3/11/2021 Quantum Matter Seminar
2021-03-11 10:30 - 12:00 (America/New_York) | MA

Exact symmetries and threshold states in two-dimensional models for QCD
2021-03-17 10:30 - 2021-03-18 12:00 | MA
Speaker: Silviu Pufu (Princeton University)
Title: Exact symmetries and threshold states in two-dimensional models for QCD
Abstract: Two-dimensional QCD models form an interesting playground for studying phenomena such as confinement and screening. In this talk I will describe one such model, namely a 2d SU(N) gauge theory with an adjoint and a fundamental fermion, and explain how to compute the spectrum of bound states using discretized light-cone quantization at large N. Surprisingly, the spectrum of the discretized theory exhibits a large number of exact degeneracies, for which I will provide two different explanations. I will also discuss how these degeneracies provide a physical picture of confinement in 2d QCD with just a massless adjoint fermion. This talk is based on joint work with R. Dempsey and I. Klebanov.

3/18/2021 Quantum Matter Seminar
2021-03-18 12:00 - 13:30 | MA

3/22/2021 Mathematical Physics Seminar
2021-03-22 10:00 - 11:00 | Virtual

3/23/2021 Computer Science for Mathematicians
2021-03-23 11:30 - 12:30 | MA

3/24/2021 Quantum Matter seminar
2021-03-24 12:00 - 13:30 | MA

Word and Graph Embeddings for Machine Learning
2021-03-24 15:00 - 16:00 | MA
Speaker: Steve Skiena, Dept. of Computer Science and AI Insititute, Stony Brook University
Title: Word and Graph Embeddings for Machine Learning
Abstract: DeepWalk is an approach we have developed to construct vertex embeddings: vector representations of vertices which be applied to a very general class of problems in data mining and information retrieval. DeepWalk exploits an appealing analogy between sentences as sequences of words and random walks as sequences of vertices to transfer deep learning (unsupervised feature learning) techniques from natural language processing to network analysis. It has become extremely popular, having been cited by over 4600 research papers since its publication at KDD 2014. In this talk, I will introduce the notion of graph embeddings, and demonstrate why they make such powerful features for machine learning applications. I will focus on more recent efforts concerning (1) fast embedding methods for very large networks, (2) techniques for embedding dynamic graphs, and (3) embedding spaces as models for knowledge generation.

3/25/2021 Quantum Matter Seminar
2021-03-25 10:30 - 12:00 | MA

CMSA Math-Science Literature Lecture: On the History of quantum cohomology and homological mirror symmetry
2021-03-30 09:00 - 22:30 | Virtual
Maxim Kontsevich (IHÉS)
Title: On the History of quantum cohomology and homological mirror symmetry
Abstract: About 30 years ago, string theorists made remarkable discoveries of hidden structures in algebraic geometry. First, the usual cup-product on the cohomology of a complex projective variety admits a canonical multi-parameter deformation to so-called quantum product, satisfying a nice system of differential equations (WDVV equations). The second discovery, even more striking, is Mirror Symmetry, a duality between families of Calabi-Yau varieties acting as a mirror reflection on the Hodge diamond.
Later it was realized that the quantum product belongs to the realm of symplectic geometry, and a half of mirror symmetry (called Homological Mirror Symmetry) is a duality between complex algebraic and symplectic varieties. The search of correct definitions and possible generalizations lead to great advances in many domains, giving mathematicians new glasses, through which they can see familiar objects in a completely new way.
I will review the history of major mathematical advances in the subject of HMS, and the swirl of ideas around it.
Talk chair: Paul Seidel
Video

3/31/2021 Quantum Matter Seminar
2021-03-31 10:30 - 12:00 | MA

3/31/2021 Random Matrix and Probability Theory Seminar
2021-03-31 14:00 - 15:00 | MA

Random Matrix & Probability Theory Seminar
2021-03-31 14:00 - 15:00 | MA
Beginning immediately, until at least December 31, all seminars will take place virtually, through Zoom.
In the 2020-2021 AY, the Random Matrix and Probability Theory Seminar will take place on select Wednesdays from 2:00 – 3:00pm virtually. This seminar is organized by Christian Brennecke (brennecke@math.harvard.edu ).
To learn how to attend this seminar, please fill out this form.
The schedule below will be updated as the details are confirmed.
Spring 2021:

