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
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
Speaker: Chung-Ming Pan, Institut de Mathématiques de Toulouse
Title: Kähler-Einstein metrics on families of Fano varieties
Abstract: This talk aims to introduce a pluripotential approach to study uniform a priori estimates of Kähler-Einstein (KE) metrics on families of Fano varieties. I will first recall basic tools in the pluripotential theory and the variational approach to complex Monge-Ampère equations. I will then define a notion of convergence of quasi-plurisubharmonic functions in families of normal varieties and extend several classical properties under this context. Last, I will explain how these elements help to obtain a purely analytic proof of the openness of existing singular KE metrics and a uniform $L^\infty$ estimate of KE potentials. This is joint work with Antonio Trusiani.
Title: Black hole microstate counting from the gravitational path integral
Abstract: Reproducing the integer count of black hole micro-states from the gravitational path integral is an important problem in quantum gravity. In the first part of the talk, I will show that, by using supersymmetric localization, the gravitational path integral for 1/16-BPS black holes in supergravity can reproduce the index obtained in the string theory construction of such black holes. A more refined argument then shows that not only the black hole index but also the total number of black hole microstates within an energy window above extremality that is polynomially suppressed in the charges also matches this string theory index. In the second part of the talk, I will present a second perspective on this state count and show how the BPS Hilbert space can be obtained by directly preparing states using the gravitational path integral. While such a preparation naively gives rise to a Hilbert space of BPS states whose dimension is much larger than expected, I will explain how non-perturbative corrections in the overlap of such states are again responsible for reproducing the correct dimension of the Hilbert space.
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
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: Optimal Dynamic Allocation: Simplicity through Information Design
Abstract: We study dynamic nonmonetary markets where objects are allocated to unit-demand agents with private types. An agent’s value for an object is supermodular in her type and the quality of the object, and her payoff is quasilinear in her waiting cost. The social planner’s objective is a linear combination of allocative efficiency (i.e., the sum of values) and welfare (i.e., the sum of payoffs). We identify the optimal mechanism in the class of direct-revelation mechanisms that elicit agents’ types and assign them to objects over time. We show that, when the social planner can design the information disclosed to the agents about the objects, the optimal mechanism has a simple implementation: a first-come first-served waitlist with deferrals. In this implementation, the information disclosed about each object is an interval containing the object quality, rather than the exact quality. These intervals partition the quality space. We also show that when the planner’s objective weight on welfare increases, these intervals become coarser, and optimal disclosure policies less informative. A direct corollary is that mechanisms that achieve higher welfare also induce lower distributional inequality, in terms of the Lorenz order.
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: Sampling from the SK and mixed p-spin measures with stochastic localization
Abstract: I will present an algorithm which efficiently samples from the Sherrington-Kirkpatrick (SK) measure with no external field at high temperature. The approach is based on the stochastic localization process of Eldan, together with a subroutine for computing the mean vectors of a family of measures tilted by an appropriate external field. Conversely, we show that no ‘stable’ algorithm can approximately sample from the SK measure at low temperature. Time permitting, we discuss extensions to the p-spin model. This is based on a joint work with Andrea Montanari and Mark Sellke.
Title: Rough solutions of the relativistic Euler equations
Abstract: I will discuss recent works on the relativistic Euler equations with dynamic vorticity and entropy. We use a new formulation of the equations, which has geo-analytic structures. In this geometric formulation, we decompose the flow into geometric “sound-wave part” and “transport-div-curl part”. This allows us to derive sharp results about the dynamics, including the existence of low-regularity solutions. Then, I will discuss the results of rough solutions of the relativistic Euler equations and the role that nonlinear geometric optics plays in the framework. Our main result is that the Sobolev norm $H^{2+}$ of the variables in the “wave-part” and the H\”older norm $C^{0,0+}$ of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption $H^{2+}$ is the optimal result for the variables in the “wave-part.” This talk will include the main ideas of the proof, as well as a comparison of the relativistic and non-relativistic scenarios.
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: Enhancing Detection of Topological Order by Local Error Correction
Abstract: The exploration of topologically-ordered states of matter is a long-standing goal at the interface of several subfields of the physical sciences. Such states feature intriguing physical properties such as long-range entanglement, emergent gauge fields and non-local correlations, and can aid in realization of scalable fault-tolerant quantum computation. However, these same features also make creation, detection, and characterization of topologically-ordered states particularly challenging. Motivated by recent experimental demonstrations, we introduce a new paradigm for quantifying topological states—locally error-corrected decoration (LED)—by combining methods of error correction with ideas of renormalization-group flow. Our approach allows for efficient and robust identification of topological order, and is applicable in the presence of incoherent noise sources, making it particularly suitable for realistic experiments. We demonstrate the power of LED using numerical simulations of the toric code under a variety of perturbations, and we subsequently apply it to an experimental realization of a quantum spin liquid using a Rydberg-atom quantum simulator. Finally, we illustrate how LED can be applied to more general phases including non-abelian topological orders.
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: Modular graph forms and iterated integrals in string amplitudes
Abstract: I will discuss string amplitudes as a laboratory for special functions and period integrals that drive fruitful cross-talk with particle physicists and mathematicians. At genus zero, integration over punctures on a disk or sphere worldsheet generates multiple zeta values in the low-energy expansion of open- and closed-string amplitudes. At genus one, closed-string amplitudes introduce infinite families of non-holomorphic modular forms through the integration over torus punctures known as modular graph forms. The latter inspired Francis Brown’s alternative construction of non-holomorphic modular forms in the mathematics literature via iterated integrals, and I will report on recent progress in clarifying their connection with modular graph forms.
Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau
Sponsored by the National Science Foundation, Harvard University Mathematics, Harvard University Center of Mathematical Sciences and Applications, and the Massachusetts Institute of Technology.
Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.
Title: Enhancing Detection of Topological Order by Local Error Correction
Abstract: The exploration of topologically-ordered states of matter is a long-standing goal at the interface of several subfields of the physical sciences. Such states feature intriguing physical properties such as long-range entanglement, emergent gauge fields and non-local correlations, and can aid in realization of scalable fault-tolerant quantum computation. However, these same features also make creation, detection, and characterization of topologically-ordered states particularly challenging. Motivated by recent experimental demonstrations, we introduce a new paradigm for quantifying topological states—locally error-corrected decoration (LED)—by combining methods of error correction with ideas of renormalization-group flow. Our approach allows for efficient and robust identification of topological order, and is applicable in the presence of incoherent noise sources, making it particularly suitable for realistic experiments. We demonstrate the power of LED using numerical simulations of the toric code under a variety of perturbations, and we subsequently apply it to an experimental realization of a quantum spin liquid using a Rydberg-atom quantum simulator. Finally, we illustrate how LED can be applied to more general phases including non-abelian topological orders.
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: Modular graph forms and iterated integrals in string amplitudes
Abstract: I will discuss string amplitudes as a laboratory for special functions and period integrals that drive fruitful cross-talk with particle physicists and mathematicians. At genus zero, integration over punctures on a disk or sphere worldsheet generates multiple zeta values in the low-energy expansion of open- and closed-string amplitudes. At genus one, closed-string amplitudes introduce infinite families of non-holomorphic modular forms through the integration over torus punctures known as modular graph forms. The latter inspired Francis Brown’s alternative construction of non-holomorphic modular forms in the mathematics literature via iterated integrals, and I will report on recent progress in clarifying their connection with modular graph forms.
Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau
Sponsored by the National Science Foundation, Harvard University Mathematics, Harvard University Center of Mathematical Sciences and Applications, and the Massachusetts Institute of Technology.
Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.
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
Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau
Sponsored by the National Science Foundation, Harvard University Mathematics, Harvard University Center of Mathematical Sciences and Applications, and the Massachusetts Institute of Technology.
Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.
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
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: Swampland bounds on the abelian gauge sectors
Abstract: In this talk we will derive various bounds on the 0-form and the 1-form abelian gauge sectors of gravitational effective theories in 6 dimensions with minimal supersymmetry. We will start by considering 6-dimensional F-theory vacua with at least one tensor multiplets, to bound for them the number of the (0-form) U(1) gauge factors as well as the cyclic orders of the 1-form discrete gauge factors. While the two abelian gauge sectors may look rather independent, we will observe that both are heavily constrained by the solitonic heterotic strings present in the spectrum, which provide a common intuition for the derived bounds. Building upon the heterotic intuition, we will also try extending the arena to address analogous bounds for all F-theory vacua in 6 dimensions and even beyond. If time permits, several applications and future directions of research will be discussed at the end of the talk.
Abstract: I will explain an approach via “an adaptive Mermin-Wagner style shift” which proves localization of N x N Gaussian random band matrices with band width W satisfying W << N^{1/4}.
Joint work with Giorgio Cipolloni, Jeffrey Schenker and Jacob Shapiro.
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: Spectral gap and two-point functions in spin glasses
Abstract: Many have worked on spin glass models over the past 50 years, including physicists, mathematicians, and computers. A question that arises is whether computers yield dependable simulation results. In this talk, I will discuss some recent mathematical progress on spectral gap and two-point functions in spin glasses models.
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
Speaker: James Halverson (Northeastern University)
Title: Unexpected Uses of Neural Networks: Field Theory and Metric Flows
Abstract: We are now quite used to the idea that deep neural networks may be trained in a variety of ways to tackle cutting-edge problems in physics and mathematics, sometimes leading to rigorous results. In this talk, however, I will argue that breakthroughs in deep learning theory are also useful for making progress, focusing on applications to field theory and metric flows. Specifically, I will introduce a neural network approach to field theory with a different statistical origin, that exhibits generalized free field behavior at infinite width and interactions at finite width, and that allows for the study of symmetries via the study of correlation functions in a different duality frame. Then, I will review recent progress in approximating Calabi-Yau metrics as neural networks and cast that story into the language of neural tangent kernel theory, yielding a theoretical understanding of neural network metric flows induced by gradient descent and recovering famous metric flows, such as Perelman’s formulation of Ricci flow, in particular limits.
Title: Large deviations of Selberg’s central limit theorem
Abstract: Selberg’s CLT concerns the typical behaviour of the Riemann zeta function and shows that the random variable $\Re \log \zeta(1/2 + i t)$, for a uniformly drawn $t$, behaves as a Gaussian random variable with a particular variance. It is natural to investigate how far into the tails this Gaussianity persists, which is the topic of this work. There are also very close connections to similar problems in circular unitary ensemble characteristic polynomials. It transpires that a `multiscale scheme’ can be applied to both situations to understand these questions of large deviations, as well as certain maxima and moments. In this talk I will focus more on the techniques we apply to approach this problem and I will assume no number theoretic knowledge. This is joint work with Louis-Pierre Arguin.
Abstract: In the past few years, the Event Horizon Telescope has released the first close-up interferometric images of two supermassive black holes, M87* and SgrA*. It is believed that within these images is embedded a fine, yet-unresolved brightness enhancement called the photon ring. The ring is a universal consequence of strong lensing by the black hole and thereby conveys information on its spacetime geometry, potentially providing a new independent avenue for tests of general relativity in the strong-field regime. In the talk I will briefly review the theory of the photon ring and its corresponding spacetime region, the photon shell, which governs the universal lensing structure. I will then describe some current efforts and future prospects for resolving the ring, which include both the construction of transformative new instruments and the development of novel analysis methods. Focusing on the latter, I will present an upcoming proposal to use spectro-temporal autocorrelations in signals emitted from black hole environs as a probe of strong lensing effects.
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: Control of actin cable length by decelerated growth and network geometry
Abstract: The sizes of many subcellular structures are coordinated with cell size to ensure that these structures meet the functional demands of the cell. In eukaryotic cells, these subcellular structures are often membrane-bound organelles, whose volume is the physiologically important aspect of their size. Scaling organelle volume with cell volume can be explained by limiting pool mechanisms, wherein a constant concentration of molecular building blocks enables subcellular structures to increase in size proportionally with cell volume. However, limiting pool mechanisms cannot explain how the size of linear subcellular structures, such as cytoskeletal filaments, scale with the linear dimensions of the cell. Recently, we discovered that the length of actin cables in budding yeast (used for intracellular transport) precisely matches the length of the cell in which they are assembled. Using mathematical modeling and quantitative imaging of actin cable growth dynamics, we found that as the actin cables grow longer, their extension rates slow (or decelerate), enabling cable length to match cell length. Importantly, this deceleration behavior is cell-length dependent, allowing cables in longer cells to grow faster, and therefore reach a longer length before growth stops at the back of the cell. In addition, we have unexpectedly found that cable length is specified by cable shape. Our imaging analysis reveals that cables progressively taper as they extend from the bud neck into the mother cell, and further, this tapering scales with cell length. Integrating observations made for tapering actin networks in other systems, we have developed a novel mathematical model for cable length control that recapitulates our quantitative experimental observations. Unlike other models of size control, this model does not require length-dependent rates of assembly or disassembly. Instead, feedback control over the length of the cable is an emergent property due to the cross-linked and bundled architecture of the actin filaments within the cable. This work reveals a new strategy that cells use to coordinate the size of their internal parts with their linear dimensions. Similar design principles may control the size and scaling of other subcellular structures whose physiologically important dimension is their length.
Speaker: Guanyu Zhu (IBM Quantum, T. J. Watson Research Center)
Title: Fault-tolerant quantum computation via topological order on fractals and emergent symmetries
Abstract: Topological quantum error correcting codes in integer spatial dimensions have been widely studied in the field of quantum information. A remaining major challenge is to reduce the space-time overhead for universal fault-tolerant quantum computation with topological codes. In the first part of my talk, I will present a theory of topological order and quantum codes on fractals embedded in three and higher dimensions and its connection to systolic geometry. The construction of such fractal codes can hence significantly reduce the space overhead. In the second part, I will show how to perform fault-tolerant non-Clifford logical gates in such fractal codes using the idea of emergent symmetries. In particular, I will discuss the existence of higher-form symmetries corresponding to sweeping of certain codimension-2 invertible defects and exotic gapped boundaries which condense such defects.
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
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
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
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
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: Recent progress on coupled Einstein-Yang-Mills dynamics
Abstract: I will discuss my recent work with prof. S.T. Yau on Einstein-Yang-Mills theory. I’ll discuss the main ideas behind the stability of the Minkowski and Milne space time under coupled gauge-gravity perturbations. Construction of the gauge invariant energies that control the suitable norms of the physical degrees of freedom in a double-null framework play an important role. I’ll sketch how the special structure of the Einstein and Yang-Mills nonlinearities play a crucial role in obtaining the stability result of non-expanding spacetimes in particular AF spacetimes.
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
Harvard Science Center, 1 Oxford Street, Cambridge MA 02138
Probability Seminar
Speaker: Evita Nestoridi (Stonybrook)
Title: Diagonalizing Transition Matrices of Card Shuffles
Abstract: In their seminal work, Diaconis and Shahshahani used representation theory of the symmetric group to diagonalize the transition matrix of random transpositions. More recently, Dieker and Saliola introduced another technique to diagonalize the random-to-random card shuffle. In this talk we will discuss connections between these techniques as well as application to card shuffling.
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: Despite more than a century since the development of Einstein’s theory, the general relativistic two-body problem remains unsolved. A precise description of its solution is now essential, as it is necessary for understanding the strong-gravity dynamics of compact binaries observed at LIGO/VIRGO/KAGRA and in future gravitational wave observatories. In this talk, I will describe how considering the scattering of black holes and gravitons can shed new light on this problem. I will explain how using modern ideas from collider and particle physics we can calculate scattering observables in classical gravity, and extract the basic ingredients that describe the bound binary dynamics. Such calculations have produced state-of-art predictions for current and future gravitational wave observatories, which open the door for further discovery as we enter this new era of precision gravitational physics.
Title: A model of the cuprates: from the pseudogap metal to d-wave superconductivity and charge order
Abstract: Soon after the discovery of high-temperature superconductivity in the cuprates, Anderson proposed a connection to quantum spin liquids. But observations since then have shown that the low-temperature phase diagram is dominated by conventional states, with a competition between superconductivity and charge-ordered states which break translational symmetry. We employ the “pseudogap metal” phase, found at intermediate temperatures and low hole doping, as the parent to the phases found at lower temperatures. The pseudogap metal is described as a fractionalized phase of a single-band model, with small pocket Fermi surfaces of electron-like quasiparticles whose enclosed area is not equal to the free electron value, and an underlying pi-flux spin liquid with an emergent SU(2) gauge field. This pi-flux spin liquid is now known to be unstable to confinement at sufficiently low energies. We develop a theory of the different routes to confinement of the pi-flux spin liquid and show that d-wave superconductivity, antiferromagnetism, and charge order are natural outcomes. We argue that this theory provides routes to resolving a number of open puzzles on the cuprate phase diagram. As a side result, at half-filling, we propose a deconfined quantum critical point between an antiferromagnet and a d-wave superconductor described by a conformal gauge theory of 2 flavors of massless Dirac fermions and 2 flavors of complex scalars coupled as fundamentals to a SU(2) gauge field. This talk is based on Maine Christos, Zhu-Xi Luo, Henry Shackleton, Mathias S. Scheurer, and S. S., arXiv:2302.07885
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
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
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
Jefferson Hall Room 368, 17 Oxford St, Cambridge, MA 02138
Swampland Seminar
Speaker: Thomas Grimm (Utrecht U.)
Title: The Tameness of Quantum Field Theories
Abstract: Tameness is a generalized notion of finiteness that is restricting the geometric complexity of sets and functions. The underlying mathematical foundation lies in tame geometry, which is built from o-minimal structures introduced in mathematical logic. In this talk I formalize the connection between quantum field theories and logical structures and argue that the tameness of a quantum field theory relies on its UV definition. I quantify our expectations on the tameness of effective theories that can be coupled to quantum gravity and on CFTs. In particular, I present tameness conjectures about CFT observables and propose universal constraints that render spaces of CFTs to be tame sets. I then highlight the relation of these conjectures to other swampland conjectures, e.g., by arguing that the tameness of CFT observables restricts having parametrical gaps in the operator spectrum.
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
Speaker: Tin Yau Tsang, University of California Irvine
Title: Recent advances in scalar curvature and positive mass theorems
Abstract: First, we have a review of classical tools for studying scalar curvature and positive mass theorem. Then we are going to discuss some advances and new perspectives on these tools which lead to a deeper understanding of geometry and initial data sets.
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: On complete Calabi-Yau metrics and some related Monge-Ampere equations
Abstract: We will give a basic introduction to constructions for complete Calabi-Yau metrics. A systematic approach to construct such metrics using PDE methods was proposed in the work of Tian-Yau in the 90s and have attracted a lot of attention in recent years. I will discuss some joint work with B. Guo and T. Collins on a singular version of such a construction, as well as some ongoing work with Prof. Yau on some related boundary value problems.
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: Boundary behavior at classical and quantum phase transitions
Abstract: There has been a lot of recent interest in the boundary behavior of materials. This interest is driven in part by the field of topological states of quantum matter, where exotic protected boundary states are ubiquitous. In this talk, I’ll ask: what happens at a boundary of a system, when the bulk goes through a phase transition. While this question was studied in the context of classical statistical mechanics in the 70s and 80s, basic aspects of the boundary phase diagram for the simplest classical phase transitions have been missed until recently. I’ll describe progress in this field, as well as some extensions to quantum phase transitions.
Title: Toolformer: Language Models Can Teach Themselves to Use Tools
Abstract: Language models exhibit remarkable abilities to solve new tasks from just a few examples or textual instructions, especially at scale. They also, paradoxically, struggle with basic functionality, such as arithmetic or factual lookup, where much simpler and smaller models excel. In this talk, we show how these limitations can be overcome by letting language models teach themselves to use external tools via simple APIs. We discuss Toolformer, a model trained to independently decide which APIs to call, when to call them, what arguments to pass, and how to best incorporate the results into future token prediction. Through this, it achieves substantially improved zero-shot performance across a variety of downstream tasks without sacrificing its core language modeling abilities.
Title: Boundary current fluctuations for the half space ASEP
Abstract: The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
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: Boundary behavior at classical and quantum phase transitions
Abstract: There has been a lot of recent interest in the boundary behavior of materials. This interest is driven in part by the field of topological states of quantum matter, where exotic protected boundary states are ubiquitous. In this talk, I’ll ask: what happens at a boundary of a system, when the bulk goes through a phase transition. While this question was studied in the context of classical statistical mechanics in the 70s and 80s, basic aspects of the boundary phase diagram for the simplest classical phase transitions have been missed until recently. I’ll describe progress in this field, as well as some extensions to quantum phase transitions.
Title: Toolformer: Language Models Can Teach Themselves to Use Tools
Abstract: Language models exhibit remarkable abilities to solve new tasks from just a few examples or textual instructions, especially at scale. They also, paradoxically, struggle with basic functionality, such as arithmetic or factual lookup, where much simpler and smaller models excel. In this talk, we show how these limitations can be overcome by letting language models teach themselves to use external tools via simple APIs. We discuss Toolformer, a model trained to independently decide which APIs to call, when to call them, what arguments to pass, and how to best incorporate the results into future token prediction. Through this, it achieves substantially improved zero-shot performance across a variety of downstream tasks without sacrificing its core language modeling abilities.
