that violates quantum theory. Starting in 1997, it was observed that black holes in string theory did not have the form expected from general relativity: in place of “empty space will all the mass at the center,” one finds a “fuzzball” where the mass is distributed throughout the interior of the horizon. This resolves the paradox, but opposition to this resolution came from groups who sought to extrapolate some ideas in holography. In 2009 it was shown, using some theorems from quantum information theory, that these extrapolations were incorrect, and the fuzzball structure was essential for resolving the puzzle. Opposition continued along different lines, with a postulate that information would leak out through wormholes. Recently, it was shown that this wormhole idea had some basic flaws, leaving the fuzzball paradigm as the natural resolution of Hawking’s puzzle.
Abstract: Consider an agency holding a large database of sensitive personal information—say, medical records, census survey answers, web searches, or genetic data. The agency would like to discover and publicly release global characteristics of the data while protecting the privacy of individuals’ records.
I will discuss recent (and not-so-recent) results on this problem with a focus on the release of statistical models. I will first explain some of the fundamental limitations on the release of machine learning models—specifically, why such models must sometimes memorize training data points nearly completely. On the more positive side, I will present differential privacy, a rigorous definition of privacy in statistical databases that is now widely studied, and increasingly used to analyze and design deployed systems. I will explain some of the challenges of sound statistical inference based on differentially private statistics, and lay out directions for future investigation.
2/8/2022
Wenbin Yan (Tsinghua University) (special time: 9:30 pm ET)
Abstract: We introduce and study tetrahedron instantons. Physically they capture instantons on $\mathbb{C}^{3}$ in the presence of the most general intersecting codimension-two supersymmetric defects. In this talk, we will review instanton moduli spaces, explain the construction, moduli space and partition functions of tetrahedron instantons. We will also point out possible relations with M-theory index which could be a generalization of Gupakuma-Vafa theory.
Abstract: In 1960’s, Narasimhan and Seshadri discovered the equivalence between irreducible unitary flat bundles and stable bundles of degree $0$ on compact Riemann surfaces. In 1980’s, Donaldson, Uhlenbeck and Yau generalized it to the equivalence between irreducible Hermitian-Einstein bundles and stable bundles on smooth projective varieties. This is a surprising bridge connecting differential geometry and algebraic geometry. Since then, many interesting generalizations have been studied.
In this talk, we would like to review a stream in the study of such correspondences for Higgs bundles, integrable connections, $D$-modules and periodic monopoles.
Abstract: In the AdS/CFT correspondence, the holographic entropy cone, which identifies von Neumann entropies of CFT regions that are consistent with a semiclassical bulk dual, is currently known only up to n=5 regions. I explain that average entropies of p-partite subsystems can be checked for consistency with a semiclassical bulk dual far more easily, for an arbitrary number of regions n. This analysis defines the “Holographic Cone of Average Entropies” (HCAE). I conjecture the exact form of HCAE, and find that it has the following properties: (1) HCAE is the simplest it could be, namely it is a simplicial cone. (2) Its extremal rays represent stages of thermalization (black hole formation). (3) In a time-reversed picture, the extremal rays of HCAE represent stages of unitary black hole evaporation, as stipulated by the island solution of the black hole information paradox. (4) HCAE is bound by a novel, infinite family of holographic entropy inequalities. (5) HCAE is the simplest it could be also in its dependence on the number of regions n, namely its bounding inequalities are n-independent. (6) In a precise sense I describe, the bounding inequalities of HCAE unify (almost) all previously discovered holographic inequalities and strongly constrain future inequalities yet to be discovered. I also sketch an interpretation of HCAE in terms of error correction and the holographic Renormalization Group. The big lesson that HCAE seems to be teaching us is about the universality of black hole physics.
Abstract: We consider SL_n-local systems on graphs on surfaces and show how the associated Kasteleyn matrix can be used to compute probabilities of various topological events involving the overlay of n independent dimer covers (or “n-webs”).