Date
Speaker
Title/Abstract

3/31/2021
Philippe Sosoe, Cornell University
Title: Fluctuation bounds for O’Connell-Yor type systems
Abstract: The O’Connell-Yor polymer is a fundamental model of a polymer in a random environment. It corresponds to the positive temperature version of Brownian Last Passage percolation. Although much is known about this model thanks to remarkable algebraic structure uncovered by O’Connell, Yor and others, basic estimates for the behavior of the tails of the centered partition function for finite N that are available for zero temperature models are missing. I will present an iterative estimate to obtain strong concentration and localization bounds for the O’Connell-Yor polymer on an almost optimal scale N^{1/3+epsilon}.
In the second part of the talk, I will introduce a system of interacting diffusions describing the successive increments of partition functions of different sizes. For this system, the N^{2/3} variance upper bound known for the OY polymer can be proved for a general class of interactions which are not expected to correspond to integrable models.
Joint work with Christian Noack and Benjamin Landon.

4/7/2021
Yue M. Lu, Harvard
Title: Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Random Matrices

Abstract: In many problems in statistical learning, random matrix theory, and statistical physics, one needs to simulate dynamics on random matrix ensembles. A classical example is to use iterative methods to compute the extremal eigenvalues/eigenvectors of a (spiked) random matrix. Other examples include approximate message passing on dense random graphs, and gradient descent algorithms for solving learning and estimation problems with random initialization. We will show that all such dynamics can be simulated by an efficient matrix-free scheme, if the random matrix is drawn from an ensemble with translation-invariant properties. Examples of such ensembles include the i.i.d. Gaussian (i.e. the rectangular Ginibre) ensemble, the Haar-distributed random orthogonal ensemble, the Gaussian orthogonal ensemble, and their complex-valued counterparts.A “direct” approach to the simulation, where one first generates a dense n × n matrix from the ensemble, requires at least O(n^2) resource in space and time. The new algorithm, named Householder Dice (HD), overcomes this O(n^2) bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are O(nT) and O(n T^2), respectively, with T being the number of iterations. When T ≪ n, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint.Finally, the HD algorithm is not just a computational trick. I will show how its construction can serve as a simple proof technique for several problems in high-dimensional estimation

4/14/2021
Canceled

4/16/2021
Friday
Patrick Lopatto (IAS)
Title: Fluctuations in local quantum unique ergodicity for generalized Wigner matrices
Abstract: In a disordered quantum system, delocalization can be understood in many ways. One of these is quantum unique ergodicity, which was proven in the random matrix context by Bourgade and Yau. It states that for a given eigenvector and set of coordinates J, the mass placed on J by the eigenvector tends to N^{-1}|J|, the mass placed on those coordinates by the uniform distribution. Notably, this convergence holds for any size of J, showing that the eigenvectors distribute evenly on all scales.
I will present a result which establishes that the fluctuations of these averages are Gaussian on scales where |J| is asymptotically less than N, for generalized Wigner matrices with smooth entries. The proof uses new eigenvector observables, which are analyzed dynamically using the eigenvector moment flow and the maximum principle.
This is joint work with Lucas Benigni.

4/21/2021
Jean-Christophe Mourrat, Courant Institute, NYU
Title: Mean-field spin glasses: beyond Parisi’s formula?

Abstract: Spin glasses are models of statistical mechanics encoding disordered interactions between many simple units. One of the fundamental quantities of interest is the free energy of the model, in the limit when the number of units tends to infinity. For a restricted class of models, this limit was predicted by Parisi, and later rigorously proved by Guerra and Talagrand. I will first show how to rephrase this result using an infinite-dimensional Hamilton-Jacobi equation. I will then present partial results suggesting that this new point of view may allow to understand limit free energies for a larger class of models, focusing in particular on the case in which the units are organized over two layers, and only interact across layers.

Fall 2020:

Date
Speaker
Title/Abstract

9/9/2020
Yukun He (Zurich)
Title: Single eigenvalue fluctuations of sparse Erdős–Rényi graphs
Abstract: I discuss the fluctuations of individual eigenvalues of the adjacency matrix of the Erdös-Rényi graph $G(N,p)$. I show that if $N^{-1}ll p ll N^{-2/3}, then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. The main technical tool of the proof is a rigidity bound of accuracy $N^{-1-varepsilon}p^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from previous works. Joint work with Antti Knowles.