Title: Boundary current fluctuations for the half space ASEP
Abstract: The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
Speaker: Philippe G. LeFloch, Sorbonne University and CNRS
Title: The localized seed-to-solution method for the Einstein constraints
Abstract: I will discuss advances on asymptotically Euclidian initial data sets and the variational method introduced by J. Corvino and R. Schoen. This talk is based on joint papers with The-Cang Nguyen (Montpellier) and Bruno Le Floch (Sorbonne Univ. and CNRS). In the vicinity of any given reference data set, we define a “localized seed-to-solution” map, which allows us to parametrize the initial data sets satisfying the Einstein constraints (possibly with matter fields). The parametrization is defined over classes of data sets understood modulo the image of the dual linearized constraints. Our main contribution concerns the sharp behavior of solutions at infinity, which we can arbitrarily localize in asymptotic cones in the sense of A. Carlotto and R. Schoen. Most importantly, as we prove it, the solutions enjoy sharp decay estimates at the harmonic and super-harmonic levels. In the course of this analysis, we discover the notion of ‘asymptotic modulators’, as we call them, or “correctors” to the standard ADM invariants.
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: Competition at the front of expanding populations
Abstract: When competing species grow into new territory, the population is dominated by descendants of successful ancestors at the expansion front. Successful ancestry depends on the reproductive advantage (fitness), as well as ability and opportunity to colonize new domains. (1) Based on symmetry considerations, we present a model that integrates both elements by coupling the classic description of one-dimensional competition (Fisher equation) to the minimal model of front shape (KPZ equation). Macroscopic manifestations of these equations on growth morphology are explored, providing a framework to study spatial competition, fixation, and differentiation, In particular, we find that ability to expand in space may overcome reproductive advantage in colonizing new territory. (2) Variations of fitness, as well as fixation time upon differentiation, are shown to belong to distinct universality classes depending on limits to gain of fitness.
Abstract: Fracton order describes novel quantum phases of matter that host quasiparticles with restricted mobility, and thus lies beyond the existing paradigm of topological order. In particular, excitations that cannot move without creating other excitations are called fractons. Here we address a fundamental open question — can the notion of self-exchange statistics be naturally defined for fractons, given their complete immobility as isolated excitations? Surprisingly, we demonstrate how fractons can be exchanged, and show their self-statistics is a key part of the characterization of fracton orders. We derive general constraints satisfied by the fracton self-statistics in a large class of abelian fracton orders. Finally, we show the existence of semionic or fermionic fracton self-statistics in some twisted variants of the checkerboard model and Haah’s code, establishing that these models are in distinct quantum phases as compared to their untwisted cousins.
References: H Song, N Tantivasadakarn, W Shirley, M Hermele, arXiv:2304.00028.
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
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
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
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.
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.
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: 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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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 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.
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 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.
Title: Synthetic Regression Discontinuity: Estimating Treatment Effects using Machine Learning
Abstract: In the standard regression discontinuity setting, treatment assignment is based on whether a unit’s observable score (running variable) crosses a known threshold. We propose a two-stage method to estimate the treatment effect when the score is unobservable to the econometrician while the treatment status is known for all units. In the first stage, we use a statistical model to predict a unit’s treatment status based on a continuous synthetic score. In the second stage, we apply a regression discontinuity design using the predicted synthetic score as the running variable to estimate the treatment effect on an outcome of interest. We establish conditions under which the method identifies the local treatment effect for a unit at the threshold of the unobservable score, the same parameter that a standard regression discontinuity design with known score would identify. We also examine the properties of the estimator using simulations, and propose the use machine learning algorithms to achieve high prediction accuracy. Finally, we apply the method to measure the effect of an investment grade rating on corporate bond prices by any of the three largest credit ratings agencies. We find an average 1% increase in the prices of corporate bonds that received an investment grade as opposed to a non-investment grade rating.
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
On August 2–5, the CMSA hosted a workshop on Phase Transitions and Topological Defects in the Early Universe.
The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA and online via Zoom webinar.
The next decade will see a wealth of new cosmological data, which can lead to new insights into fundamental physics. Upcoming facilities (such as LISA) will be able to probe signals of fascinating phenomena in the early universe. These include signals from “Phase Transitions and Topological Defects,” which are ubiquitously given rise to in well-motivated UV models. In-depth studies of such signals requires cross-talks between experts from a wide spectrum of fields.
The workshop aims to provide a platform for efficient exchange of new ideas related to these topics. It will start with an overview of some of the past and future experimental efforts. Next, there will be a substantial number of talks probing different aspects of phenomenology of phase transitions and topological defects in the early universe. It will finally close with discussions on recent formal development in the field.
Scientific Advisory: Julian B. Muñoz, Lisa Randall, Matthew Reece, Tracy Slatyer, Shing-Tung Yau
Organizers: Harvard: Nick DePorzio, Katie Fraser, Sam Homiller, Rashmish Mishra, & Aditya Parikh MIT: Pouya Asadi, Marianne Moore, & Yitian Sun
Schedule/Format There will be 20+ 10 minute talks, ample discussion time, and lightning chalkboard talks.
Speakers:
Nancy Aggarwal (Northwestern)
Jae Hyeok Chang (UMD – JHU)
Yanou Cui (UC Riverside)
David Dunsky (UC Berkeley)
Isabel Garcia-Garcia (KITP – UCSB)
Oliver Gould (Nottingham)
Yann Gouttenoire (Tel Aviv)
Eleanor Hall (UC Berkeley)
Sungwoo Hong (Chicago)
Anson Hook (UMD)
Jessica Howard (UC Irvine)
Seth Koren (Chicago)
Mrunal Korwar (Wisconsin)
Soubhik Kumar (UC Berkeley)
Vuk Mandic (Minnesota)
Yuto Minami (Osaka)
Michael Nee (Oxford)
Kai Schmitz (CERN)
Stephen R. Taylor (Vanderbilt)
Ofri Telem (UC Berkeley)
Juven Wang (Harvard)
Yikun Wang (Caltech)
Participants:
Manuel Buen Abad (UMD)
Pouya Asadi (MIT)
Sean Benevedes (MIT)
Sandipan Bhattacherjee (Birla Institute of Technology Mesra Ranchi India)
Xingang Chen (Harvard University)
Nicholas DePorzio (Harvard University)
Peizhi Du (Stony Brook University)
Nicolas Fernandez (University of Illinois Urbana-Champaign)
Joshua Foster (MIT)
Katherine Fraser (Harvard University)
Sarah Geller (MIT)
Aurora Ireland (University of Chicago)
Marius Kongsore (New York University)
Ho Tat Lam (Massachusetts Institute of Technology)
Lingfeng Li (Brown University)
Yingying Li (Fermilab)
Gustavo Marques-Tavares (UMD)
Rashmish Mishra (Harvard University)
Siddharth Mishra-Sharma (MIT/Harvard University)
Toby Opferkuch (UC Berkeley)
Tong Ou (University of Chicago)
Aditya Parikh (Harvard University)
Yitian Sun (MIT)
Juan Sebastian Valbuena-Bermudez (Ludwig Maximilian University of Munich and Max Planck Institute for Physics)
Isaac Wang (Rutgers)
Wei Xue (University of Florida)
Winston Yin (UC Berkeley)
Quratulain Zahoor (The Islamia University of Bahwalpur Punjab (Pakistan)
Schedule
Tuesday, August 2, 2022
9:00–9:20 am
Breakfast
9:20–9:30 am
Rashmish Mishra
Opening Remarks
9:30–10:00 am
Vuk Mandic
Title: Searching for the Stochastic Gravitational Wave Background with LISA
Abstract: The upcoming space-borne gravitational wave detector Laser Interferometer Space Antenna (LISA) will open a window into the milliHertz band of the gravitational wave spectrum. Among the many sources in this band is the stochastic gravitational wave background (SGWB), arising as an incoherent superposition of many uncorrelated gravitational wave sources. The SGWB could be of cosmological origin, carrying unique information about the physical processes that took place within the first minute after the big bang, including possible phase transitions and topological defects. LISA therefore has the potential to illuminate particle physics at very high energy scales that may be inaccessible in laboratories. I will discuss how LISA can be used to search for the SGWB, highlighting a new pipeline developed for this purpose as well as several challenges and limitations that such a search will encounter.
10:00–10:30 am
Nancy Aggarwal
Title: Gravitational waves at frequencies above 10 kHz
Abstract: Gravitational waves (GWs) at frequencies higher than the LIGO band can bring us completely new information about the universe. Besides being the most-interesting frequency region for looking at cosmological phenomena, they can also convey signatures of ultralight bosons through blackhole superradiance and light primordial blackholes (PBHs). I will introduce a new global initiative to study GW sources and detectors at ultra-high-frequencies (MHz-GHz), as well as a new experiment at Northwestern University to look for GWs in the frequency band of 10 kHz to 300 kHz using levitated optomechanical sensors. I will summarize the design, the current experimental progress, as well as a path forward for future improvements.
10:30–11:00 am
Yuto Minami
Title: New measurements of the cosmic birefringence
Abstract: Polarised light of the cosmic microwave background, the remnant light of the Big Bang, is sensitive to parity-violating physics, cosmic birefringence. In this presentation I report on a new measurement of cosmic birefringence from polarisation data of the European Space Agency (ESA)’s Planck satellite released in 2018. The statistical significance of the measured signal is 2.4 sigma. Recently, we found a signal with 3.3 sigma statistical significance when we use the latest Planck data and consider an effect of polarised foreground emission. If confirmed with higher statistical significance in future, it would have important implications for the elusive nature of dark matter and dark energy.
11:00–1:30 pm
Break
1:30–3:00 pm
Lighting Talks 1
Lingfeng Li Winston Yin Marius Kongsore Nick DePorzio
3:00–3:30 pm
Jae Hyeok Chang
Title: Correlating gravitational wave and gamma-ray signals from primordial black holes
Abstract: Asteroid-mass primordial black holes (PBHs) can explain the observed dark matter abundance while being consistent with the current indirect detection constraints. These PBHs can produce gamma-ray signals from Hawking radiation that are within the sensitivity of future measurements by the AMEGO and e-ASTROGAM experiments. PBHs which give rise to such observable gamma-ray signals have a cosmic origin from large primordial curvature fluctuations. There must then be a companion, stochastic gravitational wave (GW) background produced by the same curvature fluctuations. I will demonstrate that the resulting GW signals will be well within the sensitivity of future detectors such as LISA, DECIGO, BBO, and the Einstein Telescope. The multimessenger signal from the observed gamma-rays and GWs will allow a precise measurement of the primordial curvature perturbation that produces the PBH. I will also argue that the resulting correlation between the two types of observations can provide a smoking-gun signal of PBHs.
3:30–4:00 pm
Anson Hook (Virtual via Zoom)
Title: Early Universe Cosmology from Stochastic Gravitational Waves
Abstract: The causal tail of stochastic gravitational waves can be used to probe the energy density in free streaming relativistic species as well as measure gstar and beta functions as a function of temperature. In the event of the discovery of loud stochastic gravitational waves, we demonstrate that LISA can measure the free streaming fraction of the universe down to the 10^-3 level, 100 times more sensitive than current constraints. Additionally, it would be sensitive to O(1) deviations of gstar and the QCD beta function from their Standard Model value at temperatures ~ 10^5 GeV. In this case, many motivated models such as split SUSY and other solutions to the Electroweak Hierarchy problem would be tested. Future detectors, such as DECIGO, would be 100 times more sensitive than LISA to these effects and be capable of testing other motivated scenarios such as WIMPs and axions. The amazing prospect of using precision gravitational wave measurements to test such well motivated theories provides a benchmark to aim for when developing a precise understanding of the gravitational wave spectrum both experimentally and theoretically.
Wednesday, August 3, 2022
9:00–9:30 am
Breakfast
9:30–10:00 am
Kai Schmitz (Virtual via Zoom)
Title: Gravitational waves from metastable cosmic strings
Abstract: Cosmic strings are predicted by many Standard Model extensions involving the cosmological breaking of an Abelian symmetry and represent a potential source of primordial gravitational waves (GWs). In many Grand Unified Theories (GUTs), cosmic strings especially turn out to be metastable, as the nucleation of GUT monopoles along strings after a finite lifetime eventually leads to the collapse of the entire string network. In this talk, I will discuss the theoretical description of such a network and its individual components as well as the consequences for the emitted GW spectrum. Remarkably, the GW signal from metastable strings may well explain the common-spectrum process recently observed in pulsar timing data, while at the same time and in contrast to stable cosmic strings predicting a signal at higher frequencies that is still within the reach of current-generation ground-based interferometers. On their way to design sensitivity, existing GW experiments will thus have a realistic chance to probe particle physics processes at energies close to the GUT scale via the observation of GWs from metastable strings. This talk is based on 2107.04578 in collaboration with Wilfried Buchmüller and Valerie Domcke.
10:00–10:30 am
Oliver Gould (Virtual via Zoom)
Title: Effective field theory for cosmological phase transitions
Abstract: Phase transitions are driven by thermal loop fluctuations, which modify background fields at leading order. This breaks the loop expansion and leads to large theoretical uncertainties in typical calculations, especially for gravitational wave predictions. I will give an overview of our present understanding of these uncertainties, and of the tools that have been developed to overcome them. Effective field theory has been at the forefront of this development, and I will outline how it can be used to solve a number of decades-long-standing theoretical problems.
10:30–11:00 am
Isabel Garcia-Garcia
Title: The Rocket Science of Expanding Bubbles
11:00–1:30 pm
Break
1:30–3:00 pm
Lightning Talks 2
Sarah Geller Peizhi Du Tong Ou Isaac Wang Katie Fraser
3:00–3:30 pm
David Dunsky (Virtual via Zoom)
Title: Gravitational Wave Gastronomy
Abstract: The symmetry breaking of grand unified gauge groups in the early universe often leaves behind relic topological defects such as cosmic strings, domain walls, or monopoles. For some symmetry breaking chains, hybrid defects can form where cosmic strings attach to domain walls or monopoles attach to strings. In general, such hybrid defects are unstable and can leave behind unique gravitational wave fingerprints. In this talk, I will discuss the gravitational wave spectrum from 1) the destruction of a cosmic string network by the nucleation of monopoles which cut up and “eat” the strings, 2) the collapse and decay of a monopole-string network by strings that “eat” the monopoles, 3) the destruction of a domain wall network by the nucleation of string-bounded holes on the wall that expand and “eat” the wall, and 4) the collapse and decay of a string-bounded wall network by walls that “eat” the strings. We call the gravitational wave signals produced from the “eating” of one topological defect by another “gravitational wave gastronomy”. The gravitational wave gastronomy signals considered yield unique spectra that can be used to narrow down the SO(10) symmetry breaking chain to the Standard Model and the scales of symmetry breaking associated with the consumed topological defects.
3:30–4:00 pm
Yanou Cui (Virtual via Zoom)
Title: Cosmic Archaeology with gravitational waves from (axion) cosmic strings
Abstract: In this talk I will discuss important aspects of cosmology and particle physics that can be probed with GW signals from cosmic strings: probing the pre-BBN primordial dark age and axion physics. Gravitational waves (GWs) originating from the dynamics of a cosmic string network have the ability to probe many otherwise inaccessible properties of the early universe. In particular, I will discuss how the frequency spectrum of a stochastic GW background (SGWB) from a cosmic string network can be used to probe Hubble expansion rate of the early universe prior to Big Bang Nucleosynthesis (BBN), during the “primordial dark age”. Furthermore I will show that in contrast to the standard expectation, cosmic strings formed before inflation could regrow back into the horizon and leave imprints, with GW bursts potentially being the leading signal. In relation to axion physics I will also demonstrate the detection prospect for SGWB from global/axion strings which may provide a new probe for axion-like dark matter models, considering various scenarios of cosmic history.
4:00–4:30 pm
Michael Nee
Title: The Boring Monopole
Abstract: First order phase transitions play an important role in the cosmology of many theories of BSM physics. In this talk I will discuss how a population of magnetic monopoles present in the early universe can seed first order phase transitions, causing them to proceed much more rapidly than in the usual case. The field profiles describing the decay do not have the typically assumed O(3)/O(4) symmetry, thus requiring an extension of the usual decay rate calculation. To numerically determine the saddle point solutions which describe the decay we use a new algorithm based on the mountain pass theorem. Our results show that monopole-catalysed tunnelling can dominate over the homogeneous decay for a wide range of parameters.
Thursday, August 4, 2022
9:00–9:30 am
Breakfast
9:30–10:00 am
Yikun Wang
Title: A New Approach to Electroweak Symmetry Non-Restoration
Abstract: Electroweak symmetry non-restoration up to high temperatures well above the electroweak scale has intriguing implications for (electroweak) baryogenesis and early universe thermal histories. In this talk, I will discuss such a possible fate of the electroweak symmetry in the early universe and a new approach to realize it, via an inert Higgs sector that couples to the Standard Model Higgs as well as an extended scalar singlet sector. Examples of benchmark scenarios that allow for electroweak symmetry non-restoration all the way up to hundreds of TeV temperatures, at the same time featuring suppressed sphaleron washout factors down to the electroweak scale, will be presented. Renormalization group improvements and thermal resummation, necessary to evaluate the effective potential spanning over a broad range of energy scales and temperatures, have been implemented calculating the thermal history. This method for transmitting the Standard Model broken electroweak symmetry to an inert Higgs sector can be scrutinized through Higgs physics phenomenology and electroweak precision measurements at the HL-LHC.
10:00–10:30 am
Soubhik Kumar
Title: Probing primordial fluctuations through stochastic gravitational wave background anisotropies
Abstract: Stochastic gravitational wave backgrounds are expected to be anisotropic. While such anisotropies can be of astrophysical origin, a cosmological component of such anisotropies can carry rich information about primordial perturbations. Focusing on the case of a cosmological phase transition, I will talk about how such anisotropies can give us a powerful probe of primordial non-Gaussianities, complementary to current and future CMB and LSS searches. In the scenario where astrophysical foregrounds are also present, I will then discuss some strategies using which we can extract the cosmological signal, focusing on the case of LISA, Taiji and BBO, in particular.
10:30–11:00 am
Jessica Howard (Virtual via Zoom)
Title: Dark Matter Freeze-out during SU(2)_L Confinement
Abstract: We explore the possibility that dark matter is a pair of SU(2)_L doublets and propose a novel mechanism of dark matter production that proceeds through the confinement of the weak sector of the Standard Model. This phase of confinement causes the Standard Model doublets and dark matter to confine into pion-like objects. Before the weak sector deconfines, the dark pions freezeout and generate a relic abundance of dark matter. We solve the Boltzmann equations for this scenario to determine the scale of confinement and constituent dark matter mass required to produce the observed relic density. We determine which regions of this parameter space evade direct detection and collider bounds.
11:00–11:30 am
Juven Wang
Title: Quantum Matter Adventure to Beyond the Standard Model Prediction
Abstract: Ideas developed from the quantum matter and quantum field theory frontier may guide us to explore new physics beyond the 4d Standard Model. I propose a few such ideas. First, new physics for neutrinos: right-handed neutrinos carry a Z_{16} class mixed gauge-gravitational global anomaly index, which could be replaced by 4d or 5d topological quantum field theory, or 4d interacting conformal field theory. These theories provide possible new neutrino mass mechanisms [arXiv:2012.15860]. Second, deconfined quantum criticality between Grand Unified Theories: dictated by a Z_2 class global anomaly, a gapless quantum critical region can happen between Georgi-Glashow and Pati-Salam models as deformation of the Standard Model, where Beyond the Standard Model physics and Dark Gauge sector occur as neighbor phases [arXiv:2106.16248, arXiv:2112.14765, arXiv:2204.08393]. Third, the Strong CP problem can be solved by a new solution involving Symmetric Mass Generation [arXiv:2204.14271].
11:30–1:30 pm
Break
1:30–4:00 pm
Stephen R. Taylor
Title: Pulsar Timing Arrays: The Next Window onto the Low-frequency Gravitational-wave Universe
Abstract: The nanohertz-frequency band of gravitational waves should be awash with signals from supermassive black-hole binaries, as well as cosmological signatures of phase transitions, cosmic strings, and other relics of the early Universe. Pulsar-timing arrays (PTAs) like the North American Nanohertz Observatory for Gravitational waves (NANOGrav) and the International Pulsar Timing Array are poised to chart this new frontier of gravitational wave discovery within the next several years. I will present exciting new results from recent cutting-edge searches, discuss some milestones on the road to the next decade of PTA discovery, and take workshop attendees through a guided tutorial of how the broader community can use our production-level analysis pipeline to extract new science with ease.