This is joint work with Dan Douglas and Haolin Shi.
Abstract: The low dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. We analyze the cases where the training data are distributed in a linear subspace of Rd. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network’s depth and noise in the codimension of the data manifold.
Abstract: Over the last century, ecologists, statisticians, physicists, financial quants, and other scientists discovered that, in many examples, the sample variance approximates a power of the sample mean of each of a set of samples of nonnegative quantities. This power-law relationship of variance to mean is known as a power variance function in statistics, as Taylor’s law in ecology, and as fluctuation scaling in physics and financial mathematics. This survey talk will emphasize ideas, motivations, recent theoretical results, and applications rather than detailed proofs. Many models intended to explain Taylor’s law assume the probability distribution underlying each sample has finite mean and variance. Recently, colleagues and I generalized Taylor’s law to samples from probability distributions with infinite mean or infinite variance and higher moments. For such heavy-tailed distributions, we extended Taylor’s law to higher moments than the mean and variance and to upper and lower semivariances (measures of upside and downside portfolio risk). In unpublished work, we suggest that U.S. COVID-19 cases and deaths illustrate Taylor’s law arising from a distribution with finite mean and infinite variance. This model has practical implications. Collaborators in this work are Mark Brown, Richard A. Davis, Victor de la Peña, Gennady Samorodnitsky, Chuan-Fa Tang, and Sheung Chi Phillip Yam.
Abstract: In this talk I first review some of the many appearances of localized degrees of freedom — edge modes — in a variety of physical systems. Edge modes are implicated for example in quantum entanglement and in various topological and holographic dualities. I then review recent work in which it has been realized that a careful treatment of such modes, paying attention to relevant symmetries, is required in order to properly understand such basic physical quantities as Noether charges. From many points of view, it is conjectured that this physics may be pointing at basic properties of quantum spacetimes and gravity.
Abstract: In the last three decades, the problem of consciousness – how and why physical systems such as the brain have conscious experiences – has received increasing attention among neuroscientists, psychologists, and philosophers. Recently, a decidedly mathematical perspective has emerged as well, which is now called Mathematical Consciousness Science. In this talk, I will give an introduction and overview of Mathematical Consciousness Science for mathematicians, including a bottom-up introduction to the problem of consciousness and how it is amenable to mathematical tools and methods.
Abstract: The standard definition of symmetries of a structure given on a set S (in the sense of Bourbaki) is the group of bijective maps S to S, compatible with this structure. But in fact, symmetries of various structures related to storing and transmitting information (such as information spaces) are naturally embodied in various classes of loops such as Moufang loops, – nonassociative analogs of groups.
The idea of symmetry as a group is closely related to classical physics, in a very definite sense, going back at least to Archimedes. When quantum physics started to replace classical, it turned out that classical symmetries must also be replaced by their quantum versions, e.g. quantum groups.
In this talk we explain how to define and study quantum versions of symmetries, relevant to information theory and other contexts
Abstract: Suppose we transmit n bits on a noisy channel that deletes some fraction of the bits arbitrarily. What’s the supremum p* of deletion fractions that can be corrected with a binary code of non-vanishing rate? Evidently p* is at most 1/2 as the adversary can delete all occurrences of the minority bit. It was unknown whether this simple upper bound could be improved, or one could in fact correct deletion fractions approaching 1/2.
We show that there exist absolute constants A and delta > 0 such that any subset of n-bit strings of size exp((log n)^A) must contain two strings with a common subsequence of length (1/2+delta)n. This immediately implies that the zero-rate threshold p* of worst-case bit deletions is bounded away from 1/2.
Our techniques include string regularity arguments and a structural lemma that classifies bit-strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence.