10/14/2020
David Belius (University of Basel)
Title: The TAP approach to mean field spin glasses
Abstract: The Thouless-Anderson-Palmer (TAP) approach to the Sherrington-Kirkpatrick mean field spin glass model was proposed in one of the earliest papers on this model. Since then it has complemented subsequently elaborated methods in theoretical physics and mathematics, such as the replica method, which are largely orthogonal to the TAP approach. The TAP approach has the advantage of being interpretable as a variational principle optimizing an energy/entropy trade-off, as commonly encountered in statistical physics and large deviations theory, and potentially allowing for a more direct characterization of the Gibbs measure and its “pure states”. In this talk I will recall the TAP approach, and present preliminary steps towards a solution of mean field spin glass models entirely within a TAP framework.

10/28/2020
Giuseppe Genovese (University of Basel)
Title: Non-convex variational principles for the RS free energy of restricted Boltzmann machines
Abstract: From the viewpoint of spin glass theory, restricted Boltzmann machines represent a veritable challenge, as to the lack of convexity prevents us to use Guerra’s bounds. Therefore even the replica symmetric approximation for the free energy presents some challenges. I will present old and new results around the topic along with some open problems.

11/4/2020
Benjamin Landon (MIT)
Title: Fluctuations of the spherical Sherrington-Kirkpatrick model
Abstract: The SSK model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual SK model with Ising spins. Fluctuations of its observables may be related to quantities from random matrix theory using integral representations. In this informal talk we discuss some results on fluctuations of this model at critical temperature and with a magnetic field

11/11/2020
3:00 – 4:00pm
Lucas Benigni (University of Chicago)
Title: Optimal delocalization for generalized Wigner matrices
Abstract: We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.

11/18/2020
Simone Warzel (Technical University of Munich)
Title: Hierarchical quantum spin glasses

Abstract: Hierarchical spin glasses such as the generalised random energy model are known to faithfully model typical energy landscapes in the classical theory of mean-field spin glasses. Their built-in hierarchical structure is known to emerge spontaneously in the spin-glass phase of, e.g., the Sherrington-Kirkpatrick model. In this talk, I will review recent results on the effects of a transversal magnetic field on such hierarchical quantum spin glasses.
In particular, I will present a formula of Parisi-type for their free energy which allows to make predictions about the phase diagram.

12/2/2020
Sabine Jansen (LMU Munich)
Title: Thermodynamics of a hierarchical mixture of cubes

Abstract: The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes of side-lengths 2^j, jin
_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).

For information on previous seminars, click here
The schedule will be updated as details are confirmed.

Doing Mathematics with Simple Types: Infinitary Combinatorics in Isabelle/HOL
2021-03-31 15:00 - 16:00 | MA
Speaker: Lawrence Paulson, University of Cambridge Computer Laboratory
Title: Doing Mathematics with Simple Types: Infinitary Combinatorics in Isabelle/HOL
Abstract: Are proof assistants relevant to mathematics? One approach to this question is to explore the breadth of mathematical topics that can be formalised. The partition calculus was introduced by Erdös and R. Rado in 1956 as the study of “analogues and extensions of Ramsey’s theorem”. Highly technical results were obtained by Erdös-Milner, Specker and Larson (among many others) for the particular case of ordinal partition relations, which is concerned with countable ordinals and order types. Much of this material was formalised last year (with the assistance of Džamonja and Koutsoukou-Argyraki). Some highlights of this work will be presented along with general observations about the formalisation of mathematics, including ZFC, in simple type theory.

4/1/2021 Strongly Correlated Quantum Materials
2021-04-01 09:00 - 10:00 | MA

4/5/2021 Interdisciplinary Science Seminar
2021-04-05 10:00 - 11:00 | MA

4/5/2021 Math Physics Seminar
2021-04-05 10:00 - 11:00 | MA

CMSA Math-Science Literature Lecture: Isadore Singer’s Work on Analytic Torsion
2021-04-06 09:00 - 10:30 | Virtual
Edward Witten (IAS)
Title: Isadore Singer’s Work on Analytic Torsion
Abstract: I will review two famous papers of Ray and Singer on analytic torsion written approximately half a century ago. Then I will sketch the influence of analytic torsion in a variety of areas of physics including anomalies, topological field theory, and string theory.
This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer.
Talk chair: Cumrun Vafa
Slides | Video

Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer
2021-04-06 09:00 - 2021-04-27 10:30 | Virtual
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.