Friday, August 5, 2022
9:00–9:30 am
Breakfast
9:30–10:00 am
Ofri Telem
Title: Charge-Monopole Pairwise Phases from Dressed Quantum States
10:00–10:30 am
Sungwoo Hong
Title: Coupling a Cosmic String to a TQFT
Abstract: In the last few years, the notion of symmetry has been enlarged to “generalized symmetry” or “higher-form symmetry” and these more generalized symmetries have played a critical role in deepening our understanding of QFT, notably IR phases of QFT. In this talk, I will discuss a various ways of coupling the axion-Maxwell theory to a topological field theory (TQFT). Contrary to a common wisdom, I will show that such topological modifications can lead to direct changes in the local physics with possible observable consequences. This surprise can be realized by a dimensional reduction, namely, a coupling to a TQFT in 4d leads to a non-trivial and local impact on the 2d string world-sheet QFT. There also exists a topological modification of the theory, i.e. gauging a discrete subgroup of 0-form shift symmetry, and this time it results in a alteration of spectrum of cosmic strings. If time permits, I will also discuss generalized symmetries and associated higher-groups of these theories.
10:30–11:00 am
Eleanor Hall (Virtual via Zoom)
Title: Non-perturbative methods for false vacuum decay
Abstract: Gravitational waves from phase transitions in the early universe are one of our most promising signal channels of BSM physics; however, existing methods for predicting these signals are limited to weakly-coupled theories. In this talk, I present the quasi-stationary effective action, a new non-perturbative formalism for false vacuum decay that integrates over local fluctuations in field space using the functional renormalization group. This method opens the door to reliable calculation of gravitational wave signals and false vacuum decay rates for strongly-interacting theories. I will also discuss recent developments and ongoing extensions of the QSEA.
11:00–1:30 pm
Break
1:30–2:00 pm
Mrunal Korwar
Title: Electroweak Symmetric Balls
Abstract: Electroweak symmetric balls are macroscopic objects with electroweak symmetry restored inside. Such an object can arise in models where dark sectors contain monopole or non-topological soliton with a Higgs portal interaction to the Standard Model. It could be produced in the early universe via phase transition or parametric resonance, accounting for all dark matter. In a scenario where the balls are allowed to evaporate, the observed baryon asymmetry in our universe could be explained by a mechanism of “catalyzed baryogenesis.” In this mechanism, the motion of a ball-like catalyst provides the necessary out-of-equilibrium condition, its outer wall has CP-violating interactions with the Standard Model particles, and its interior has baryon number violating interactions via electroweak Sphaleron. Because of electroweak symmetric cores, such objects have a large geometric cross-section off a nucleus, generating a multi-hit signature in large volume detectors. These objects could radiatively capture a nucleus and release GeV-scale energy for each interaction. The IceCube detector can probe dark matter balls with masses up to a gram.
2:00–2:30 pm
Seth Koren
Title: Discrete Gauged Baryon Minus Lepton Number and the Cosmological Lithium Problem
Abstract: We study the baryon minus lepton number gauge theory broken by a scalar with charge six. The infrared discrete vestige of the gauge symmetry demands the existence of cosmic string solutions, and their production as dynamical objects in the early universe is guaranteed by causality. These topological defects can support interactions which convert three protons into three positrons, and we argue an `electric’-`magnetic’ interplay can lead to an amplified, strong-scale cross-section in an analogue of the Callan-Rubakov effect. The cosmological lithium problem—that theory predicts a primordial abundance thrice as high as that observed—has resisted decades of attempts by cosmologists, nuclear physicists, and astronomers alike to root out systematics. We suggest cosmic strings have disintegrated O(1) of the primordial lithium nuclei and estimate the rate in a benchmark scenario. To our knowledge this is the first new physics mechanism with microphysical justification for the abundance of lithium uniquely to be modified after Big Bang Nucleosynthesis.
2:30–3:00 pm
Yann Gouttenoire
Title: Supercool Composite Dark Matter beyond 100 TeV
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.
Title: Recent Advances on Maximum Flows and Minimum-Cost Flows
Abstract: We survey recent advances on computing flows in graphs, culminating in an almost linear time algorithm for solving minimum-cost flow and several other problems to high accuracy on directed graphs. Along the way, we will discuss intuitions from linear programming, graph theory, and data structures that influence these works, and the resulting natural open problems.
Bio: Yang P. Liu is a final-year graduate student at Stanford University. He is broadly interested in the efficient design of algorithms, particularly flows, convex optimization, and online algorithms. For his work, he has been awarded STOC and ITCS best student papers.
Title: Gifts from anomalies: new results on quantum critical transport in non-Fermi liquids
Abstract: Non-Fermi liquid phenomena arise naturally near Landau ordering transitions in metallic systems. Here, we leverage quantum anomalies as a powerful nonperturbative tool to calculate optical transport in these models in the infrared limit. While the simplest such models with a single boson flavor (N=1) have zero incoherent conductivity, a recently proposed large N deformation involving flavor-random Yukawa couplings between N flavors of bosons and fermions admits a nontrivial incoherent conductivity (z is the boson dynamical exponent) when the order parameter is odd under inversion. The presence of incoherent conductivity in the random flavor model is a consequence of its unusual anomaly structure. From this we conclude that the large N deformation does not share important nonperturbative features with the physical N = 1 model, though it remains an interesting theory in its own right. Going beyond the IR fixed point, we also consider the effects of irrelevant operators and show, within the scope of the RPA expansion, that the old result due to Kim et al. is incorrect for inversion-odd order parameters.
On August 26, 2022 the CMSA hosted our eighth annual Conference on Big Data. The Big Data Conference features 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 2022 Big Data Conference took place virtually on Zoom.
Organizers:
Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business
Horng-Tzer Yau, Professor of Mathematics, Harvard University
Title: On ANN optimal estimation and inference for policy functionals of nonparametric conditional moment restrictions
Abstract: Many causal/policy parameters of interest are expectation functionals of unknown infinite-dimensional structural functions identified via conditional moment restrictions. Artificial Neural Networks (ANNs) can be viewed as nonlinear sieves that can approximate complex functions of high dimensional covariates more effectively than linear sieves. In this talk we present ANN optimal estimation and inference on policy functionals, such as average elasticities or value functions, of unknown structural functions of endogenous covariates. We provide ANN efficient estimation and optimal t based confidence interval for regular policy functionals such as average derivatives in nonparametric instrumental variables regressions. We also present ANN quasi likelihood ratio based inference for possibly irregular policy functionals of general nonparametric conditional moment restrictions (such as quantile instrumental variables models or Bellman equations) for time series data. We conduct intensive Monte Carlo studies to investigate computational issues with ANN based optimal estimation and inference in economic structural models with endogeneity. For economic data sets that do not have very high signal to noise ratios, there are current gaps between theoretical advantage of ANN approximation theory vs inferential performance in finite samples. Some of the results are applied to efficient estimation and optimal inference for average price elasticity in consumer demand and BLP type demand.
The talk is based on two co-authored papers: (1) Efficient Estimation of Average Derivatives in NPIV Models: Simulation Comparisons of Neural Network Estimators (Authors: Jiafeng Chen, Xiaohong Chen and Elie Tamer) https://arxiv.org/abs/2110.06763
(2) Neural network Inference on Nonparametric conditional moment restrictions with weakly dependent data (Authors: Xiaohong Chen, Yuan Liao and Weichen Wang).
Title: Labor Reactions to Credit Deterioration: Evidence from LinkedIn Activity
Abstract: We analyze worker reactions to their firms’ credit deterioration. Using weekly networking activity on LinkedIn, we show workers initiate more connections immediately following a negative credit event, even at firms far from bankruptcy. Our results suggest that workers are driven by concerns about both unemployment and future prospects at their firm. Heightened networking activity is associated with contemporaneous and future departures, especially at financially healthy firms. Other negative events like missed earnings and equity downgrades do not trigger similar reactions. Overall, our results indicate that the build-up of connections triggered by credit deterioration represents a source of fragility for firms.
10:50 am – 11:35 am
Miles Cranmer
Title: Interpretable Machine Learning for Physics
Abstract: Would Kepler have discovered his laws if machine learning had been around in 1609? Or would he have been satisfied with the accuracy of some black box regression model, leaving Newton without the inspiration to discover the law of gravitation? In this talk I will explore the compatibility of industry-oriented machine learning algorithms with discovery in the natural sciences. I will describe recent approaches developed with collaborators for addressing this, based on a strategy of “translating” neural networks into symbolic models via evolutionary algorithms. I will discuss the inner workings of the open-source symbolic regression library PySR (github.com/MilesCranmer/PySR), which forms a central part of this interpretable learning toolkit. Finally, I will present examples of how these methods have been used in the past two years in scientific discovery, and outline some current efforts.
Abstract: Large language models of a huge number of parameters and trained on near internet-sized number of tokens have been empirically shown to obey “neural scaling laws” for which their performance behaves predictably as a power law in either parameters or dataset size until bottlenecked by the other resource. To understand this better, we first identify the necessary properties allowing such scaling laws to arise and then propose a statistical model — a joint generative data model and random feature model — that captures this neural scaling phenomenology. By solving this model using tools from random matrix theory, we gain insight into (i) the statistical structure of datasets and tasks that lead to scaling laws (ii) how nonlinear feature maps, i.e the role played by the deep neural network, enable scaling laws when trained on these datasets, and (iii) how such scaling laws can break down, and what their behavior is when they do. A key feature is the manner in which the power laws that occur in the statistics of natural datasets are translated into power law scalings of the test loss, and how the finite extent of such power laws leads to both bottlenecks and breakdowns.
Title: Exploring and Exploiting the Universality Phenomena in High-Dimensional Estimation and Learning
Abstract: Universality is a fascinating high-dimensional phenomenon. It points to the existence of universal laws that govern the macroscopic behavior of wide classes of large and complex systems, despite their differences in microscopic details. The notion of universality originated in statistical mechanics, especially in the study of phase transitions. Similar phenomena have been observed in probability theory, dynamical systems, random matrix theory, and number theory. In this talk, I will present some recent progresses in rigorously understanding and exploiting the universality phenomena in the context of statistical estimation and learning on high-dimensional data. Examples include spectral methods for high-dimensional projection pursuit, statistical learning based on kernel and random feature models, and approximate message passing algorithms on highly structured, strongly correlated, and even (nearly) deterministic data matrices. Together, they demonstrate the robustness and wide applicability of the universality phenomena.
Bio: Yue M. Lu attended the University of Illinois at Urbana-Champaign, where he received the M.Sc. degree in mathematics and the Ph.D. degree in electrical engineering, both in 2007. He is currently Gordon McKay Professor of Electrical Engineering and of Applied Mathematics at Harvard University. He is also fortunate to have held visiting appointments at Duke University in 2016 and at the École Normale Supérieure (ENS) in 2019. His research interests include the mathematical foundations of statistical signal processing and machine learning in high dimensions.
Abstract: In 1973, Lemmens and Seidel asked to determine N_alpha(r), the maximum number of equiangular lines in R^r with common angle arccos(alpha). Recently, this problem has been almost completely settled when r is exponentially large relative to 1/alpha, with the approach both relying on Ramsey’s theorem, as well as being limited by it. In this talk, we will show how orthogonal projections of matrices with respect to the Frobenius inner product can be used to overcome this limitation, thereby obtaining significantly improved upper bounds on N_alpha(r) when r is polynomial in 1/alpha. In particular, our results imply that N_alpha(r) = Theta(r) for alpha >= Omega(1 / r^1/5).
Our projection method generalizes to complex equiangular lines in C^r, which may be of independent interest in quantum theory. Applying this method also allows us to obtain the first universal bound on the maximum number of complex equiangular lines in C^r with common Hermitian angle arccos(alpha), an extension of the Alon-Boppana theorem to dense regular graphs, which is tight for strongly regular graphs corresponding to r(r+1)/2 equiangular lines in R^r, an improvement to Welch’s bound in coding theory.
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.
Speaker: Semyon Klevtsov, University of Strasbourg
Title: Geometric test for topological states of matter
Abstract: We generalize the flux insertion argument due to Laughlin, Niu-Thouless-Tao-Wu, and Avron-Seiler-Zograf to the case of fractional quantum Hall states on a higher-genus surface. We propose this setting as a test to characterise the robustness, or topologicity, of the quantum state of matter and apply our test to the Laughlin states. Laughlin states form a vector bundle, the Laughlin bundle, over the Jacobian – the space of Aharonov-Bohm fluxes through the holes of the surface. The rank of the Laughlin bundle is the
degeneracy of Laughlin states or, in presence of quasiholes, the dimension of the corresponding full many-body Hilbert space; its slope, which is the first Chern class divided by the rank, is the Hall conductance. We compute the rank and all the Chern classes of Laughlin bundles for any genus and any number of quasiholes, settling, in particular, the Wen-Niu conjecture. Then we show that Laughlin bundles with non-localized quasiholes are not projectively flat and that the Hall current is precisely quantized only for the states with localized quasiholes. Hence our test distinguishes these states from the full many-body Hilbert space. Based on joint work with Dimitri Zvonkine (CNRS, University of Paris-Versaille).
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: 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.
Speaker: Avner Karasik (University of Cambridge, UK)
Title: Candidates for Non-Supersymmetric Dualities
Abstract: In the talk I will discuss the possibility and the obstructions of finding non-supersymmetric dualities for 4d gauge theories. I will review consistency conditions based on Weingarten inequalities, anomalies and large N, and clarify some subtle points and misconceptions about them. Later I will go over some old and new examples of candidates for non-supersymmetric dualities. The will be based on 2208.07842
Title: Statistical Mechanical theory for spatio-temporal evolution of Intra-tumor heterogeneity in cancers: Analysis of Multiregion sequencing data (https://arxiv.org/abs/2202.10595)
Abstract: Variations in characteristics from one region (sub-population) to another are commonly observed in complex systems, such as glasses and a collection of cells. Such variations are manifestations of heterogeneity, whose spatial and temporal behavior is hard to describe theoretically. In the context of cancer, intra-tumor heterogeneity (ITH), characterized by cells with genetic and phenotypic variability that co-exist within a single tumor, is often the cause of ineffective therapy and recurrence of cancer. Next-generation sequencing, obtained by sampling multiple regions of a single tumor (multi-region sequencing, M-Seq), has vividly demonstrated the pervasive nature of ITH, raising the need for a theory that accounts for evolution of tumor heterogeneity. Here, we develop a statistical mechanical theory to quantify ITH, using the Hamming distance, between genetic mutations in distinct regions within a single tumor. An analytic expression for ITH, expressed in terms of cell division probability (α) and mutation probability (p), is validated using cellular-automaton type simulations. Application of the theory successfully captures ITH extracted from M-seq data in patients with exogenous cancers (melanoma and lung). The theory, based on punctuated evolution at the early stages of the tumor followed by neutral evolution, is accurate provided the spatial variation in the tumor mutation burden is not large. We show that there are substantial variations in ITH in distinct regions of a single solid tumor, which supports the notion that distinct subclones could co-exist. The simulations show that there are substantial variations in the sub-populations, with the ITH increasing as the distance between the regions increases. The analytical and simulation framework developed here could be used in the quantitative analyses of the experimental (M-Seq) data. More broadly, our theory is likely to be useful in analyzing dynamic heterogeneity in complex systems such as supercooled liquids.
Bio: I am a postdoctoral fellow in Harvard SEAS (Applied Mathematics) and Dana Farber Cancer Institute (Data Science) beginning Feb 2022. I finished my PhD in Physics (Theoretical Biophysics) from UT Austin (Jan 2022) on “Theoretical and computational studies of growing tissue”. I pursued my undergraduate degree in Physics from the Indian Institute of Technology, Kanpur in India (2015). Boradly, I am interested in developing theoretical models, inspired from many-body statistical physics, for biological processes at different length and time scales.
Title: Extracting the quantum Hall conductance from a single bulk wavefunction from the modular flow
Abstract: One question in the study of topological phases is to identify the topological data from the ground state wavefunction without accessing the Hamiltonian. Since local measurement is not enough, entanglement becomes an indispensable tool. Here, we use modular Hamiltonian (entanglement Hamiltonian) and modular flow to rephrase previous studies on topological entanglement entropy and motivate a natural generalization, which we call the entanglement linear response. We will show how it embraces a previous work by Kim&Shi et al on the chiral central charge, and furthermore, inspires a new formula for the quantum Hall conductance.
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
Title: Holomorphic Twists and Confinement in N=1 SYM
Abstract: Supersymmetric QFT’s are of long-standing interest for their high degree of solvability, phenomenological implications, and rich connections to mathematics. In my talk, I will describe how the holomorphic twist isolates the protected quantities which give SUSY QFTs their potency by restricting to the cohomology of one supercharge. I will briefly introduce infinite dimensional symmetry algebras, generalizing Virasoro and Kac-Moody symmetries, which emerge. Finally, I will explain a potential novel UV manifestation of confinement, dubbed “holomorphic confinement,” in the example of pure SU(N) super Yang-Mills. Based on arXiv:2207.14321 and 2 forthcoming works with Kasia Budzik, Davide Gaiotto, Brian Williams, Jingxiang Wu, and Matthew Yu.
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.
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.
Speaker: Jennifer Cano (Stony Brook and Flatiron Institute)
Title: Engineering topological phases with a superlattice potential
Abstract: We propose an externally imposed superlattice potential as a platform for manipulating topological phases, which has both advantages and disadvantages compared to a moire superlattice. In the first example, we apply the superlattice potential to the 2D surface of a 3D topological insulator. The superlattice potential creates tunable van Hove singularities, which, when combined with strong spin-orbit coupling and Coulomb repulsion give rise to a topological meron lattice spin texture. Thus, the superlattice potential provides a new route to the long sought-after goal of realizing spontaneous magnetic order on the surface of a 3D TI. In the second example, we show that a superlattice potential applied to Bernal-stacked bilayer graphene can generate flat Chern bands, similar to in twisted bilayer graphene, whose bandwidth can be as small as a few meV. The superlattice potential offers flexibility in both lattice size and geometry, making it a promising alternative to achieve designer flat bands without a moire heterostructure.
A Conference in Honor of Elliott H. Lieb on his 90th Birthday
On July 30 – Aug 1, 2022 the Harvard Mathematics Department and the CMSA co-hosted a birthday conference in honor of Elliott Lieb.
This meeting highlights Elliott’s vast contribution to math and physics. Additionally, this meeting features Prof. Lieb’s more recent impact in strong subadditivity of entropy and integrable systems (ice model, Temperley-Lieb algebra etc.).
Venue:
July 30–31, 2022: Hall B, Science Center, 1 Oxford Street, Cambridge, MA, 02138 August 1, 2022: Hall C, Science Center, 1 Oxford Street, Cambridge, MA, 02138
Organizers: Michael Aizenman, Princeton University Joel Lebowitz, Rutgers University Ruedi Seiler, Technische Universität Berlin Herbert Spohn, Technical University of Munich Horng-Tzer Yau, Harvard University Shing-Tung Yau, Harvard University Jakob Yngvason, University of Vienna
SPEAKERS: Rafael Benguria, Pontificia Universidad Catolica de Chile Eric Carlen, Rutgers University Philippe Di Francesco, University of Illinois Hugo Duminil-Copin, IHES László Erdös, Institute of Science and Technology Austria Rupert Frank, Ludwig Maximilian University of Munich Jürg Fröhlich, ETH Zurich Alessandro Giuliani, Università degli Studi Roma Tre Bertrand Halperin, Harvard University Klaus Hepp, Institute for Theoretical Physics, ETH Zurich Sabine Jansen, Ludwig Maximilian University of Munich Mathieu Lewin, Université Paris-Dauphine Bruno Nachtergaele, The University of California, Davis Yoshiko Ogata, University of Tokyo Ron Peled, Tel Aviv University Benjamin Schlein, University of Zurich Robert Seiringer, Institute of Science and Technology Austria Jan Philip Solovej, University of Copenhagen Hal Tasaki, Gakushuin University Simone Warzel, Technical University of Munich Jun Yin, The University of California, Los Angeles
Abstract: The establishment of neural circuitry during early infancy is critical for developing visual, auditory, and motor functions. However, how cortical tissue develops postnatally is largely unknown. By combining T1 relaxation time from quantitative MRI and mean diffusivity (MD) from diffusion MRI, we tracked cortical tissue development in infants across three timepoints (newborn, 3 months, and 6 months). Lower T1 and MD indicate higher microstructural tissue density and more developed cortex. Our data reveal three main findings: First, primary sensory-motor areas (V1: visual, A1: auditory, S1: somatosensory, M1: motor) have lower T1 and MD at birth than higher-level cortical areas. However, all primary areas show significant reductions in T1 and MD in the first six months of life, illustrating profound tissue growth after birth. Second, significant reductions in T1 and MD from newborns to 6-month-olds occur in all visual areas of the ventral and dorsal visual streams. Strikingly, this development was heterogenous across the visual hierarchies: Earlier areas are more developed with denser tissue at birth than higher-order areas, but higher-order areas had faster rates of development. Finally, analysis of transcriptomic gene data that compares gene expression in postnatal vs. prenatal tissue samples showed strong postnatal expression of genes associated with myelination, synaptic signaling, and dendritic processes. Our results indicate that these cellular processes may contribute to profound postnatal tissue growth in sensory cortices observed in our in-vivo measurements. We propose a novel principle of postnatal maturation of sensory systems: development of cortical tissue proceeds in a hierarchical manner, enabling the lower-level areas to develop first to provide scaffolding for higher-order areas, which begin to develop more rapidly following birth to perform complex computations for vision and audition.