Abstract: Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, a precursor of general relativity characterized by a dimensionless coupling constant (the “Foeppl-von Karman number”) that can easily reach vK = 10^7 in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics, as exemplified by experiments that twist and bend individual atomically-thin free-standing graphene sheets (with vK = 10^13!) A crumpling transition out of the flat phase for thermalized elastic membranes has been predicted when kT is large compared to the microscopic bending stiffness, which could have interesting consequences for Dirac cones of electrons embedded in graphene. It may be possible to lower the crumpling temperature for graphene to more readily accessible range by inserting a regular lattice of laser-cut perforations, an expectation an confirmed by extensive molecular dynamics simulations. We then move on to analyze the physics of sheets mutilated with puckers and stitches. Puckers and stitches lead to Ising-like phase transitions riding on a background of flexural phonons, as well as an anomalous coefficient of thermal expansion. Finally, we argue that thin membranes with a background curvature lead to thermalized spherical shells that must collapse beyond a critical size at room temperature, even in the absence of an external pressure.
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.
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.
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.
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.
Abstract: I will give a survey of the program of categorification for quantum groups, some of its recent development and applications to representation theory.
10/27/2021
Karim Adiprasito, Hebrew University and University of Copenhagen
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.
Abstract: We will explain how to model the Hitchin integrable system on a certain Lagrangian upward flow as the spectrum of equivariant cohomology of a Grassmannian.
Abstract: Many combinatorial objects can be thought of as a hypergraph decomposition, i.e. a partition of (the edge set of) one hypergraph into (the edge sets of) copies of some other hypergraphs. For example, a Steiner Triple System is equivalent to a decomposition of a complete graph into triangles. In general, Steiner Systems are equivalent to decompositions of complete uniform hypergraphs into other complete uniform hypergraphs (of some specified sizes). The Existence Conjecture for Combinatorial Designs, which I proved in 2014, states that, bar finitely many exceptions, such decompositions exist whenever the necessary ‘divisibility conditions’ hold. I also obtained a generalisation to the quasirandom setting, which implies an approximate formula for the number of designs; in particular, this resolved Wilson’s Conjecture on the number of Steiner Triple Systems. A more general result that I proved in 2018 on decomposing lattice-valued vectors indexed by labelled complexes provides many further existence and counting results for a wide range of combinatorial objects, such as resolvable designs (the generalised form of Kirkman’s Schoolgirl Problem), whist tournaments or generalised Sudoku squares. In this talk, I plan to review this background and then describe some more recent and ongoing applications of these results and developments of the ideas behind them.
Abstract: Enumerative geometry is a venerable subfield of Mathematics, with roots dating back to Greek Antiquity and a present inextricably linked with developments in other domains. Since the early 90s, in particular, the interaction with String Theory has sent shockwaves through the subject, giving both unexpected new perspectives and a remarkably powerful, physics-motivated toolkit to tackle several traditionally hard questions in the field. I will survey some recent developments in this vein for the case of enumerative invariants associated to a pair (X, D), with X a complex algebraic surface and D a singular anticanonical divisor in it. I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to (X, D), including the log Gromov-Witten invariants of the pair, the Gromov-Witten invariants of an associated higher dimensional Calabi-Yau variety, the open Gromov-Witten invariants of certain special Lagrangians in toric Calabi–Yau threefolds, the Donaldson–Thomas theory of a class of symmetric quivers, and certain open and closed Gopakumar-Vafa-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.
Abstract: For a simple and simply connected complex group G, I will discuss some elements of the proof of the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over families of smooth projective curves with marked points. Under the isomorphism with the bundle of conformal blocks, this connection is equivalent to the one constructed by conformal field theory. This is joint work with Indranil Biswas and Swarnava Mukhopadhyay.
Abstract: Tree decompositions are a powerful tool in both structural graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph.
Tree decompositions have traditionally been used in the context of forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.
Abstract: I will present a construction of the object in the title which, applied to the classical Toda system, controls the theory of categorical representations of compact Lie groups, along with applications (some conjectural, some rigorous) to gauged Gromov-Witten theory. Time permitting, we will review applications to Coulomb branches and the categorified Weyl character formula.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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.