In the Spring 2021 semester, the CMSA hosted a sub-program on this series titled A Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch and Singer. Below is the schedule for talks in that subprogram
April 6, 2021 | 9:00 – 10:30am ET

Edward Witten (IAS)
Title: Isadore Singer’s Work on Analytic Torsion

April 13, 2021 | 9:00 – 10:30am ET

Claire Voisin (College de France)
Title: K-theory and characteristic classes in topology and complex geometry (a tribute to Atiyah and Hirzebruch)

April 20, 2021 | 9:00 – 10:30am ET

Dan Freed (the University of Texas at Austin)
Title: The Atiyah-Singer Index Theorem

April 27, 2021 | 9:00 – 10:30am ET

Frances Kirwan (University of Oxford)
Title: Moment maps and the Yang-Mills functional

4/6/2021 Computer Science for Mathematicians
2021-04-06 11:30 - 12:30 | MA

4/7/2021 Quantum Matter in Mathematics and Physics
2021-04-07 10:30 - 12:00 | MA

4/7/2021 Random Matrix and Probability Theory Seminar
2021-04-07 14:00 - 15:00 | MA

Type theory from the perspective of artificial intelligence
2021-04-07 15:00 - 16:00 | MA
Speaker: David McAllester – Toyota Technological Institute at Chicago
Title: Type theory from the perspective of artificial intelligence
Abstract: This talk will discuss dependent type theory from the perspective of artificial intelligence and cognitive science. From an artificial intelligence perspective it will be argued that type theory is central to defining the “game” of mathematics — an action space and reward structure for pure mathematics. From a cognitive science perspective type theory provides a model of the grammar of the colloquial (natural) language of mathematics. Of particular interest is the notion of a signature-axiom structure class and the three fundamental notions of equality in mathematics — set-theoretic equality between structure elements, isomorphism between structures, and Birkoff and Rota’s notion of cryptomorphism between structure classes. This talk will present a version of type theory based on set-theoretic semantics and the 1930’s notion of structure and isomorphism given by the Bourbaki group of mathematicians. It will be argued that this “Bourbaki type theory” (BTT) is more natural and accessible to classically trained mathematicians than Martin-Löf type theory (MLTT). BTT avoids the Curry-Howard isomorphism and axiom J of MLTT. The talk will also discuss BTT as a model of MLTT. The BTT model is similar to the groupoid model in that propositional equality is interpreted as isomorphism but different in various details. The talk will also briefly mention initial thoughts in defining an action space and reward structure for a game of mathematics.

CMSA Math-Science Literature Lecture: Quantum error correcting codes and fault tolerance
2021-04-08 09:00 - 10:30 | Virtual
Peter Shor (MIT)
Title: Quantum error correcting codes and fault tolerance
Abstract: We will go over the fundamentals of quantum error correction and fault tolerance and survey some of the recent developments in the field.

Talk chair: Zhengwei Liu
Video

4/8/2021 Interdisciplinary Science Seminar
2021-04-08 12:00 - 13:00 | MA

4/8/2021 Quantum Matter Seminar
2021-04-08 13:00 - 14:30 | MA

4/12/2021 Mathematical Physics Seminar
2021-04-12 10:00 - 11:00 | MA

4/14/2021 Quantum Matter Seminar
2021-04-14 10:30 - 12:00 | MA

A Bayesian neural network predicts the dissolution of compact planetary systems
2021-04-14 15:00 - 16:00 | Virtual
Speaker: Miles Cranmer – Princeton University
Title: A Bayesian neural network predicts the dissolution of compact planetary systems
Abstract: Despite over three hundred years of effort, no solutions exist for predicting when a general planetary configuration will become unstable. I will discuss our deep learning architecture (arxiv:2101.04117) which pushes forward this problem for compact systems. While current machine learning algorithms in this area rely on scientist-derived instability metrics, our new technique learns its own metrics from scratch, enabled by a novel internal structure inspired from dynamics theory. The Bayesian neural network model can accurately predict not only if, but also when a compact planetary system with three or more planets will go unstable. Our model, trained directly from short N-body time series of raw orbital elements, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on three-planet configurations, the model demonstrates robust generalization to five-planet systems, even outperforming models designed for that specific set of integrations. I will also discuss some work on recovering symbolic representations of such models using arxiv:2006.11287.

4/15/2021 Interdisciplinary Science Seminar
2021-04-15 09:00 - 10:00 | Virtual

4/15/2021 Quantum Matter in Mathematics Seminar
2021-04-15 10:30 - 12:00 | MA