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
Location: Room G10, 20 Garden Street, Cambridge, MA 02138.
Organizers: Michael R. Douglas (CMSA/Stony Brook/IAIFI) and Peter Chin (CMSA/BU).
Machine learning has driven many exciting recent scientific advances. It has enabled progress on long-standing challenges such as protein folding, and it has helped mathematicians and mathematical physicists create new conjectures and theorems in knot theory, algebraic geometry, and representation theory.
At this workshop, we will bring together mathematicians, theoretical physicists, and machine learning researchers to review the state of the art in machine learning, discuss how ML results can be used to inspire, test and refine precise conjectures, and identify mathematical questions which may be suitable for this approach.
Speakers:
James Halverson, Northeastern University Dept. of Physics and IAIFI
Fabian Ruehle, Northeastern University Dept. of Physics and Mathematics and IAIFI
Abstract: A main challenge in analyzing single-cell RNA sequencing (scRNA-seq) data is to reduce technical variations yet retain cell heterogeneity. Due to low mRNAs content per cell and molecule losses during the experiment (called ‘dropout’), the gene expression matrix has a substantial amount of zero read counts. Existing imputation methods treat either each cell or each gene as independently and identically distributed, which oversimplifies the gene correlation and cell type structure. We propose a statistical model-based approach, called SIMPLEs (SIngle-cell RNA-seq iMPutation and celL clustErings), which iteratively identifies correlated gene modules and cell clusters and imputes dropouts customized for individual gene module and cell type. Simultaneously, it quantifies the uncertainty of imputation and cell clustering via multiple imputations. In simulations, SIMPLEs performed significantly better than prevailing scRNA-seq imputation methods according to various metrics. By applying SIMPLEs to several real datasets, we discovered gene modules that can further classify subtypes of cells. Our imputations successfully recovered the expression trends of marker genes in stem cell differentiation and can discover putative pathways regulating biological processes.
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.
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.
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.
Title: Diffusive growth sourced by topological defects
Abstract: In this talk, we develop a minimal model of morphogenesis of a surface where the dynamics of the intrinsic geometry is diffusive growth sourced by topological defects. We show that a positive (negative) defect can dynamically generate a cone (hyperbolic cone). We analytically explain features of the growth profile as a function of position and time, and predict that in the presence of a positive defect, a bump forms with height profile h(t) ~ t^(1/2) for early times t. To incorporate the effect of the mean curvature, we exploit the fact that for axisymmetric surfaces, the extrinsic geometry can be deduced entirely by the intrinsic geometry. We find that the resulting stationary geometry, for polar order and small bending modulus, is a deformed football. We apply our framework to various biological systems. In an ex-vivo setting of cultured murine neural progenitor cells, we show that our framework is consistent with the observed cell accumulation at positive defects and depletion at negative defects. In an in-vivo setting, we show that the defect configuration consisting of a bound +1 defect state, which is stabilized by activity, surrounded by two -1/2 defects can create a stationary ring configuration of tentacles, consistent with observations of a basal marine invertebrate Hydra
During the 2021–22 academic year, the CMSA will be hosting a seminar on Combinatorics, Physics and Probability, organized by Matteo Parisi and Michael Simkin. This seminar will take place on Tuesdays at 9:00 am – 10:00 am (Boston time). The meetings will take place virtually on Zoom. To learn how to attend, please fill out this form, or contact the organizers Matteo (mparisi@cmsa.fas.harvard.edu) and Michael (msimkin@cmsa.fas.harvard.edu).
The schedule below will be updated as talks are confirmed.
Spring 2022
Date
Speaker
Title/Abstract
1/25/2022 *note special time 9:00–10:00 AM ET
Jacob Bourjaily (Penn State University, Eberly College of Science
Abstract: Recent years have seen tremendous advances in our understanding of perturbative quantum field theory—fueled largely by discoveries (and eventual explanations and exploitation) of shocking simplicity in the mathematical form of the predictions made for experiment. Among the most important frontiers in this progress is the understanding of loop amplitudes—their mathematical form, underlying geometric structure, and how best to manifest the physical properties of finite observables in general quantum field theories. This work is motivated in part by the desire to simplify the difficult work of doing Feynman integrals. I review some of the examples of this progress, and describe some ongoing efforts to recast perturbation theory in terms that expose as much simplicity (and as much physics) as possible.
Abstract: The (tree) amplituhedron was introduced in 2013 by Arkani-Hamed and Trnka in their study of N=4 SYM scattering amplitudes. A central conjecture in the field was to prove that the m=4 amplituhedron is triangulated by the images of certain positroid cells, called the BCFW cells. In this talk I will describe a resolution of this conjecture. The seminar is based on a recent joint work with Chaim Even-Zohar and Tsviqa Lakrec.
Abstract: I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.
2/15/2022
Igor Balla, Hebrew University of Jerusalem
Title: Equiangular lines and regular graphs
Abstract: In 1973, Lemmens and Seidel asked to determine N_alpha(r), the maximum number of equiangular lines in R^r with common angle arccos(alpha). Recently, this problem has been almost completely settled when r is exponentially large relative to 1/alpha, with the approach both relying on Ramsey’s theorem, as well as being limited by it. In this talk, we will show how orthogonal projections of matrices with respect to the Frobenius inner product can be used to overcome this limitation, thereby obtaining significantly improved upper bounds on N_alpha(r) when r is polynomial in 1/alpha. In particular, our results imply that N_alpha(r) = Theta(r) for alpha >= Omega(1 / r^1/5).
Our projection method generalizes to complex equiangular lines in C^r, which may be of independent interest in quantum theory. Applying this method also allows us to obtain the first universal bound on the maximum number of complex equiangular lines in C^r with common Hermitian angle arccos(alpha), an extension of the Alon-Boppana theorem to dense regular graphs, which is tight for strongly regular graphs corresponding to r(r+1)/2 equiangular lines in R^r, an improvement to Welch’s bound in coding theory.
Fall 2021
Date
Speaker
Title/Abstract
9/21/2021
Nima Arkani-Hamed IAS (Institute for Advanced Study), School of Natural Sciences
Title: Surfacehedra and the Binary Positive Geometry of Particle and “String” Amplitudes
9/28/2021
Melissa Sherman-Bennett University of Michigan, Department of Mathematics
Title: The hypersimplex and the m=2 amplituhedron
Abstract: I’ll discuss a curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). Along the way, we prove the sign-flip description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers (inspired by an analogous hypersimplex decomposition).
10/5/2021
Daniel Cizma, Hebrew University
Title: Geodesic Geometry on Graphs
Abstract: In a graph G = (V, E) we consider a system of paths S so that for every two vertices u,v in V there is a unique uv path in S connecting them. The path system is said to be consistent if it is closed under taking subpaths, i.e. if P is a path in S then any subpath of P is also in S. Every positive weight function w: E–>R^+ gives rise to a consistent path system in G by taking the paths in S to be geodesics w.r.t. w. In this case, we say w induces S. We say a graph G is metrizable if every consistent path system in G is induced by some such w.
We’ll discuss the concept of graph metrizability, and, in particular, we’ll see that while metrizability is a rare property, there exists infinitely many 2-connected metrizable graphs.
Joint work with Nati Linial.
10/12/2021
Lisa Sauermann, MIT
Title: On counting algebraically defined graphs
Abstract: For many classes of graphs that arise naturally in discrete geometry (for example intersection graphs of segments or disks in the plane), the edges of these graphs can be defined algebraically using the signs of a finite list of fixed polynomials. We investigate the number of n-vertex graphs in such an algebraically defined class of graphs. Warren’s theorem (a variant of a theorem of Milnor and Thom) implies upper bounds for the number of n-vertex graphs in such graph classes, but all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving a lower bound for this number (under some reasonable assumptions on the fixed list of polynomials), and this lower bound essentially matches the upper bound from Warren’s theorem.
10/19/2021
Pavel Galashin UCLA, Department of Mathematics
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.
10/26/2021
Candida Bowtell, University of Oxford
Title: The n-queens problem
Abstract: The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n).
In this talk we’ll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost’ configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.
This is joint work with Peter Keevash.
11/9/2021
Steven Karp Universite du Quebec a Montreal, LaCIM (Laboratoire de combinatoire et d’informatique mathématique)
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: From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees. Joint work with Omer Angel and Gourab Ray.
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).
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: 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.
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.
12/21/2021
01/25/2022
Jacob Bourjaily Penn State University, Department of Physics
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: We discuss the algebraic geometry of maximum likelihood estimation from the perspective of scattering amplitudes in particle physics. A guiding examples the moduli space of n-pointed rational curves. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions to rational function equations. Their number is an Euler characteristic. Soft limit degenerations are combined with certified numerical methods for concrete computations.
**This talk will be hybrid. Talk will be held at CMSA (20 Garden St) Room G10.
All non-Harvard affiliated visitors to the CMSA building will need to complete this covid form prior to arrival.
Abstract: I will first review the construction of the moduli space of tropical curves (or metric graphs), and its relation to graph complexes. The graph Laplacian may be interpreted as a tropical version of the classical Torelli map and its determinant is the Kirchhoff graph polynomial (also called 1st Symanzik), which is one of the two key components in Feynman integrals in high energy physics.The other component is the so-called 2nd Symanzik polynomial, which is defined for graphs with external half edges and involves particle masses (edge colourings). I will explain how this too may be interpreted as the determinant of a generalised graph Laplacian, and how it leads to a volumetric interpretation of a certain class of Feynman integrals.
Abstract: The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics, probability, computer science, and chemistry. The algorithm builds a maximal independent set by inspecting the graph’s vertices one at a time according to a random order, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge.
In this talk, I will present a simple yet general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a valuable notion of local convergence. I will demonstrate the applicability of this framework by giving short and straightforward proofs for results on previously studied families of graphs, such as paths and various random graphs, and by providing new results for other models such as random trees.
If time allows, I will discuss a more delicate (and combinatorial) result, according to which, in expectation, the cardinality of a random greedy independent set in the path is no larger than that in any other tree of the same order.
The talk is based on joint work with Michael Krivelevich, Tamás Mészáros and Clara Shikhelman.
Abstract: Eugene Wachspress introduced polypols as real bounded semialgebraic sets in the plane that generalize polygons. He aimed to generalize barycentric coordinates from triangles to arbitrary polygons and further to polypols. For this, he defined the adjoint curve of a rational polypol. In the study of scattering amplitudes in physics, positive geometries are real semialgebraic sets together with a rational canonical form. We combine these two worlds by providing an explicit formula for the canonical form of a rational polypol in terms of defining equations of the adjoint curve and the facets of the polypol. For the special case of polygons, we show that the adjoint curve is hyperbolic and provide an explicit description of its nested ovals. Finally, we discuss the map that associates the adjoint curve to a given rational polypol, in particular the cases where this map is finite. For instance, using monodromy we find that a general quartic curve is the adjoint of 864 heptagons.
This talk is based on joint work with R. Piene, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn, M.-S. Sorea, and S. Telen.
Abstract: There are several properties of closed geodesics which are proven using its Hamiltonian formulation, which has no analogue for minimal surfaces. I will talk about some recent progress in proving some of these properties for minimal surfaces.
Abstract: Recent years have seen tremendous advances in our understanding of perturbative quantum field theory—fueled largely by discoveries (and eventual explanations and exploitation) of shocking simplicity in the mathematical form of the predictions made for experiment. Among the most important frontiers in this progress is the understanding of loop amplitudes—their mathematical form, underlying geometric structure, and how best to manifest the physical properties of finite observables in general quantum field theories. This work is motivated in part by the desire to simplify the difficult work of doing Feynman integrals. I review some of the examples of this progress, and describe some ongoing efforts to recast perturbation theory in terms that expose as much simplicity (and as much physics) as possible.
Abstract: I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.
Abstract: The (tree) amplituhedron was introduced in 2013 by Arkani-Hamed and Trnka in their study of N=4 SYM scattering amplitudes. A central conjecture in the field was to prove that the m=4 amplituhedron is triangulated by the images of certain positroid cells, called the BCFW cells. In this talk I will describe a resolution of this conjecture. The seminar is based on a recent joint work with Chaim Even-Zohar and Tsviqa Lakrec.
Abstract: Encryption is the backbone of cybersecurity. While encryption can secure data both in transit and at rest, in the new era of ubiquitous computing, modern cryptography also aims to protect data during computation. Secure multi-party computation (MPC) is a powerful technology to tackle this problem, which enables distrustful parties to jointly perform computation over their private data without revealing their data to each other. Although it is theoretically feasible and provably secure, the adoption of MPC in real industry is still very much limited as of today, the biggest obstacle of which boils down to its efficiency.
My research goal is to bridge the gap between the theoretical feasibility and practical efficiency of MPC. Towards this goal, my research spans both theoretical and applied cryptography. In theory, I develop new techniques for achieving general MPC with the optimal complexity, bringing theory closer to practice. In practice, I design tailored MPC to achieve the best concrete efficiency for specific real-world applications. In this talk, I will discuss the challenges in both directions and how to overcome these challenges using cryptographic approaches. I will also show strong connections between theory and practice.
Biography: Peihan Miao is an assistant professor of computer science at the University of Illinois Chicago (UIC). Before coming to UIC, she received her Ph.D. from the University of California, Berkeley in 2019 and had brief stints at Google, Facebook, Microsoft Research, and Visa Research. Her research interests lie broadly in cryptography, theory, and security, with a focus on secure multi-party computation — especially in incorporating her industry experiences into academic research.
On April 27–29, 2022, the CMSA hosted a workshop on Nonlinear Algebra and Combinatorics.
Organizers: Bernd Sturmfels (MPI Leipzig) and Lauren Williams (Harvard).
In recent years, ideas from integrable systems and scattering amplitudes have led to advances in nonlinear algebra and combinatorics. In this short workshop, aimed at younger participants in the field, we will explore some of the interactions between the above topics.
Abstract: Matroid theory provides a combinatorial model for linearity, but it plays useful roles beyond linearity. In the classical setup, a linear subspace V of an n-dimensional vector space gives rise to a matroid M(V) on {1,…,n}. However, the matroid M(V) also knows about some nonlinear geometric spaces related to V. Conversely, those nonlinear spaces teach us things we didn’t know about matroids. My talk will discuss some examples.
10:30 am–11:00 am
COFFEE BREAK
11:00 am–11:45 am
Chris Eur
Title: Tautological classes of matroids
Abstract: Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call “tautological bundles (classes)” of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.
11:45 am–2:00 pm
LUNCH BREAK
2:00 pm–2:45 pm
Nick Early
Title: Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes
Abstract: The associahedron is known to encapsulate physical properties such as the notion of tree-level factorization for one of the simplest Quantum Field Theories, the biadjoint scalar, which has only cubic interactions. I will discuss novel instances of the associahedron and the biadjoint scalar in a class of generalized amplitudes, discovered by Cachazo, Early, Guevara and Mizera, by taking certain limits thereof. This connection leads to a simple proof of a new realization of the associahedron involving a Minkowski sum of certain positroid polytopes in the second hypersimplex.
2:45 pm–3:30 pm
Anna Seigal
Title: Invariant theory for maximum likelihood estimation
Abstract: I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.
3:30 pm–4:00 pm
COFFEE BREAK
4:00 pm–4:45 pm
Matteo Parisi
Title: Amplituhedra, Scattering Amplitudes, and Triangulations
Abstract: In this talk I will discuss about Amplituhedra – generalizations of polytopes inside the Grassmannian – introduced by physicists to encode interactions of elementary particles in certain Quantum Field Theories. In particular, I will explain how the problem of finding triangulations of Amplituhedra is connected to computing scattering amplitudes of N=4 super Yang-Mills theory. Triangulations of polygons are encoded in the associahedron, studied by Stasheff in the sixties; in the case of polytopes, triangulations are captured by secondary polytopes, constructed by Gelfand et al. in the nineties. Whereas a “secondary” geometry describing triangulations of Amplituhedra is still not known, and we pave the way for such studies. I will discuss how the combinatorics of triangulations interplays with T-duality from String Theory, in connection with the Momentum Amplituhedron. A generalization of T-duality led us to discover a striking duality between Amplituhedra of “m=2” type and a seemingly unrelated object – the Hypersimplex. The latter is a polytope which appears in many contexts, from matroid theory to tropical geometry. Based on joint works with Lauren Williams, Melissa Sherman-Bennett, Tomasz Lukowski.
4:45 pm–5:30 pm
Melissa Sherman-Bennett
Title: The hypersimplex and the m=2 amplituhedron
Abstract: In this talk, I’ll continue where Matteo left off. I’ll give some more details about the curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). The hypersimplex decompositions are closely related to matroidal subdivisions. Along the way, we prove a nice description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers, inspired by an analogous triangulation of the hypersimplex into Eulerian-number-many simplices.
Thursday, April 28, 2022
9:30 am–10:30 am
Claudia Fevola
Title: Nonlinear Algebra meets Feynman integrals
Abstract: Feynman integrals play a central role in particle physics in the theory of scattering amplitudes. They form a finite-dimensional vector space and the elements of a basis are named “master integrals” in the physics literature. The number of master integrals has been interpreted in different ways: it equals the dimension of a twisted de Rham cohomology group, the Euler characteristic of a very affine variety, and the holonomic rank of a D-module. In this talk, we are interested in a more general family of integrals that contains Feynman integrals as a special case. We explore this setting using tools coming from nonlinear algebra. This is an ongoing project with Daniele Agostini, Anna-Laura Sattelberger, and Simon Telen.
10:30 am–11:00 am
COFFEE BREAK
11:00 am–11:45 am
Simon Telen
Title: Landau discriminants
Abstract: The Landau discriminant is a projective variety containing kinematic parameters for which a Feynman integral can have singularities. We present a definition and geometric properties. We discuss how to compute Landau discriminants using symbolic and numerical methods. Our methods can be used, for instance, to compute the Landau discriminant of the pentabox diagram, which is a degree 12 hypersurface in 6-space. This is joint work with Sebastian Mizera.
11:45 am–2:00 pm
LUNCH BREAK
2:00 pm–2:45 pm
Christian Gaetz
Title: 1-skeleton posets of Bruhat interval polytopes
Abstract: Bruhat interval polytopes are a well-studied class of generalized permutohedra which arise as moment map images of various toric varieties and totally positive spaces in the flag variety. I will describe work in progress in which I study the 1-skeleta of these polytopes, viewed as posets interpolating between weak order and Bruhat order. In many cases these posets are lattices and the polytopes, despite not being simple, have interesting h-vectors. In a special case, work of Williams shows that Bruhat interval polytopes are isomorphic to bridge polytopes, so that chains in the 1-skeleton poset correspond to BCFW-bridge decompositions of plabic graphs.
2:45 pm–3:30 pm
Madeleine Brandt
Title: Top Weight Cohomology of $A_g$
Abstract: I will discuss a recent project in computing the top weight cohomology of the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. This piece of the cohomology is controlled by the combinatorics of the boundary strata of a compactification of $A_g$. Thus, it can be computed combinatorially. This is joint work with Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.
3:30 pm–4:00 pm
COFFEE BREAK
4:00 pm–5:00 pm
Emma Previato
Title: Sigma function on curves with non-symmetric semigroup
Abstract: We start with an overview of the correspondence between spectral curves and commutative rings of differential operators, integrable hierarchies of non-linear PDEs and Jacobian vector fields. The coefficients of the operators can be written explicitly in terms of the Kleinian sigma function: Weierstrass’ sigma function was generalized to genus greater than one by Klein, and is a ubiquitous tool in integrability. The most accessible case is the sigma function of telescopic curves. In joint work with J. Komeda and S. Matsutani, we construct a curve with non-symmetric Weierstrass semigroup (equivalently, Young tableau), consequently non-telescopic, and its sigma function. We conclude with possible applications to commutative rings of differential operators.
Abstract: It is well-known that soliton solutions of the KdV hierarchy are obtained by singular limits of hyper-elliptic curves. However, there is no general results for soliton solutions of the KP hierarchy, KP solitons. In this talk, I will show that some of the KP solitons are related to the singular space curves associated with certain class of numerical semigroups.