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.
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 Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
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 envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.
The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:
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. Click here for information.
The methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using topological invariants has allowed for a better understanding and even predictions of new phases of matter. The use of topology could lead to the discovery of new electronic, photonic, and ultracold atomic states of matter previously unknown. The concrete problems in the physical phenomena could inspire new developments in the study of topological invariants in mathematics.
Here is a list of the scholars participating in this program.
A major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis.
Here is a list of the scholars participating in this program.
In the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum many-body theory.
String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as dramatic progress in old branches.
Several examples from the recent past exemplifying this synergy include the prediction from string theory of mirror symmetry, a highly unexpected mathematical equivalence between distinct pairs of Calabi-Yau manifolds. This fueled exciting developments in algebraic, enumerative and symplectic geometry. At the same time the realization of string theory as a phenomenologically viable physical theory depends crucially on detailed mathematical properties of these manifolds. In Einstein’s theory of general relativity the proofs of the positive energy theorem and the stability of flat spacetime were accompanied by fundamental new results in functional analysis, differential geometry and minimal surface theory. In the coming decades we expect many more important discoveries to arise from the interface of mathematics and physics. The Cheng Fund will foster these efforts.
Here is a partial list of the mathematicians who have indicated that they will attend part or all of this special program
Most physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations.
The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims to spur activity in this subject, to provide a natural setting for the most cutting edge results to be communicated, and to facilitate interaction among researchers of different backgrounds.
During the year, there will be two weekly seminar programs. Each program participants will be asked to give a talk on geometric analysis, or the evolution of equations, hyperbolic equations, and fluid dynamics.
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
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.
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
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.
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).
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.
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.
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.
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.
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.
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.
Title: Why do some universities have separate departments of statistics? And are they all anachronisms, destined to follow the path of other dinosaurs?
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.
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.
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.
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.
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.
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 inroom G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
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.
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.
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 sciences
On 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.
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.
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.
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 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.
On 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.
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.
The 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
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.
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.
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 Liu
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.
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.
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 G1020 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 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.
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.
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.
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.
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
On 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.
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.
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, aMemorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer.
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.
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, aMemorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
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.
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.
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.
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.
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.
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, aMemorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer.
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.
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.
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.
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 model
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 G1020 Garden Street, Cambridge, MA 02138
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”
On 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.
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.
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.
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.
Jointly 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.
Title: Noise stability of the spectrum of large matrices
Abstract: The spectrum of large non-normal matrices is notoriously sensitive to perturbations, as the example of nilpotent matrices shows. Remarkably, the spectrum of these matrices perturbed by polynomially(in the dimension) vanishing additive noise is remarkably stable. I will describe some results and the beginning of a theory.
The talk is based on joint work with Anirban Basak and Elliot Paquette, and earlier works with Feldheim, Guionnet, Paquette and Wood.
10:20 am – 11:20 am:Andrea Montanari
Title: Algorithms for estimating low-rank matrices
Abstract: Many interesting problems in statistics can be formulated as follows. The signal of interest is a large low-rank matrix with additional structure, and we are given a single noisy view of this matrix. We would like to estimate the low rank signal by taking into account optimally the signal structure. I will discuss two types of efficient estimation procedures based on message-passing algorithms and semidefinite programming relaxations, with an emphasis on asymptotically exact results.
11:20 am – 11:45 am: Break
11:45 am – 12:45 pm:Paul Bourgade
Title: Random matrices, the Riemann zeta function and trees
Abstract: Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on joint works with Arguin, Belius, Radziwill and Soundararajan.
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: Weak universality and Singular SPDEs
Abstract: Mesoscopic fluctuations of microscopic (discrete or continuous) dynamics can be described in terms of nonlinear stochastic partial differential equations which are universal: they depend on very few details of the microscopic model. This universality comes at a price: due to the extreme irregular nature of the random field sample paths, these equations turn out to not be well-posed in any classical analytic sense. I will review recent progress in the mathematical understanding of such singular equations and of their (weak) universality and their relation with the Wilsonian renormalisation group framework of theoretical physics.