10:00 am–10:30 am
COFFEE BREAK
10:30 am–11:15 am
Yelena Mandelshtam
Title: Curves, degenerations, and Hirota varieties
Abstract: The Kadomtsev-Petviashvili (KP) equation is a differential equation whose study yields interesting connections between integrable systems and algebraic geometry. In this talk I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes all KP solutions arising from such a sum. I will then discuss a special case, studying the Hirota variety of a rational nodal curve. Of particular interest is an irreducible subvariety that is the image of a parameterization map. Proving that this is a component of the Hirota variety entails solving a weak Schottky problem for rational nodal curves. This talk is based on joint work with Daniele Agostini, Claudia Fevola, and Bernd Sturmfels.
11:15 am–12:00 pm
Charles Wang
Title: Differential Algebra of Commuting Operators
Abstract: In this talk, we will give an overview of the problem of finding the centralizer of a fixed differential operator in a ring of differential operators, along with connections to integrable hierarchies and soliton solutions to e.g. the KdV or KP equations. Given these interesting connections, it is important to be able to compute centralizers of differential operators, and we discuss how to use techniques from differential algebra to approach this question, as well as how having these computational tools can help in understanding the structure of soliton solutions to these equations.
12:00 pm–2:00 pm
LUNCH BREAK
2:00 pm–3:00 pm
Sebastian Mizera
Title: Feynman Polytopes
Abstract: I will give an introduction to a class of polytopes that recently emerged in the study of scattering amplitudes in quantum field theory.
3:00 pm–3:30 pm
COFFEE BREAK
3:30 pm–4:30 pm
Nima Arkani-Hamed
Title: Spacetime, Quantum Mechanics and Combinatorial Geometries at Infinity
Abstract: Introducing internal degrees of freedom in the description of crystalline insulators has led to a myriad of theoretical and experimental advances. Of particular interest are the effects of periodic perturbations, either in time or space, as they considerably enrich the variety of electronic responses. Here, we present a semiclassical approach to transport and accumulation of general spinor degrees of freedom in adiabatically driven, weakly inhomogeneous crystals of dimensions one, two and three under external electromagnetic fields. Our approach shows that spatio-temporal modulations of the system induce a spinor current and density that is related to geometrical and topological objects — the spinor-Chern fluxes and numbers — defined over the higher-dimensional phase-space of the system, i.e., its combined momentum-position-time coordinates.
Bio: Ioannis Petrides is a postdoctoral fellow at the School of Engineering and Applied Sciences at Harvard University. He received his Ph.D. from the Institute for Theoretical Physics at ETH Zurich. His research focuses on the topological and geometrical aspects of condensed matter systems.
Title: The phenotype of the last universal common ancestor and the evolution of complexity
Abstract: A fundamental concept in evolutionary theory is the last universal common ancestor (LUCA) from which all living organisms originated. While some authors have suggested a relatively complex LUCA it is still widely assumed that LUCA must have been a very simple cell and that life has subsequently increased in complexity through time. However, while current thought does tend towards a general increase in complexity through time in Eukaryotes, there is increasing evidence that bacteria and archaea have undergone considerable genome reduction during their evolution. This raises the surprising possibility that LUCA, as the ancestor of bacteria and archaea may have been a considerably complex cell. While hypotheses regarding the phenotype of LUCA do exist, all are founded on gene presence/absence. Yet, despite recent attempts to link genes and phenotypic traits in prokaryotes, it is still inherently difficult to predict phenotype based on the presence or absence of genes alone. In response to this, we used Bayesian phylogenetic comparative methods to predict ancestral traits. Testing for robustness to horizontal gene transfer (HGT) we inferred the phenotypic traits of LUCA using two robust published phylogenetic trees and a dataset of 3,128 bacterial and archaeal species.
Our results depict LUCA as a far more complex cell than has previously been proposed, challenging the evolutionary model of increased complexity through time in prokaryotes. Given current estimates for the emergence of LUCA we suggest that early life very rapidly evolved cellular complexity.
Abstract: The statistical analysis of genomic data has incubated many innovations for computational method development. This talk will discuss some simple algorithms that may be useful in analyzing such data. Examples include algorithms for efficient resampling-based hypothesis testing, minimizing the sum of truncated convex functions, and fitting equality-constrained lasso problems. These algorithms have the potential to be used in other applications beyond statistical genomics.
Bio: Hui Jiang is an Associate Professor in the Department of Biostatistics at the University of Michigan. He received his Ph.D. in Computational and Mathematical Engineering from Stanford University. Before joining the University of Michigan, he was a postdoc in the Department of Statistics and Stanford Genome Technology Center at Stanford University. He is interested in developing statistical and computational methods for analyzing large-scale biological data generated using modern high-throughput technologies.
On June 21–24, 2022, the Harvard Black Hole Initiative and the CMSA hosted the Joint BHI/CMSA Conference on Flat Holography (and related topics).
The recent discovery of infinitely-many soft symmetries for all quantum theories of gravity in asymptotically flat space has provided a promising starting point for a bottom-up construction of a holographic dual for the real world. Recent developments have brought together previously disparate studies of soft theorems, asymptotic symmetries, twistor theory, asymptotically flat black holes and their microscopic duals, self-dual gravity, and celestial scattering amplitudes, and link directly to AdS/CFT.
The conference was held in room G10 of the CMSA, 20 Garden Street, Cambridge, MA.
Organizers:
Daniel Kapec, CMSA
Andrew Strominger, BHI
Shing-Tung Yau, Harvard & Tsinghua
Confirmed Speakers:
Nima Arkani-Hamed, IAS
Shamik Banerjee, Bhubaneswar, Inst. Phys.
Miguel Campiglia, Republica U., Montevido
Geoffrey Compere, Brussels
Laura Donnay, Vienna
Netta Engelhardt, MIT
Laurent Freidel, Perimeter
Alex Lupsasca, Princeton
Juan Maldacena, IAS
Lionel Mason, Oxford
Natalie Paquette, U. Washington
Sabrina Pasterski, Princeton/Perimeter
Andrea Puhm, Ecole Polytechnique
Ana-Maria Raclariu, Perimeter
Marcus Spradlin, Brown
Tomasz Taylor, Northeastern
Herman Verlinde, Princeton
Anastasia Volovich, Brown
Bin Zhu, Northeastern
Short talks by: Gonçalo Araujo-Regado (Cambridge), Adam Ball (Harvard), Eduardo Casali (Harvard), Jordan Cotler (Harvard), Erin Crawley (Harvard), Stéphane Detournay (Brussels), Alfredo Guevara (Harvard), Temple He (UC Davis), Elizabeth Himwich (Harvard), Yangrui Hu (Brown), Daniel Kapec (Harvard), Rifath Khan (Cambridge), Albert Law (Harvard), Luke Lippstreu (Brown), Noah Miller (Harvard), Sruthi Narayanan (Harvard), Lecheng Ren (Brown), Francisco Rojas (UAI), Romain Ruzziconi (Vienna), Andrew Strominger (Harvard), Adam Tropper (Harvard), Tianli Wang (Harvard), Walker Melton (Harvard)
Schedule
Monday, June 20, 2022
Arrival
7:00–9:00 pm
Welcome Reception at Andy’s residence
Tuesday, June 21, 2022
9:00–9:30 am
Breakfast
light breakfast provided
Morning Session
Chair: Dan Kapec
9:30–10:00 am
Herman Verlinde
Title: Comments on Celestial Dynamics
10:00–10:30 am
Juan Maldacena
Title: What happens when you spend too much time looking at supersymmetric black holes?
10:30–11:00
Coffee break
11:00–11:30 am
Miguel Campiglia
Title: Asymptotic symmetries and loop corrections to soft theorems
11:30–12:00 pm
Geoffrey Compere
Title: Metric reconstruction from $Lw_{1+\infty}$ multipoles
Abstract: The most general vacuum solution to Einstein’s field equations with no incoming radiation can be constructed perturbatively from two infinite sets of canonical multipole moments, which are found to be exchanged under gravitational electric-magnetic duality at the non-linear level. We demonstrate that in non-radiative regions such spacetimes are completely determined by a set of conserved celestial charges, which uniquely label transitions among non-radiative regions caused by radiative processes. The algebra of the conserved celestial charges is derived from the real $Lw_{1+\infty}$ algebra. The celestial charges are expressed in terms of multipole moments, which allows to holographically reconstruct the metric in de Donder, Newman-Unti or Bondi gauge outside of sources.
12:00–2:00 pm
Lunch break
Afternoon Session
Chair: Eduardo Casali
2:00–2:30 pm
Natalie Paquette
Title: New thoughts on old gauge amplitudes
2:30–3:00 pm
Lionel Mason
Title: An open sigma model for celestial gravity
Abstract: A global twistor construction for conformally self-dual split signature metrics on $S2\times S2$ was developed 15 years ago by Claude LeBrun and the speaker. This encodes the conformal metric into the location of a finite deformation of the real twistor space inside the flat complex twistor space, $\mathbb{CP}3$. This talk adapts the construction to construct global SD Einstein metrics from conformal boundary data and perturbations around the self-dual sector. The construction entails determining a family of holomorphic discs in $\mathbb{CP}3$ whose boundaries lie on the deformed real slice and the (chiral) sigma model controls these discs in the Einstein case and provides amplitude formulae.
3:00–3:30 pm
Coffee break
3:30–4:30 pm
Short Talks
Daniel Kapec: Soft Scalars and the Geometry of the Space of Celestial CFTs
Albert Law: Soft Scalars and the Geometry of the Space of Celestial CFTs
Sruthi Narayanan: Soft Scalars and the Geometry of the Space of Celestial CFTs
Stéphane Detournay: Non-conformal symmetries and near-extremal black holes
Francisco Rojas: Celestial string amplitudes beyond tree level
Temple He: An effective description of energy transport from holography
4:30–5:00 pm
Nima Arkani-Hamed
(Dual) surfacehedra and flow particles know about strings
Wednesday, June 22, 2022
9:00–9:30 am
Breakfast
light breakfast provided
Morning Session
Chair: Alfredo Guevara
9:30–10:00 am
Laurent Freidel
Title: Higher spin symmetry in gravity
Abstract: In this talk, I will review how the gravitational conservation laws at infinity reveal a tower of symmetry charges in an asymptotically flat spacetime. I will show how the conservation laws, at spacelike infinity, give a tower of soft theorems that connect to the ones revealed by celestial holography. I’ll present the expression for the symmetry charges in the radiative phase space, which opens the way to reveal the structure of the algebra beyond the positive helicity sector. Then, if time permits I’ll browse through many questions that these results raise: such as the nature of the spacetime symmetry these charges represent, the nature of the relationship with multipole moments, and the insights their presence provides for quantum gravity.
10:00–10:30 am
Ana-Maria Raclariu
Title: Eikonal approximation in celestial CFT
10:30–11:00 am
Coffee break
11:00–11:30 am
Anastasia Volovich
Title: Effective Field Theories with Celestial Duals
11:30–12:00 pm
Marcus Spradlin
Title: Loop level gluon OPE’s in celestial holography
12:00–2:00 pm
Lunch break
Afternoon Session
Chair: Chiara Toldo
2:00–2:30 pm
Netta Engelhardt
Title: Wormholes from entanglement: true or false?
2:30–3:00 pm
Short Talks
Luke Lippstreu: Loop corrections to the OPE of celestial gluons
Yangrui Hu: Light transforms of celestial amplitudes
Lecheng Ren: All-order OPE expansion of celestial gluon and graviton primaries from MHV amplitudes
3:00–3:30 pm
Coffee break
3:30–4:30 pm
Short Talks
Noah Miller: C Metric Thermodynamics
Erin Crawley: Kleinian black holes
Rifath Khan: Cauchy Slice Holography: A New AdS/CFT Dictionary
Gonçalo Araujo-Regado: Cauchy Slice Holography: A New AdS/CFT Dictionary
Tianli Wang: Soft Theorem in the BFSS Matrix Model
Adam Tropper: Soft Theorem in the BFSS Matrix Model
Title: Celestial wave scattering on Kerr-Schild backgrounds
10:30–11:00 am
Coffee break
11:00–11:30 am
Sabrina Pasterski
Title: Mining Celestial Symmetries
Abstract: The aim of this talk is to delve into the common thread that ties together recent work with H. Verlinde, L. Donnay, A. Puhm, and S. Banerjee exploring, explaining, and exploiting the symmetries encoded in the conformally soft sector.
Come prepared to debate the central charge, loop corrections, contour prescriptions, and orders of limits!
11:30–12:00 pm
Shamik Banerjee
Title: Virasoro and other symmetries in CCFT
Abstract: In this talk I will briefly describe my ongoing work with Sabrina Pasterski. In this work we revisit the standard construction of the celestial stress tensor as a shadow of the subleading conformally soft graviton. In its original formulation, we find that there is an obstruction to reproducing the expected $TT$ OPE in the double soft limit. This obstruction is related to the existence of the $SL_2$ current algebra symmetry of the CCFT. We propose a modification to the definition of the stress tensor which circumvents this obstruction and also discuss its implications for the existence of other current algebra (w_{1+\infty}) symmetries in CCFT.
12:00–2:00 pm
Lunch break
Afternoon Session
Chair: Albert Law
2:00–2:30 pm
Tomasz Taylor
Title: Celestial Yang-Mills amplitudes and D=4 conformal blocks
2:30–3:00 pm
Bin Zhu
Title: Single-valued correlators and Banerjee-Ghosh equations
Abstract: Low-point celestial amplitudes are plagued with singularities resulting from spacetime translation. We consider a marginal deformation of the celestial CFT which is realized by coupling Yang-Mills theory to a background dilaton field, with the (complex) dilaton source localized on the celestial sphere. This picture emerges from the physical interpretation of the solutions of the system of differential equations discovered by Banerjee and Ghosh. We show that the solutions can be written as Mellin transforms of the amplitudes evaluated in such a dilaton background. The resultant three-gluon and four-gluon amplitudes are single-valued functions of celestial coordinates enjoying crossing symmetry and all other properties expected from standard CFT correlators.
3:00–3:30 pm
Coffee break
3:30–4:00 pm
Alex Lupsasca
Title: Holography of the Photon Ring
4:00–5:30 pm
Short Talks
Elizabeth Himwich: Celestial OPEs and w(1+infinity) symmetry of massless and massive amplitudes
Adam Ball: Perturbatively exact $w_{1+\infty}$ asymptotic symmetry of quantum self-dual gravity
Romain Ruzziconi: A Carrollian Perspective on Celestial Holography
Jordan Cotler: Soft Gravitons in 3D
Alfredo Guevara: Comments on w_1+\inf
Andrew Strominger: Top-down celestial holograms
Eduardo Casali: Celestial amplitudes as AdS-Witten diagrams
Speaker: Jianfeng Lu, Duke UniversityTitle: Surface hopping algorithms for non-adiabatic quantum systems
Abstract: Surface hopping algorithm is widely used in chemistry for mixed quantum-classical dynamics. In this talk, we will discuss some of our recent works in mathematical understanding and algorithm development for surface hopping methods. These methods are based on stochastic approximations of semiclassical path-integral representation to the solution of multi-level Schrodinger equations; such methodology also extends to other high-dimensional transport systems.
On June 6-8, 2022, the CMSA hosted the 3rd annual Symposium on Foundations of Responsible Computing (FORC).
The Symposium on Foundations of Responsible Computing (FORC) is a forum for mathematical research in computation and society writ large. The Symposium aims to catalyze the formation of a community supportive of the application of theoretical computer science, statistics, economics and other relevant analytical fields to problems of pressing and anticipated societal concern.
Title: From Theory to Impact: Why Better Data Systems are Necessary for Criminal Legal Reform
Abstract: This talk will dive into the messy, archaic, and siloed world of local criminal justice data in America. We will start with a 30,000 foot discussion about the current state of criminal legal data systems, then transition to the challenges of this broken paradigm, and conclude with a call to measure new things – and to measure them better! This talk will leave you with an understanding of criminal justice data infrastructure and transparency in the US, and will discuss how expensive case management software and other technology are built on outdated normative values which impede efforts to reform the system. The result is an infuriating paradox: an abundance of tech products built without theoretical grounding, in a space rich with research and evidence.
10:15 am–10:45 am
Coffee Break
10:45 am–12:15 pm
Paper Session 1
Session Chair: Ruth Urner
Georgy Noarov, University of Pennsylvania
Title: Online Minimax Multiobjective Optimization
Abstract: We introduce a simple but general online learning framework in which a learner plays against an adversary in a vector-valued game that changes every round. The learner’s objective is to minimize the maximum cumulative loss over all coordinates. We give a simple algorithm that lets the learner do almost as well as if she knew the adversary’s actions in advance. We demonstrate the power of our framework by using it to (re)derive optimal bounds and efficient algorithms across a variety of domains, ranging from multicalibration to a large set of no-regret algorithms, to a variant of Blackwell’s approachability theorem for polytopes with fast convergence rates. As a new application, we show how to “(multi)calibeat” an arbitrary collection of forecasters — achieving an exponentially improved dependence on the number of models we are competing against, compared to prior work.
Matthew Eichhorn, Cornell University
Title: Mind your Ps and Qs: Allocation with Priorities and Quotas
Abstract: In many settings, such as university admissions, the rationing of medical supplies, and the assignment of public housing, decision-makers use normative criteria (ethical, financial, legal, etc.) to justify who gets an allocation. These criteria can often be translated into quotas for the number of units available to particular demographics and priorities over agents who qualify in each demographic. Each agent may qualify in multiple categories at different priority levels, so many allocations may conform to a given set of quotas and priorities. Which of these allocations should be chosen? In this talk, I’ll formalize this reserve allocation problem and motivate Pareto efficiency as a natural desideratum. I’ll present an algorithm to locate efficient allocations that conform to the quota and priority constraints. This algorithm relies on beautiful techniques from integer and linear programming, and it is both faster and more straightforward than existing techniques in this space. Moreover, its clean formulation allows for further refinement, such as the secondary optimization of some heuristics for fairness.
Haewon Jeong, Harvard University
Title: Fairness without Imputation: A Decision Tree Approach for Fair Prediction with Missing Values
Abstract: We investigate the fairness concerns of training a machine learning model using data with missing values. Even though there are a number of fairness intervention methods in the literature, most of them require a complete training set as input. In practice, data can have missing values, and data missing patterns can depend on group attributes (e.g. gender or race). Simply applying off-the-shelf fair learning algorithms to an imputed dataset may lead to an unfair model. In this paper, we first theoretically analyze different sources of discrimination risks when training with an imputed dataset. Then, we propose an integrated approach based on decision trees that does not require a separate process of imputation and learning. Instead, we train a tree with missing incorporated as attribute (MIA), which does not require explicit imputation, and we optimize a fairness-regularized objective function. We demonstrate that our approach outperforms existing fairness intervention methods applied to an imputed dataset, through several experiments on real-world datasets.
Emily Diana, University of Pennsylvania
Title: Multiaccurate Proxies for Downstream Fairness
Abstract: We study the problem of training a model that must obey demographic fairness conditions when the sensitive features are not available at training time — in other words, how can we train a model to be fair by race when we don’t have data about race? We adopt a fairness pipeline perspective, in which an “upstream” learner that does have access to the sensitive features will learn a proxy model for these features from the other attributes. The goal of the proxy is to allow a general “downstream” learner — with minimal assumptions on their prediction task — to be able to use the proxy to train a model that is fair with respect to the true sensitive features. We show that obeying multiaccuracy constraints with respect to the downstream model class suffices for this purpose, provide sample- and oracle efficient-algorithms and generalization bounds for learning such proxies, and conduct an experimental evaluation. In general, multiaccuracy is much easier to satisfy than classification accuracy, and can be satisfied even when the sensitive features are hard to predict.
12:15 pm–1:45 pm
Lunch Break
1:45–3:15 pm
Paper Session 2
Session Chair: Guy Rothblum
Elbert Du, Harvard University
Title: Improved Generalization Guarantees in Restricted Data Models
Abstract: Differential privacy is known to protect against threats to validity incurred due to adaptive, or exploratory, data analysis — even when the analyst adversarially searches for a statistical estimate that diverges from the true value of the quantity of interest on the underlying population. The cost of this protection is the accuracy loss incurred by differential privacy. In this work, inspired by standard models in the genomics literature, we consider data models in which individuals are represented by a sequence of attributes with the property that where distant attributes are only weakly correlated. We show that, under this assumption, it is possible to “re-use” privacy budget on different portions of the data, significantly improving accuracy without increasing the risk of overfitting.
Ruth Urner, York University
Title: Robustness Should not be at Odds with Accuracy
Abstract: The phenomenon of adversarial examples in deep learning models has caused substantial concern over their reliability and trustworthiness: in many instances an imperceptible perturbation can falsely flip a neural network’s prediction. Applied research in this area has mostly focused on developing novel adversarial attack strategies or building better defenses against such. It has repeatedly been pointed out that adversarial robustness may be in conflict with requirements for high accuracy. In this work, we take a more principled look at modeling the phenomenon of adversarial examples. We argue that deciding whether a model’s label change under a small perturbation is justified, should be done in compliance with the underlying data-generating process. Through a series of formal constructions, systematically analyzing the the relation between standard Bayes classifiers and robust-Bayes classifiers, we make the case for adversarial robustness as a locally adaptive measure. We propose a novel way defining such a locally adaptive robust loss, show that it has a natural empirical counterpart, and develop resulting algorithmic guidance in form of data-informed adaptive robustness radius. We prove that our adaptive robust data-augmentation maintains consistency of 1-nearest neighbor classification under deterministic labels and thereby argue that robustness should not be at odds with accuracy.
Sushant Agarwal, University of Waterloo
Title: Towards the Unification and Robustness of Perturbation and Gradient Based Explanations
Abstract: As machine learning black boxes are increasingly being deployed in critical domains such as healthcare and criminal justice, there has been a growing emphasis on developing techniques for explaining these black boxes in a post hoc manner. In this work, we analyze two popular post hoc interpretation techniques: SmoothGrad which is a gradient based method, and a variant of LIME which is a perturbation based method. More specifically, we derive explicit closed form expressions for the explanations output by these two methods and show that they both converge to the same explanation in expectation, i.e., when the number of perturbed samples used by these methods is large. We then leverage this connection to establish other desirable properties, such as robustness and linearity, for these techniques. We also derive finite sample complexity bounds for the number of perturbations required for these methods to converge to their expected explanation. Finally, we empirically validate our theory using extensive experimentation on both synthetic and real world datasets.
Tijana Zrnic, University of California, Berkeley
Title: Regret Minimization with Performative Feedback
Abstract: In performative prediction, the deployment of a predictive model triggers a shift in the data distribution. As these shifts are typically unknown ahead of time, the learner needs to deploy a model to get feedback about the distribution it induces. We study the problem of finding near-optimal models under performativity while maintaining low regret. On the surface, this problem might seem equivalent to a bandit problem. However, it exhibits a fundamentally richer feedback structure that we refer to as performative feedback: after every deployment, the learner receives samples from the shifted distribution rather than only bandit feedback about the reward. Our main contribution is regret bounds that scale only with the complexity of the distribution shifts and not that of the reward function. The key algorithmic idea is careful exploration of the distribution shifts that informs a novel construction of confidence bounds on the risk of unexplored models. The construction only relies on smoothness of the shifts and does not assume convexity. More broadly, our work establishes a conceptual approach for leveraging tools from the bandits literature for the purpose of regret minimization with performative feedback.
Keynote Speaker: Isaac Kohane, Harvard Medical School
Title: What’s in a label? The case for and against monolithic group/ethnic/race labeling for machine learning
Abstract: Populations and group labels have been used and abused for thousands of years. The scale at which AI can incorporate such labels into its models and the ways in which such models can be misused are cause for significant concern. I will describe, with examples drawn from experiments in precision medicine, the task dependence of how underserved and oppressed populations can be both harmed and helped by the use of group labels. The source of the labels and the utility models underlying their use will be particularly emphasized.
10:15 am–10:45 am
Coffee Break
10:45 am–12:15 pm
Paper Session 3
Session Chair: Ruth Urner
Rojin Rezvan, University of Texas at Austin
Title: Individually-Fair Auctions for Multi-Slot Sponsored Search
Abstract: We design fair-sponsored search auctions that achieve a near-optimal tradeoff between fairness and quality. Our work builds upon the model and auction design of Chawla and Jagadeesan, who considered the special case of a single slot. We consider sponsored search settings with multiple slots and the standard model of click-through rates that are multiplicatively separable into an advertiser-specific component and a slot-specific component. When similar users have similar advertiser-specific click-through rates, our auctions achieve the same near-optimal tradeoff between fairness and quality. When similar users can have different advertiser-specific preferences, we show that a preference-based fairness guarantee holds. Finally, we provide a computationally efficient algorithm for computing payments for our auctions as well as those in previous work, resolving another open direction from Chawla and Jagadeesan.
Judy Hanwen Shen, Stanford
Title: Leximax Approximations and Representative Cohort Selection
Abstract: Finding a representative cohort from a broad pool of candidates is a goal that arises in many contexts such as choosing governing committees and consumer panels. While there are many ways to define the degree to which a cohort represents a population, a very appealing solution concept is lexicographic maximality (leximax) which offers a natural (pareto-optimal like) interpretation that the utility of no population can be increased without decreasing the utility of a population that is already worse off. However, finding a leximax solution can be highly dependent on small variations in the utility of certain groups. In this work, we explore new notions of approximate leximax solutions with three distinct motivations: better algorithmic efficiency, exploiting significant utility improvements, and robustness to noise. Among other definitional contributions, we give a new notion of an approximate leximax that satisfies a similarly appealing semantic interpretation and relate it to algorithmically-feasible approximate leximax notions. When group utilities are linear over cohort candidates, we give an efficient polynomial-time algorithm for finding a leximax distribution over cohort candidates in the exact as well as in the approximate setting. Furthermore, we show that finding an integer solution to leximax cohort selection with linear utilities is NP-Hard.
Jiayuan Ye, National University of Singapore
Title: Differentially Private Learning Needs Hidden State (or Much Faster Convergence)
Abstract: Differential privacy analysis of randomized learning algorithms typically relies on composition theorems, where the implicit assumption is that the internal state of the iterative algorithm is revealed to the adversary. However, by assuming hidden states for DP algorithms (when only the last-iterate is observable), recent works prove a converging privacy bound for noisy gradient descent (on strongly convex smooth loss function) that is significantly smaller than composition bounds after a few epochs. In this talk, we extend this hidden-state analysis to various stochastic minibatch gradient descent schemes (such as under “shuffle and partition” and “sample without replacement”), by deriving novel bounds for the privacy amplification by random post-processing and subsampling. We prove that, in these settings, our privacy bound is much smaller than composition for training with a large number of iterations (which is the case for learning from high-dimensional data). Our converging privacy analysis, thus, shows that differentially private learning, with a tight bound, needs hidden state privacy analysis or a fast convergence. To complement our theoretical results, we present experiments for training classification models on MNIST, FMNIST and CIFAR-10 datasets, and observe a better accuracy given fixed privacy budgets, under the hidden-state analysis.
Mahbod Majid, University of Waterloo
Title: Efficient Mean Estimation with Pure Differential Privacy via a Sum-of-Squares Exponential Mechanism
Abstract: We give the first polynomial-time algorithm to estimate the mean of a d-variate probability distribution from O(d) independent samples (up to logarithmic factors) subject to pure differential privacy.
Our main technique is a new approach to use the powerful Sum of Squares method (SoS) to design differentially private algorithms. SoS proofs to algorithms is a key theme in numerous recent works in high-dimensional algorithmic statistics – estimators which apparently require exponential running time but whose analysis can be captured by low-degree Sum of Squares proofs can be automatically turned into polynomial-time algorithms with the same provable guarantees. We demonstrate a similar proofs to private algorithms phenomenon: instances of the workhorse exponential mechanism which apparently require exponential time but which can be analyzed with low-degree SoS proofs can be automatically turned into polynomial-time differentially private algorithms. We prove a meta-theorem capturing this phenomenon, which we expect to be of broad use in private algorithm design.
12:15 pm–1:45 pm
Lunch Break
1:45–3:15 pm
Paper Session 4
Session Chair: Kunal Talwar
Kunal Talwar, Apple
Title: Differential Secrecy for Distributed Data and Applications to Robust Differentially Secure Vector Summation
Abstract: Computing the noisy sum of real-valued vectors is an important primitive in differentially private learning and statistics. In private federated learning applications, these vectors are held by client devices, leading to a distributed summation problem. Standard Secure Multiparty Computation (SMC) protocols for this problem are susceptible to poisoning attacks, where a client may have a large influence on the sum, without being detected. In this work, we propose a poisoning-robust private summation protocol in the multiple-server setting, recently studied in PRIO. We present a protocol for vector summation that verifies that the Euclidean norm of each contribution is approximately bounded. We show that by relaxing the security constraint in SMC to a differential privacy like guarantee, one can improve over PRIO in terms of communication requirements as well as the client-side computation. Unlike SMC algorithms that inevitably cast integers to elements of a large finite field, our algorithms work over integers/reals, which may allow for additional efficiencies.
Giuseppe Vietri, University of Minnesota
Title: Improved Regret for Differentially Private Exploration in Linear MDP
Abstract: We study privacy-preserving exploration in sequential decision-making for environments that rely on sensitive data such as medical records. In particular, we focus on solving the problem of reinforcement learning (RL) subject to the constraint of (joint) differential privacy in the linear MDP setting, where both dynamics and rewards are given by linear functions. Prior work on this problem due to Luyo et al. (2021) achieves a regret rate that has a dependence of O(K^{3/5}) on the number of episodes K. We provide a private algorithm with an improved regret rate with an optimal dependence of O(K^{1/2}) on the number of episodes. The key recipe for our stronger regret guarantee is the adaptivity in the policy update schedule, in which an update only occurs when sufficient changes in the data are detected. As a result, our algorithm benefits from low switching cost and only performs O(log(K)) updates, which greatly reduces the amount of privacy noise. Finally, in the most prevalent privacy regimes where the privacy parameter ? is a constant, our algorithm incurs negligible privacy cost — in comparison with the existing non-private regret bounds, the additional regret due to privacy appears in lower-order terms.
Mingxun Zhou, Carnegie Mellon University
Title: The Power of the Differentially Oblivious Shuffle in Distributed Privacy MechanismsAbstract: The shuffle model has been extensively investigated in the distributed differential privacy (DP) literature. For a class of useful computational tasks, the shuffle model allows us to achieve privacy-utility tradeoff similar to those in the central model, while shifting the trust from a central data curator to a “trusted shuffle” which can be implemented through either trusted hardware or cryptography. Very recently, several works explored cryptographic instantiations of a new type of shuffle with relaxed security, called differentially oblivious (DO) shuffles. These works demonstrate that by relaxing the shuffler’s security from simulation-style secrecy to differential privacy, we can achieve asymptotical efficiency improvements. A natural question arises, can we replace the shuffler in distributed DP mechanisms with a DO-shuffle while retaining a similar privacy-utility tradeoff? In this paper, we prove an optimal privacy amplification theorem by composing any locally differentially private (LDP) mechanism with a DO-shuffler, achieving parameters that tightly match the shuffle model. Moreover, we explore multi-message protocols in the DO-shuffle model, and construct mechanisms for the real summation and histograph problems. Our error bounds approximate the best known results in the multi-message shuffle-model up to sub-logarithmic factors. Our results also suggest that just like in the shuffle model, allowing each client to send multiple messages is fundamentally more powerful than restricting to a single message.
Badih Ghazi, Google Research
Title: Differentially Private Ad Conversion Measurement
Abstract: In this work, we study conversion measurement, a central functionality in the digital advertising space, where an advertiser seeks to estimate advertiser site conversions attributed to ad impressions that users have interacted with on various publisher sites. We consider differential privacy (DP), a notion that has gained in popularity due to its strong and rigorous guarantees, and suggest a formal framework for DP conversion measurement, uncovering a subtle interplay between attribution and privacy. We define the notion of an operationally valid configuration of the attribution logic, DP adjacency relation, privacy budget scope and enforcement point, and provide, for a natural space of configurations, a complete characterization.
3:15 pm–3:45 pm
Coffee Break
3:45 pm–5:00 pm
Open Poster Session
June 8, 2022
9:15 am–10:15 am
Keynote Speaker: Nuria Oliver, Data-Pop Alliance
Title: Data Science against COVID-19
Abstract: In my talk, I will describe the work that I have been doing since March 2020, leading a multi-disciplinary team of 20+ volunteer scientists working very closely with the Presidency of the Valencian Government in Spain on 4 large areas: (1) human mobility modeling; (2) computational epidemiological models (both metapopulation, individual and LSTM-based models); (3) predictive models; and (4) citizen surveys via the COVID19impactsurvey with over 600,000 answers worldwide.
I will describe the results that we have produced in each of these areas, including winning the 500K XPRIZE Pandemic Response Challenge and best paper award at ECML-PKDD 2021. I will share the lessons learned in this very special initiative of collaboration between the civil society at large (through the survey), the scientific community (through the Expert Group) and a public administration (through the Commissioner at the Presidency level). WIRED magazine just published an article describing our story.
10:15 am–10:45 am
Coffee Break
10:45 am–12:15 pm
Paper Session 5
Session Chair: Kunal Talwar
Shengyuan Hu, Carnegie Mellon University
Title: Private Multi-Task Learning: Formulation and Applications to Federated Learning
Abstract: Many problems in machine learning rely on multi-task learning (MTL), in which the goal is to solve multiple related machine learning tasks simultaneously. MTL is particularly relevant for privacy-sensitive applications in areas such as healthcare, finance, and IoT computing, where sensitive data from multiple, varied sources are shared for the purpose of learning. In this work, we formalize notions of task-level privacy for MTL via joint differential privacy (JDP), a relaxation of differential privacy for mechanism design and distributed optimization. We then propose an algorithm for mean-regularized MTL, an objective commonly used for applications in personalized federated learning, subject to JDP. We analyze our objective and solver, providing certifiable guarantees on both privacy and utility. Empirically, our method allows for improved privacy/utility trade-offs relative to global baselines across common federated learning benchmarks
Christina Yu, Cornell University
Title: Sequential Fair Allocation: Achieving the Optimal Envy-Efficiency Tradeoff Curve
Abstract: We consider the problem of dividing limited resources to individuals arriving over T rounds with a goal of achieving fairness across individuals. In general there may be multiple resources and multiple types of individuals with different utilities. A standard definition of `fairness’ requires an allocation to simultaneously satisfy envy-freeness and Pareto efficiency. However, in the online sequential setting, the social planner must decide on a current allocation before the downstream demand is realized, such that no policy can guarantee these desiderata simultaneously with probability 1, requiring a modified metric of measuring fairness for online policies. We show that in the online setting, the two desired properties (envy-freeness and efficiency) are in direct contention, in that any algorithm achieving additive counterfactual envy-freeness up to L_T necessarily suffers an efficiency loss of at least 1 / L_T. We complement this uncertainty principle with a simple algorithm, HopeGuardrail, which allocates resources based on an adaptive threshold policy and is able to achieve any fairness-efficiency point on this frontier. Our result is the first to provide guarantees for fair online resource allocation with high probability for multiple resource and multiple type settings. In simulation results, our algorithm provides allocations close to the optimal fair solution in hindsight, motivating its use in practical applications as the algorithm is able to adapt to any desired fairness efficiency trade-off.
Hedyeh Beyhaghi, Carnegie Mellon University
Title: On classification of strategic agents who can both game and improve
Abstract: In this work, we consider classification of agents who can both game and improve. For example, people wishing to get a loan may be able to take some actions that increase their perceived credit-worthiness and others that also increase their true credit-worthiness. A decision-maker would like to define a classification rule with few false-positives (does not give out many bad loans) while yielding many true positives (giving out many good loans), which includes encouraging agents to improve to become true positives if possible. We consider two models for this problem, a general discrete model and a linear model, and prove algorithmic, learning, and hardness results for each. For the general discrete model, we give an efficient algorithm for the problem of maximizing the number of true positives subject to no false positives, and show how to extend this to a partial-information learning setting. We also show hardness for the problem of maximizing the number of true positives subject to a nonzero bound on the number of false positives, and that this hardness holds even for a finite-point version of our linear model. We also show that maximizing the number of true positives subject to no false positive is NP-hard in our full linear model. We additionally provide an algorithm that determines whether there exists a linear classifier that classifies all agents accurately and causes all improvable agents to become qualified, and give additional results for low-dimensional data.
Keegan Harris, Carnegie Mellon University
Title: Bayesian Persuasion for Algorithmic Recourse
Abstract: When subjected to automated decision-making, decision subjects may strategically modify their observable features in ways they believe will maximize their chances of receiving a favorable decision. In many practical situations, the underlying assessment rule is deliberately kept secret to avoid gaming and maintain competitive advantage. The resulting opacity forces the decision subjects to rely on incomplete information when making strategic feature modifications. We capture such settings as a game of Bayesian persuasion, in which the decision maker offers a form of recourse to the decision subject by providing them with an action recommendation (or signal) to incentivize them to modify their features in desirable ways. We show that when using persuasion, both the decision maker and decision subject are never worse off in expectation, while the decision maker can be significantly better off. While the decision maker’s problem of finding the optimal Bayesian incentive-compatible (BIC) signaling policy takes the form of optimization over infinitely-many variables, we show that this optimization can be cast as a linear program over finitely-many regions of the space of possible assessment rules. While this reformulation simplifies the problem dramatically, solving the linear program requires reasoning about exponentially-many variables, even under relatively simple settings. Motivated by this observation, we provide a polynomial-time approximation scheme that recovers a near-optimal signaling policy. Finally, our numerical simulations on semi-synthetic data empirically illustrate the benefits of using persuasion in the algorithmic recourse setting.
12:15 pm–1:45 pm
Lunch Break
1:45–3:15 pm
Paper Session 6
Session Chair: Elisa Celis
Mark Bun, Boston University
Title: Controlling Privacy Loss in Sampling Schemes: An Analysis of Stratified and Cluster Sampling
Abstract: Sampling schemes are fundamental tools in statistics, survey design, and algorithm design. A fundamental result in differential privacy is that a differentially private mechanism run on a simple random sample of a population provides stronger privacy guarantees than the same algorithm run on the entire population. However, in practice, sampling designs are often more complex than the simple, data-independent sampling schemes that are addressed in prior work. In this work, we extend the study of privacy amplification results to more complex, data-dependent sampling schemes. We find that not only do these sampling schemes often fail to amplify privacy, they can actually result in privacy degradation. We analyze the privacy implications of the pervasive cluster sampling and stratified sampling paradigms, as well as provide some insight into the study of more general sampling designs.
Samson Zhou, Carnegie Mellon University
Title: Private Data Stream Analysis for Universal Symmetric Norm Estimation
Abstract: We study how to release summary statistics on a data stream subject to the constraint of differential privacy. In particular, we focus on releasing the family of symmetric norms, which are invariant under sign-flips and coordinate-wise permutations on an input data stream and include L_p norms, k-support norms, top-k norms, and the box norm as special cases. Although it may be possible to design and analyze a separate mechanism for each symmetric norm, we propose a general parametrizable framework that differentially privately releases a number of sufficient statistics from which the approximation of all symmetric norms can be simultaneously computed. Our framework partitions the coordinates of the underlying frequency vector into different levels based on their magnitude and releases approximate frequencies for the “heavy” coordinates in important levels and releases approximate level sizes for the “light” coordinates in important levels. Surprisingly, our mechanism allows for the release of an arbitrary number of symmetric norm approximations without any overhead or additional loss in privacy. Moreover, our mechanism permits (1+\alpha)-approximation to each of the symmetric norms and can be implemented using sublinear space in the streaming model for many regimes of the accuracy and privacy parameters.
Aloni Cohen, University of Chicago
Title: Attacks on Deidentification’s Defenses
Abstract: Quasi-identifier-based deidentification techniques (QI-deidentification) are widely used in practice, including k-anonymity, ?-diversity, and t-closeness. We present three new attacks on QI-deidentification: two theoretical attacks and one practical attack on a real dataset. In contrast to prior work, our theoretical attacks work even if every attribute is a quasi-identifier. Hence, they apply to k-anonymity, ?-diversity, t-closeness, and most other QI-deidentification techniques. First, we introduce a new class of privacy attacks called downcoding attacks, and prove that every QI-deidentification scheme is vulnerable to downcoding attacks if it is minimal and hierarchical. Second, we convert the downcoding attacks into powerful predicate singling-out (PSO) attacks, which were recently proposed as a way to demonstrate that a privacy mechanism fails to legally anonymize under Europe’s General Data Protection Regulation. Third, we use LinkedIn.com to reidentify 3 students in a k-anonymized dataset published by EdX (and show thousands are potentially vulnerable), undermining EdX’s claimed compliance with the Family Educational Rights and Privacy Act.
The significance of this work is both scientific and political. Our theoretical attacks demonstrate that QI-deidentification may offer no protection even if every attribute is treated as a quasi-identifier. Our practical attack demonstrates that even deidentification experts acting in accordance with strict privacy regulations fail to prevent real-world reidentification. Together, they rebut a foundational tenet of QI-deidentification and challenge the actual arguments made to justify the continued use of k-anonymity and other QI-deidentification techniques.