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).
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 17
Time
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.
(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.
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.
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.
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.
Abstract: One can view a partial flag variety in C^n as an adjoint orbit inside the Lie algebra of n x n skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. We classify gradient flows on adjoint orbits in various metrics which are compatible with total positivity. As applications, we show how the classical Toda flow fits into this framework, and prove that a new family of amplituhedra are homeomorphic to closed balls. This is joint work with Anthony Bloch.
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).
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
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.
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.
Abstract: The last few decades have seen a surge of interest in building towards a theory of discrete curvature that attempts to translate the key properties of curvature in differential geometry to the setting of discrete objects and spaces. In the case of graphs there have been several successful proposals, for instance by Lin-Lu-Yau, Forman and Ollivier, that replicate important curvature theorems and have inspired applications in a variety of practical settings. In this talk, I will introduce a new notion of discrete curvature on graphs, which we call the resistance curvature, and discuss some of its basic properties. The resistance curvature is defined based on the concept of effective resistance which is a metric between the vertices of a graph and has many other properties such as a close relation to random spanning trees. The rich theory of these effective resistances allows to study the resistance curvature in great detail; I will for instance show that “Lin-Lu-Yau >= resistance >= Forman curvature” in a specific sense, show strong evidence that the resistance curvature converges to zero in expectation for Euclidean random graphs, and give a connectivity theorem for positively curved graphs. The resistance curvature also has a naturally associated discrete Ricci flow which is a gradient flow and has a closed-form solution in the case of vertex-transitive and path graphs. Finally, if time permits I will draw a connection with the geometry of hyperacute simplices, following the work of Miroslav Fiedler. This work was done in collaboration with Renaud Lambiotte.
Abstract: The Prague dimension of graphs was introduced by Nesetril, Pultr and Rodl in the 1970s: as a combinatorial measure of complexity, it is closely related to clique edges coverings and partitions. Proving a conjecture of Furedi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order n/(log n) for constant edge-probabilities. The main new proof ingredient is a Pippenger-Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size O(log n).
Title: China’s financial regulatory reform, financial opening-up, and Central Bank Digital Currency (CBDC)
Abstract: In this talk, I will explain the overall situation of China’s financial industry and review the development of China’s financial regulatory system reform from 1949 to 2021. Then, I will explain the policies of the 3 stages of financial opening-up, 2001–08, 2008–18, 2018≠present. In particular, the latest round of opening-up from 2018 has brought great opportunities for foreign institutions. China has the world’s largest banking industry with assets totaling $53 trillion, and accounts for 1/3 of the growth in global insurance premiums over the next 10 years. I will also introduce the progress of research & development of China’s Central Bank Digital Currency (CBDC, or E-CNY). By October 2021, 140 million people had opened E-CNY wallets, and 1.6 million merchants could accept payments using eCNY wallets, including utilities, catering services, transportation, retail, and government services.
Abstract: I will describe an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. This is joint work with Hülya Argüz, Rahul Pandharipande, and Dimitri Zvonkine (arxiv:2109.13323).
Title: Universal relations between entanglement, symmetries, and entropy
Abstract: Entanglement is an essential property of quantum systems that distinguishes them from classical ones. It is responsible for the nonlocal character of quantum information and provides a resource for quantum teleportation and quantum computation. In this talk I will provide an introduction to quantum entanglement and explain the essential role it plays in two seemingly unrelated subjects: implementation of measurement-based quantum computation and microstate counting of black holes in quantum gravity. Time permitting, I will also discuss attempts to characterize entanglement in string theory. A unifying theme that illuminates the entanglement structure of these diverse systems is the role of surface symmetries and (entanglement) edge modes. We will explain how these universal aspects of entanglement are captured in the framework of extended topological quantum field theory.
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).
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.
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.