Steven Wu, Carnegie Mellon University
Title: Fully Adaptive Composition in Differential Privacy
Abstract: Composition is a key feature of differential privacy. Well-known advanced composition theorems allow one to query a private database quadratically more times than basic privacy composition would permit. However, these results require that the privacy parameters of all algorithms be fixed before interacting with the data. To address this, Rogers et al. introduced fully adaptive composition, wherein both algorithms and their privacy parameters can be selected adaptively. The authors introduce two probabilistic objects to measure privacy in adaptive composition: privacy filters, which provide differential privacy guarantees for composed interactions, and privacy odometers, time-uniform bounds on privacy loss. There are substantial gaps between advanced composition and existing filters and odometers. First, existing filters place stronger assumptions on the algorithms being composed. Second, these odometers and filters suffer from large constants, making them impractical. We construct filters that match the tightness of advanced composition, including constants, despite allowing for adaptively chosen privacy parameters. We also construct several general families of odometers. These odometers can match the tightness of advanced composition at an arbitrary, preselected point in time, or at all points in time simultaneously, up to a doubly-logarithmic factor. We obtain our results by leveraging recent advances in time-uniform martingale concentration. In sum, we show that fully adaptive privacy is obtainable at almost no loss, and conjecture that our results are essentially not improvable (even in constants) in general.
Abstract: In spatial population genetics, it is important to understand the probability of extinction in multi-species interactions such as growing bacterial colonies, cancer tumor evolution and human migration. This is because extinction probabilities are instrumental in determining the probability of coexistence and the genealogies of populations. A key challenge is the complication due to spatial effect and different sources of stochasticity. In this talk, I will discuss about methods to compute the probability of extinction and other long-time behaviors for stochastic reaction-diffusion equations on metric graphs that flexibly parametrizes the underlying space. Based on recent joint work with Adrian Gonzalez-Casanova and Yifan (Johnny) Yang.
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 six-dimensional origin of non-invertible symmetries
Abstract: I will present a review about recent progress in charting non-invertible symmetries for four-dimensional quantum field theories that have a six-dimensional origin. These include in particular N=4 supersymmetric Yang-Mills theories, and also a large class of N=2 supersymmetric theories which are conformal and do not have a conventional Lagrangian description (the so-called theories of “class S”). Among the main results, I will explain criteria for identifying examples of systems with intrinsic and non-intrinsic non-invertible symmetries, as well as explore their higher dimensional origin. This seminar is based on joint works with Vladimir Bashmakov, Azeem Hasan, and Justin Kaidi.
The Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description:
Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department)
In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint. We will conclude the course with a gentle introduction to cryptography, introducing some of the iconic topics: Yao’s Millionaires’ problem, zero-knowledge proof, the multi-party computation algorithm, and its proof.
The program hopes to provide several research mentors from various disciplines who will give some of the course lectures. Students will have the opportunity to work with one of the research mentors offered by the program.
Prerequisites: Basic coding ability in some programming language (C/Python/Matlab or CS50 experience). Some background in calculus and linear algebra is needed too. If you wish to work with a research mentor on differential geometry, more background in geometry such as from Math 132 or 136 will be useful. If you wish to work with a research mentor on computer science, coding experience mentioned above will be very useful. If you wish to work with a medical scientist, some background in life science or basic organic chemistry is recommended.
The course will meet 3 hours per week for 7 weeks via Zoom on days and times that will be scheduled for the convenience of the participants. There may be other times to be arranged for special events.
This program is only open to current Harvard undergraduates; both Mathematics concentrators and non-math concentrators are invited to apply. People already enrolled in a Math Department summer tutorial are welcome to partake in this program also. As with the summer tutorials, there is no association with the Harvard Summer School; and neither Math concentration credit nor Harvard College credit will be given for completing this course. This course has no official Harvard status and enrollment does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston over the summer.)
However: As with the summer tutorials, those enrolled are eligible* to receive a stipend of $700, and if you are a Mathematics concentrator, any written paper for the course can be submitted to fulfill the Math Concentration third-year paper requirement. (*The stipend is not available for people already receiving a stipend via the Math Department’s summer tutorial program, nor is it available for PRISE participants or participants in the Herchel Smith program.)
If you wish to join this program, please email Cliff Taubes (chtaubes@math.harvard.edu). The enrollment is limited, so don’t wait too long to apply.
Abstract: Conditional independence (CI) is an important tool instatistical modeling, as, for example, it gives a statistical interpretation to graphical models. In general, given a list of dependencies among random variables, it is difficult to say which constraints are implied by them. Moreover, it is important to know what constraints on the random variables are caused by hidden variables. On the other hand, such constraints are corresponding to some determinantal conditions on the tensor of joint probabilities of the observed random variables. Hence, the inference question in statistics relates to understanding the algebraic and geometric properties of determinantal varieties such as their irreducible decompositions or determining their defining equations. I will explain some recent progress that arises by uncovering the link to point configurations in matroid theory and incidence geometry. This connection, in particular, leads to effective computational approaches for (1) giving a decomposition for each CI variety; (2) identifying each component in the decomposition as a matroid variety; (3) determining whether the variety has a real point or equivalently there is a statistical model satisfying a given collection of dependencies. The talk is based on joint works with Oliver Clarke, Kevin Grace, and Harshit Motwani.
SMaSH: Symposium for Mathematical Sciences at Harvard
On Tuesday, May 17, 2022, from 9:00 am – 5:30 pm, the Harvard John A Paulson School of Engineering and Applied Sciences (SEAS) and the Harvard Center of Mathematical Sciences and Applications (CMSA) held a Symposium for Mathematical Sciences for the mathematical sciences community at Harvard.
Organizing Committee
Michael Brenner, Applied Mathematics (SEAS)
Michael Desai, Organismic and Evolutionary Biology (FAS)
Sam Gershman, Psychology (FAS)
Michael Hopkins, Mathematics (FAS)
Gary King, Government (FAS)
Peter Koellner, Philosophy (FAS)
Scott Kominers, Economics (FAS) & Entrepreneurial Management (HBS)
Coffee and Breakfast West Atrium (ground floor of the SEC)
9:30–10:30 am
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumKosuke Imai, Government & Statistics (FAS): Use of Simulation Algorithms for Legislative Redistricting Analysis and EvaluationYannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS): The Sample Complexity of Up-to-ε Multi-Dimensional Revenue Maximization
10:30–11:00 am
Coffee Break West Atrium (ground floor of the SEC)
11:00–12:00 pm
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumSeth Neel, Technology & Operations Management (HBS): “Machine (Un)Learning” or Why Your Deployed Model Might Violate Existing Privacy LawDemba Ba, Electrical Engineering & Bioengineering (SEAS): Geometry, AI, and the Brain
12:00–1:00 pm
Lunch Break Engineering Yard Tent
1:00–2:30 pm
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumMelanie Matchett Wood, Mathematics (FAS): Understanding distributions of algebraic structures through their momentsMorgane Austern, Statistics (FAS): Limit theorems for structured random objectsAnurag Anshu, Computer Science (SEAS): Operator-valued polynomial approximations and their use.
2:30–3:00 pm
Coffee Break West Atrium (ground floor of the SEC)
3:00–4:30 pm
Faculty Talks Winokur Family Hall Classroom (Room 1.321) located just off of the West AtriumMichael Brenner, Applied Mathematics (SEAS): Towards living synthetic materialsRui Duan, Biostatistics (HSPH): Federated and transfer learning for healthcare data integrationSham M. Kakade, Computer Science (SEAS) & Statistics (FAS): What is the Statistical Complexity of Reinforcement Learning?
4:30–5:30 pm
Reception with Jazz musicians & Poster Session Engineering Yard Tent
Faculty Talks
Speaker
Title / Abstract / Bio
Anurag Anshu, Computer Science (SEAS)
Title: Operator-valued polynomial approximations and their use.
Abstract: Approximation of complicated functions with low degree polynomials is an indispensable tool in mathematics. This becomes particularly relevant in computer science, where the complexity of interesting functions is often captured by the degree of the approximating polynomials. This talk concerns the approximation of operator-valued functions (such as the exponential of a hermitian matrix, or the intersection of two projectors) with low-degree operator-valued polynomials. We will highlight the challenges that arise in achieving similarly good approximations as real-valued functions, as well as recent methods to overcome them. We will discuss applications to the ground states in physics and quantum complexity theory: correlation lengths, area laws and concentration bounds.
Bio: Anurag Anshu is an Assistant Professor of computer science at Harvard University. He spends a lot of time exploring the rich structure of quantum many-body systems from the viewpoint of quantum complexity theory, quantum learning theory and quantum information theory. He held postdoctoral positions at University of California, Berkeley and University of Waterloo and received his PhD from National University of Singapore, focusing on quantum communication complexity.
Morgane Austern, Statistics (FAS)
Title: Limit theorems for structured random objects
Abstract: Statistical inference relies on numerous tools from probability theory to study the properties of estimators. Some of the most central ones are the central limit theorem and the free central limit theorem. However, these same tools are often inadequate to study modern machine problems that frequently involve structured data (e.g networks) or complicated dependence structures (e.g dependent random matrices). In this talk, we extend universal limit theorems beyond the classical setting. We consider distributionally “structured’ and dependent random object i.e random objects whose distribution is invariant under the action of an amenable group. We show, under mild moment and mixing conditions, a series of universal second and third order limit theorems: central-limit theorems, concentration inequalities, Wigner semi-circular law and Berry-Esseen bounds. The utility of these will be illustrated by a series of examples in machine learning, network and information theory.
Bio: Morgane Austern is an assistant professor in the Statistics Department of Harvard University. Broadly, she is interested in developing probability tools for modern machine learning and in establishing the properties of learning algorithms in structured and dependent data contexts. She graduated with a PhD in statistics from Columbia University in 2019 where she worked in collaboration with Peter Orbanz and Arian Maleki on limit theorems for dependent and structured data. She was a postdoctoral researcher at Microsoft Research New England from 2019 to 2021.
Abstract: A large body of experiments suggests that neural computations reflect, in some sense, the geometry of “the world”. How do artificial and neural systems learn representations of “the world” that reflect its geometry? How, for instance, do we, as humans, learn representations of objects, e.g. fruits, that reflect the geometry of object space? Developing artificial systems that can capture/understand the geometry of the data they process may enable them to learn representations useful in many different contexts and tasks. My talk will describe an artificial neural-network architecture that, starting from a simple union-of-manifold model of data comprising objects from different categories, mimics some aspects of how primates learn, organize, and retrieve concepts, in a manner that respects the geometry of object space.
Bio: Demba Ba serves as an Associate Professor of electrical engineering and bioengineering in Harvard University’s School of Engineering and Applied Sciences, where he directs the CRISP group. Recently, he has taken a keen interest in the connection between artificial neural networks and sparse signal processing. His group leverages this connection to solve data-driven unsupervised learning problems in neuroscience, to understand the principles of hierarchical representations of sensory signals in the brain, and to develop explainable AI. In 2016, he received a Research Fellowship in Neuroscience from the Alfred P. Sloan Foundation. In 2021, Harvard’s Faculty of Arts and Sciences awarded him the Roslyn Abramson award for outstanding undergraduate teaching.
Michael Brenner, Applied Mathematics (SEAS)
Title: Towards living synthetic materials
Abstract: Biological materials are much more complicated and functional than synthetic ones. Over the past several years we have been trying to figure out why. A sensible hypothesis is that biological materials are programmable. But we are very far from being able to program materials we create with this level of sophistication. I will discuss our (largely unsuccessful) efforts to bridge this gap, though as of today I’m somewhat optimistic that we are arriving at a set of theoretical models that is rich enough to produce relevant emergent behavior.
Bio: I’ve been at Harvard for a long time. My favorite part of Harvard is the students.
Rui Duan, Biostatistics (HSPH)
Title: Federated and transfer learning for healthcare data integration
Abstract: The growth of availability and variety of healthcare data sources has provided unique opportunities for data integration and evidence synthesis, which can potentially accelerate knowledge discovery and improve clinical decision-making. However, many practical and technical challenges, such as data privacy, high dimensionality, and heterogeneity across different datasets, remain to be addressed. In this talk, I will introduce several methods for the effective and efficient integration of multiple healthcare datasets in order to train statistical or machine learning models with improved generalizability and transferability. Specifically, we develop communication-efficient federated learning algorithms for jointly analyzing multiple datasets without the need of sharing patient-level data, as well as transfer learning approaches that leverage shared knowledge learned across multiple datasets to improve the performance of statistical models in target populations of interest. We will discuss both the theoretical properties and examples of implementation of our methods in real-world research networks and data consortia.
Bio: Rui Duan is an Assistant Professor of Biostatistics at the Harvard T.H. Chan School of Public Health. She received her Ph.D. in Biostatistics in May 2020 from the University of Pennsylvania. Her research interests focus on developing statistical, machine learning, and informatics tools for (1) efficient data integration in biomedical research, (2) understanding and accounting for the heterogeneity of biomedical data, and improving the generalizability and transferability of models across populations (3) advancing precision medicine research on rare diseases and underrepresented populations.
Yannai A. Gonczarowski, Economics (FAS) & Computer Science (SEAS)
Title: The Sample Complexity of Up-to-ε Multi-Dimensional Revenue Maximization
Abstract: We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of n additive bidders whose values for m heterogeneous items are drawn independently. For any such instance and any ε > 0, we show that it is possible to learn an ε-Bayesian Incentive Compatible auction whose expected revenue is within ε of the optimal ε-BIC auction from only polynomially many samples.
Our fully nonparametric approach is based on ideas that hold quite generally, and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive, and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well-understood, our corollary for this case extends slightly the state-of-the-art.
Joint work with S. Matthew Weinberg
Bio: Yannai A. Gonczarowski is an Assistant Professor of Economics and of Computer Science at Harvard University—the first faculty member at Harvard to have been appointed to both of these departments. Interested in both economic theory and theoretical computer science, Yannai explores computer-science-inspired economics: he harnesses approaches, aesthetics, and techniques traditionally originating in computer science to derive economically meaningful insights. Yannai received his PhD from the Departments of Math and CS, and the Center for the Study of Rationality, at the Hebrew University of Jerusalem, where he was advised by Sergiu Hart and Noam Nisan. Yannai is also a professionally-trained opera singer, having acquired a bachelor’s degree and a master’s degree in Classical Singing at the Jerusalem Academy of Music and Dance. Yannai’s doctoral dissertation was recognized with several awards, including the 2018 Michael B. Maschler Prize of the Israeli Chapter of the Game Theory Society, and the ACM SIGecom Doctoral Dissertation Award for 2018. For the design and implementation of the National Matching System for Gap-Year Programs in Israel, he was awarded the Best Paper Award at MATCH-UP’19 and the inaugural INFORMS AMD Michael H. Rothkopf Junior Researcher Paper Prize (first place) for 2020. Yannai is also the recipient of the inaugural ACM SIGecom Award for Best Presentation by a Student or Postdoctoral Researcher at EC’18. His first textbook, “Mathematical Logic through Python” (Gonczarowski and Nisan), which introduces a new approach to teaching the material of a basic Logic course to Computer Science students, tailored to the unique intuitions and strengths of this cohort of students, is forthcoming in Cambridge University Press.
Kosuke Imai, Government & Statistics (FAS)
Title: Use of Simulation Algorithms for Legislative Redistricting Analysis and Evaluation
Abstract: After the 2020 Census, many states have been redrawing the boundaries of Congressional and state legislative districts. To evaluate the partisan and racial bias of redistricting plans, scholars have developed Monte Carlo simulation algorithms. The idea is to generate a representative sample of redistricting plans under a specified set of criteria and conduct a statistical hypothesis test by comparing a proposed plan with these simulated plans. I will give a brief overview of these redistricting simulation algorithms and discuss how they are used in real-world court cases.
Bio: Kosuke Imai is Professor in the Department of Government and Department of Statistics at Harvard University. Before moving to Harvard in 2018, Imai taught at Princeton University for 15 years where he was the founding director of the Program in Statistics and Machine Learning. Imai specializes in the development of statistical methods and machine learning algorithms and their applications to social science research. His areas of expertise include causal inference, computational social science, program evaluation, and survey methodology.
Sham M. Kakade, Computer Science (SEAS) & Statistics (FAS)
Title: What is the Statistical Complexity of Reinforcement Learning?
Abstract: This talk will highlight much of the recent progress on the following fundamental question in the theory of reinforcement learning: what (representational or structural) conditions govern our ability to generalize and avoid the curse of dimensionality? With regards to supervised learning, these questions are reasonably well understood, both practically and theoretically: practically, we have overwhelming evidence on the value of representational learning (say through modern deep networks) as a means for sample efficient learning, and, theoretically, there are well-known complexity measures (e.g. the VC dimension and Rademacher complexity) that govern the statistical complexity of learning. Providing an analogous theory for reinforcement learning is far more challenging, where even characterizing structural conditions which support sample efficient generalization has been far less well understood, until recently.
This talk will survey recent advances towards characterizing when generalization is possible in RL, focusing on both necessary and sufficient conditions. In particular, we will introduce a new complexity measure, the Decision-Estimation Coefficient, that is proven to be necessary (and, essentially, sufficient) for sample-efficient interactive learning.
Bio: Sham Kakade is a professor at Harvard University and a co-director of the Kempner Institute for the Study of Artificial and Natural Intelligence. He works on the mathematical foundations of machine learning and AI. Sham’s thesis helped lay the statistical foundations of reinforcement learning. With his collaborators, his additional contributions include foundational results on: policy gradient methods in reinforcement learning; regret bounds for linear bandit and Gaussian process bandit models; the tensor and spectral methods for latent variable models; and a number of convergence analyses for convex and non-convex algorithms. He is the recipient of the ICML Test of Time Award, the IBM Pat Goldberg best paper award, and INFORMS Revenue Management and Pricing Prize. He has been program chair for COLT 2011.
Sham was an undergraduate at Caltech, where he studied physics and worked under the guidance of John Preskill in quantum computing. He completed his Ph.D. with Peter Dayan in computational neuroscience at the Gatsby Computational Neuroscience Unit. He was a postdoc with Michael Kearns at the University of Pennsylvania.
Title: “Machine (Un)Learning” or Why Your Deployed Model Might Violate Existing Privacy Law
Abstract: Businesses like Facebook and Google depend on training sophisticated models on user data. Increasingly—in part because of regulations like the European Union’s General Data Protection Act and the California Consumer Privacy Act—these organizations are receiving requests to delete the data of particular users. But what should that mean? It is straightforward to delete a customer’s data from a database and stop using it to train future models. But what about models that have already been trained using an individual’s data? These are not necessarily safe; it is known that individual training data can be exfiltrated from models trained in standard ways via model inversion attacks. In a series of papers we help formalize a rigorous notion of data-deletion and propose algorithms to efficiently delete user data from trained models with provable guarantees in both convex and non-convex settings.
Bio: Seth Neel is a first-year Assistant Professor in the TOM Unit at Harvard Business School, and Co-PI of the SAFR ML Lab in the D3 Institute, which develops methodology to incorporate privacy and fairness guarantees into techniques for machine learning and data analysis, while balancing other critical considerations like accuracy, efficiency, and interpretability. He obtained his Ph.D. from the University of Pennsylvania in 2020 where he was an NSF graduate fellow. His work has focused primarily on differential privacy, notions of fairness in a variety of machine learning settings, and adaptive data analysis.
Melanie Matchett Wood, Mathematics (FAS)
Title: Understanding distributions of algebraic structures through their moments
Abstract: A classical tool of probability and analysis is to use the moments (mean, variance, etc.) of a distribution to recognize an unknown distribution of real numbers. In recent work, we are interested in distributions of algebraic structures that can’t be captured in a single number. We will explain one example, the fundamental group, that captures something about the shapes of possibly complicated or high dimensional spaces. We are developing a new theory of the moment problem for random algebraic structures which helps to to identify distributions of such, such as fundamental groups of random three dimensional spaces. This talk is based partly on joint work with Will Sawin.
Bio: Melanie Matchett Wood is a professor of mathematics at Harvard University and a Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study. Her work spans number theory, algebraic geometry, algebraic topology, additive combinatorics, and probability. Wood has been awarded a CAREER grant, a Sloan Research Fellowship, a Packard Fellowship for Science and Engineering, and the AWM-Microsoft Research Prize in Algebra and Number Theory, and she is a Fellow of the American Mathematical Society. In 2021, Wood received the National Science Foundation’s Alan T. Waterman Award, the nation’s highest honor for early-career scientists and engineers.
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 May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.
Organizers: Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)
Title:Gromov Hausdorff convergence of filtered A infinity category
Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry.
Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature.
Title:Projective Hulls, Projective Linking, and Boundaries of Varieties
Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ|
Title:Singularities along the Lagrangian mean curvature flow.
Abstract:We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze.
Title: Glimpses of embeddings and deformations of CR manifolds
Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975.
Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard?
The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years.
Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry
Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds.
Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects.
Title: Symplectic deformations and the Type IIA flow
Abstract:The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang.
Title: Canonical metrics and stability in mirror symmetry
Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability. I will review what is known, and pose some questions for the future.
Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry
Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong.
Title: The quantum connection: familiar yet puzzling
Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions.
Title:Higgs-Coulumb correspondence for abelian gauged linear sigma models
Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.
Title: Topological Transitions of Calabi-Yau Threefolds
Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau.
Title:Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video
10:15 am–11:15 am
Yu-Shen Lin
Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob.
Title: Deformations of singular Fano and Calabi-Yau varieties
Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions.
Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.
10:40–11:40 am
Annachiara Piubello
Title: Estimates on the Bartnik mass and their geometric implications.
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
LUNCH BREAK
1:30–2:30 pm
Ryan Unger
Title: Density and positive mass theorems for black holes and incomplete manifolds
Abstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications I
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Antoine Song (virtual)
Title: The spherical Plateau problem
Abstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
Tuesday, May 3, 2022
9:30–10:30 am
Chao Li
Title: Stable minimal hypersurfaces in 4-manifolds
Abstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds I
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Zhongshan An
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications II
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Tin-Yau Tsang
Title: Dihedral rigidity, fill-in and spacetime positive mass theorem
Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.
Speakers Banquet
Wednesday, May 4, 2022
9:30–10:30 am
Tristan Ozuch
Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flows
Abstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.
10:40–11:40 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds II
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
LUNCH BREAK
1:30–2:30 pm
Christos Mantoulidis
Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality
Abstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.
2:40–3:40 pm
Zhizhang Xie
Title: Gromov’s dihedral extremality/rigidity conjectures and their applications III
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.
TEA BREAK
4:10–5:10 pm
Xin Zhou (Virtual)
Title: Min-max minimal hypersurfaces with higher multiplicity
Abstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
May 5, 2022
9:00–10:00 am
Andre Neves
Title: Metrics on spheres where all the equators are minimal
Abstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.
10:10–11:10 am
Robin Neumayer
Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds III
Abstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.
11:20–12:20 pm
Paula Burkhardt-Guim
Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach
Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.
LUNCH BREAK
1:30–2:30 pm
Jonathan Zhu
Title: Widths, minimal submanifolds and symplectic embeddings
Abstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
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.
Title: Topological Wick Rotation and Holographic duality
Abstract: I will explain a new type of holographic dualities between n+1D topological orders with a chosen boundary condition and nD (potentially gapless) quantum liquids. It is based on the idea of topological Wick rotation, a notion which was first used in arXiv:1705.01087 and was named, emphasized and generalized later in arXiv:1905.04924. Examples of these holographic dualities include the duality between 2+1D toric code model and 1+1D Ising chain and its finite-group generalizations (independently discovered by many others); those between 2+1D topological orders and 1+1D rational conformal field theories; and those between n+1D finite gauge theories with a gapped boundary and nD gapped quantum liquids. I will also briefly discuss some generalizations of this holographic duality and its relation to AdS/CFT duality.
Title: Non-Invertible Symmetries from Holography and Branes
Abstract: The notion of global symmetry in quantum field theory (QFT) has witnessed dramatic generalizations in the past few years. One of the most exciting developments has been the identification of 4d QFTs possessing non-invertible symmetries, i.e. global symmetries whose generators exhibit fusion rules that are not group-like. In this talk, I will discuss realizations of non-invertible symmetries in string theory and holography. As a concrete case study, I will consider the Klebanov-Strassler setup for holographic confinement in Type IIB string theory. The global symmetries of the holographic 4d QFT (both invertible and non-invertible) can be accessed by studying the topological couplings of the low-energy effective action of the dual 5d supergravity theory. Moreover, non-invertible symmetry defects can be realized in terms of D-branes. The D-brane picture captures non-trivial aspects of the fusion of non-invertible symmetry defects, and of their action on extended operators of the 4d QFT.
Title: The Hull-Strominger system through conifold transitions
Abstract: In this talk I discuss the geometry of C-Y manifolds outside of the Kähler regime and especially describe the Hull-Strominger system through the conifold transitions.
10:00am – 10:45am
Chenglong Yu*
Title: Commensurabilities among Lattices in PU(1,n)
Abstract: In joint work with Zhiwei Zheng, we study commensurabilities among certain subgroups in PU(1,n). Those groups arise from the monodromy of hypergeometric functions. Their discreteness and arithmeticity are classified by Deligne and Mostow. Thurston also obtained similar results via flat conic metrics. However, the classification of the lattices among them up to conjugation and finite index (commensurability) is not completed. When n=1, it is the commensurabilities of hyperbolic triangles. The cases of n=2 are almost resolved by Deligne-Mostow and Sauter’s commensurability pairs, and commensurability invariants by Kappes-Möller and McMullen. Our approach relies on the study of some higher dimensional Calabi-Yau type varieties instead of complex reflection groups. We obtain some relations and commensurability indices for higher n and also give new proofs for existing pairs in n=2.
11:00am – 11:45am
Thomas Creutzig*
Title: Shifted equivariant W-algebras
Abstract: The CDO of a compact Lie group is a family of VOAs whose top level is the space of functions on the Lie group. Similar structures appear at the intersections of boundary conditions in 4-dimensional gauge theories, I will call these new families of VOAs shifted equivariant W-algebras. I will introduce these algebras, construct them and explain how they can be used to quickly prove the GKO-coset realization of principal W-algebras.
11:45am – 1:30 pm
Lunch
01:30pm – 02:15pm
Cumrun Vafa
Title: Reflections on Mirror Symmetry
Abstract: In this talk I review some of the motivations leading to the search and discovery of mirror symmetry as well as some of the applications it has had.
02:30pm – 03:15pm
Jonathan Mboyo Esole
Title: Algebraic topology and matter representations in F-theory
Abstract: Recently, it was observed that representations appearing in geometric engineering in F-theory all satisfy a unique property: they correspond to characteristic representations of embedding of Dynkin index one between Lie algebras. However, the reason why that is the case is still being understood. In this talk, I will present new insights, giving a geometric explanation for this fact using K-theory and the topology of Lie groups and their classifying spaces. In physics, this will be interpreted as conditions on the charge of instantons and the classifications of Wess-Zumino-Witten terms.
03:15pm – 03:45 pm
Break
03:45pm – 04:30pm
Weiqiang Wang
Title: A Drinfeld presentation of affine i-quantum groups
Abstract: A quantum symmetric pair of affine type (U, U^i) consists of a Drinfeld-Jimbo affine quantum group (a quantum deformation of a loop algebra) U and its coideal subalgebra U^i (called i-quantum group). A loop presentation for U was formulated by Drinfeld and proved by Beck. In this talk, we explain how i-quantum groups can be viewed as a generalization of quantum groups, and then we give a Drinfeld type presentation for the affine quasi-split i-quantum group U^i. This is based on joint work with Ming Lu (Sichuan) and Weinan Zhang (Virginia).
04:45pm – 05:30pm
Tony Pantev
Title: Decomposition, anomalies, and quantum symmetries
Abstract: Decomposition is a phenomenon in quantum physics which converts quantum field theories with non-effectively acting gauge symmetries into equivalent more tractable theories in which the fields live on a disconnected space. I will explain the mathematical content of decomposition which turns out to be a higher categorical version of Pontryagin duality. I will examine how this duality interacts with quantum anomalies and secondary quantum symmetries and will show how the anomalies can be canceled by homotopy coherent actions of diagrams of groups. I will discuss in detail the case of 2-groupoids which plays a central role in anomaly cancellation, and will describe a new duality operation that yields decomposition in the presence of anomalies. The talk is based on joint works with Robbins, Sharpe, and Vandermeulen.
11/29 (Tuesday)
Refreshments
09:00am – 09:45am
Robert MacRae*
Title: Rationality for a large class of affine W-algebras
Abstract: One of the most important results in vertex operator algebras is Huang’s theorem that the representation category of a “strongly rational” vertex operator algebra is a semisimple modular tensor category. Conversely, it has been conjectured that every (unitary) modular tensor category is the representation category of a strongly rational (unitary) vertex operator algebra. In this talk, I will describe my results on strong rationality for a large class of affine W-algebras at admissible levels. This yields a large family of modular tensor categories which generalize those associated to affine Lie algebras at positive integer levels, as well as those associated to the Virasoro algebra.
10:00am – 10:45am
Bailin Song*
Title: The global sections of chiral de Rham complexes on compact Calabi-Yau manifolds
Abstract: Chiral de Rham complex is a sheaf of vertex algebras on a complex manifold. We will describe the space of global sections of the chiral de Rham complexes on compact Calabi-Yau manifolds.
11:00am – 11:45am
Carl Lian*
Title: Curve-counting with fixed domain
Abstract: The fixed-domain curve-counting problem asks for the number of pointed curves of fixed (general) complex structure in a target variety X subject to incidence conditions at the marked points. The question comes in two flavors: one can ask for a virtual count coming from Gromov-Witten theory, in which case the answer can be computed (in principle) from the quantum cohomology of X, or one can ask for the “honest” geometric count, which tends to be more subtle. The answers are conjectured to agree in the presence of sufficient positivity, but do not always. I will give an overview of some recent results and open directions. Some of this work is joint with Alessio Cela, Gavril Farkas, and Rahul Pandharipande.
11:45am – 01:30pm
Lunch
01:30pm – 02:15pm
Chin-Lung Wang
Title: A blowup formula in quantum cohomology
Abstract: We study analytic continuations of quantum cohomology $QH(Y)$ under a blowup $\phi: Y \to X$ of complex projective manifolds along the extremal ray variable $q^{\ell}$. Under $H(Y) = \phi^* H(X) plus K$ where $K = \ker \phi_*$, we show that (i) the restriction of quantum product along the $\phi^*H(X)$ direction, denoted by $QH(Y)_X$, is meromorphic in $x := 1/q^\ell$, (ii) $K$ deforms uniquely to a quantum ideal $\widetilde K$ in $QH(Y)_X$, (iii) the quotient ring $QH(Y)_X/\widetilde K$ is regular over $x$, and its restriction to $x = 0$ is isomorphic to $QH(X)$. This is a joint work (in progress) with Y.-P. Lee and H.-W. Lin.
02:30pm – 03:15pm
Ivan Loseu
Title: Quantizations of nilpotent orbits and their Lagrangian subvarieties
Abstract: I’ll report on some recent progress on classifying quantizations of the algebras of regular functions of nilpotent orbits (and their covers) in semisimple Lie algebras, as well as the classification of quantizations of certain Lagrangian subvarieties. An ultimate goal here is to understand the classification of unitary representations of real semisimple Lie groups.
03:15pm – 03:45pm
Break
03:45pm – 04:30pm
Matt Kerr*
Title: $K_2$ and quantum curves
Abstract: The basic objects for this talk are motives consisting of a curve together with a $K_2$ class, and their mixed Hodge-theoretic invariants.
My main objective will be to explain a connection (recently proved in joint work with C. Doran and S. Sinha Babu) between (i) Hodge-theoretically distinguished points in the moduli of such motives and (ii) eigenvalues of operators on L^2(R) obtained by quantizing the equations of the curves.
By local mirror symmetry, this gives evidence for a conjecture in topological string theory (due to M. Marino, A. Grassi, and others) relating enumerative invariants of toric CY 3-folds to spectra of quantum curves.
04:45pm – 05:30pm
Flor Orosz Hunziker
Title: Tensor structures associated to the N=1 super Virasoro algebra
Abstract: We have recently shown that there is a natural category of representations associated to the N=1 super Virasoro vertex operator algebras that have braided tensor structure. We will describe this category and discuss the problem of establishing its rigidity at particular central charges. This talk is based on joint work in progress with Thomas Creutzig, Robert McRae and Jinwei Yang.
11/30 (Wednesday)
08:30am – 09:00am
Refreshments
09:00am – 09:45am
Tomoyuki Arakawa
Title: 4D/2D duality and representation theory
Abstract: This talk is about the 4D/2D duality discovered by Beem et al. rather recently in physics. It associates a vertex operator algebra (VOA) to any 4-dimensional superconformal field theory, which is expected to be a complete invariant of thl theory. The VOAs appearing in this manner may be regarded as chiralization of various symplectic singularities and their representations are expected to be closely related with the Coulomb branch of the 4D theory. I will talk about this remarkable 4D/2D duality from a representation theoretic perspective.
10:00am – 10:45am
Shashank Kanade
Title: Combinatorics of principal W-algebras of type A
Abstract: The combinatorics of principal W_r(p,p’) algebras of type A is controlled by cylindric partitions. However, very little seems to be known in general about fermionic expressions for the corresponding characters. Welsh’s work explains the case of Virasoro minimal models W_2(p,p’). Andrews, Schilling and Warnaar invented and used an A_2 version of the usual (A_1) Bailey machinery to give fermionic characters (up to a factor of (q)_\infty) of some, but not all, W_3(3,p’) modules. In a recent joint work with Russell, we have given a complete set of conjectures encompassing all of the remaining modules for W_3(3,p’), and proved our conjectures for small values of p’. In another direction, characters of W_r(p,p’) algebras also arise as appropriate limits of certain sl_r coloured Jones invariants of torus knots T(p,p’), and we expect this to provide further insights on the underlying combinatorics.
11:00am – 11:45am
Gufang Zhao
Title: Quasimaps to quivers with potentials
Abstract: This talk concerns non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential. The construction borrows ideas from the theory of gauged linear sigma models as well as recent development in shifted symplectic geometry and Donaldson-Thomas theory of Calabi-Yau 4-folds. Examples of virtual counts arising from quivers with potentials are discussed. This is based on work in progress, in collaboration with Yalong Cao.
11:45am – 01:30pm
Group Photo, Lunch
01:30pm – 02:15pm
Yaping Yang
Title: Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
Abstract: Let X be a smooth local toric Calabi-Yau 3-fold. On the cohomology of the moduli spaces of certain sheaves on X, there is an action of the cohomological Hall algebra (COHA) of Kontsevich and Soibelman via “raising operators”. I will discuss the “double” of the COHA that acts on the cohomology of the moduli space by adding the “lowering operators”. We associate a root system to X. The double COHA is expected to be the shifted Yangian of this root system. We also give a prediction for the shift in terms of an intersection pairing. We provide evidence of the aforementioned expectation in various examples. This is based on my joint work with M. Rapcak, Y. Soibelman, and G. Zhao
02:30pm – 03:15pm
Fei Han
Title: Graded T-duality with H-flux for 2d sigma models
Abstract: T-duality in string theory can be realised as a transformation acting on the worldsheet fields in the two-dimensional nonlinear sigma model. Bouwknegt-Evslin-Mathai established the T-duality in a background flux for the first time upon compactifying spacetime in one direction to a principal circle by constructing the T-dual maps transforming the twisted cohomology of the dual spacetimes. In this talk, we will describe our recent work on how to promote the T-duality maps of Bouwknegt-Evslin-Mathai in two aspects. More precisely, we will introduce (1) graded T-duality, concerning the graded T-duality maps of all levels of twistings; (2) the 2-dimensional sigma model picture, concerning the double loop space of spacetimes. This represents our joint work with Mathai.
03:15pm – 3:45pm
Break
03:45pm – 04:30pm
Mauricio Romo
Title: Networks and BPS Counting: A-branes view point
Abstract: I will review the countings of BPS invariants via exponential/spectral networks and present an interpretation of this counting as a count of certain points in the moduli space of A-branes corresponding to degenerate Lagrangians.
04:45pm – 05:30pm
Shinobu Hosono
Title: Mirror symmetry of abelian fibered Calabi-Yau manifolds with ρ = 2
Abstract: I will describe mirror symmetry of Calabi-Yau manifolds fibered by (1,8)-polarized abelian surfaces, which have Picard number two. Finding a mirror family over a toric variety explicitly, I observe that mirror symmetry of all related Calabi-Yau manifods arises from the corresponding boundary points, which are not necessarily toric boundary points. Calculating Gromov-Witten invariants up to genus 2, I find that the generating functions are expressed elliptic (quasi-)modular forms, which reminds us the modular anomaly equation found for elliptic surfaces. This talk is based on a published work with Hiromichi Takaki (arXiv:2103.08150).
06:00pm
Banquet @ Royal East Restaurant, 782 Main St, Cambridge, MA 02139
12/1 (Thursday)
08:30am – 09:00am
Refreshments
09:00am – 09:45am
Conan Nai Chung Leung*
Title: Quantization of Kahler manifolds
Abstract: I will explain my recent work on relationships among geometric quantization, deformation quantization, Berezin-Toeplitz quantization and brane quantization.
10:00am – 10:45am
Cuipo Jiang*
Title: Cohomological varieties associated to vertex operator algebras
Abstract: We define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties. As examples, we construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational $C_2$-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. This talk is based on joint work with Antoine Caradot and Zongzhu Lin.
11:00am – 11:45am
Anne Moreau*
Title: Action of the automorphism group on the Jacobian of Klein’s quartic curve
Abstract: In a joint work with Dimitri Markouchevitch, we prove that the quotient variety of the 3-dimensional Jacobian of the plane Klein quartic curve by its full automorphism group of order 336 is isomorphic to the 3-dimensional weighted projective space with weights 1,2,4,7.
The latter isomorphism is a particular case of the general conjecture of Bernstein and Schwarzman suggesting that a quotient of the n-dimensional complex space by the action of an irreducible complex crystallographic group generated by reflections is a weighted projective space.
In this talk, I will explain this conjecture and the proof of our result. An important ingredient is the computation of the Hilbert function of the algebra of invariant theta-functions on the Jacobian.
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 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.
Title: Phase Fluctuations in Two-Dimensional Superconductors and Pseudogap Phenomenon
Abstract: We study the phase fluctuations in the normal state of a general two-dimensional (2d) superconducting system with s-wave pairing. The effect of phase fluctuations of the pairing fields can be dealt with perturbatively using disorder averaging, after we treat the local superconducting order parameter as a static disordered background. It is then confirmed that the phase fluctuations above the 2d Berenzinskii-Kosterlitz-Thouless (BKT) transition give birth to the pseudogap phenomenon, leading to a significant broadening of the single-particle spectral functions. Quantitatively, the broadening of the spectral weights at the BCS gap is characterized by the ratio of the superconducting coherence length and the spatial correlation length of the superconducting pairing order parameter. Our results are tested on the attractive-U fermion Hubbard model on the square lattice, using unbiased determinant quantum Monte Carlo method and stochastic analytic continuation. We also apply our method to 2d superconductors with d-wave pairing and observe that the phase fluctuations may lead to Fermi-arc phenomenon above the BKT transition.
Title: Neutrino Masses from Generalized Symmetry Breaking
Abstract: We explore generalized global symmetries in theories of physics beyond the Standard Model. Theories of Z′ bosons generically contain ‘non-invertible’ chiral symmetries, whose presence indicates a natural paradigm to break this symmetry by an exponentially small amount in an ultraviolet completion. For example, in models of gauged lepton family difference such as the phenomenologically well-motivated U(1)Lμ−Lτ, there is a non-invertible lepton number symmetry which protects neutrino masses. We embed these theories in gauged non-Abelian horizontal lepton symmetries, e.g. U(1)Lμ−Lτ⊂SU(3)H, where the generalized symmetries are broken nonperturbatively by the existence of lepton family magnetic monopoles. In such theories, either Majorana or Dirac neutrino masses may be generated through quantum gauge theory effects from the charged lepton Yukawas e.g. yν∼yτexp(−Sinst). These theories require no bevy of new fields nor ad hoc additional global symmetries, but are instead simple, natural, and predictive: the discovery of a lepton family Z′ at low energies will reveal the scale at which Lμ−Lτ emerges from a larger gauge symmetry.
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.
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.
Speaker: Jian Kang, School of Physical Science and Technology, ShanghaiTech University, Shanghai, China
Title: Continuum field theory of graphene bilayer system
Abstract: The Bistritzer-MacDonald (BM) model predicted the existence of the narrow bands in the magic-angle twisted bilayer graphene (MATBG), and nowadays is a starting point for most theoretical works. In this talk, I will briefly review the BM model and then present a continuum field theory [1] for graphene bilayer system allowing any smooth lattice deformation including the small twist angle. With the gradient expansion to the second order, the continuum theory for MATBG [2] produces the spectrum that almost perfectly matches the spectrum of the microscopic model, suggesting the validity of this theory. In the presence of the lattice deformation, the inclusion of the pseudo-vector potential does not destroy but shift the flat band chiral limit to a smaller twist angle. Furthermore, the continuum theory contains another important interlayer tunneling term that was overlooked in all previous works. This term non-negligibly breaks the particle-hole symmetry of the narrow bands and may be related with the experimentally observed particle-hole asymmetry.
1. O. Vafek and JK, arXiv: 2208.05933. 2. JK and O. Vafek, arXiv: 2208.05953.
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.
%09%5Csim%09%5Comega%5E%7B-2/z%7D "\sigma(mega) \sim mega^{-2/z}")
(z is the boson dynamical exponent) when the order parameter is odd under inversion. The presence of incoherent conductivity in the random flavor model is a consequence of its unusual anomaly structure. From this we conclude that the large N deformation does not share important nonperturbative features with the physical N = 1 model, though it remains an interesting theory in its own right. Going beyond the IR fixed point, we also consider the effects of irrelevant operators and show, within the scope of the RPA expansion, that the old result %09%5Csim%09%5Comega%5E%7B-2(z-2)/z%7D "\sigma(mega) \sim mega^{-2(z-2)/z}")
due to Kim et al. is incorrect for inversion-odd order parameters.
