CMSA

8/11/2020 Geometry and Physics Seminar
2020-08-11 09:30 - 10:30 (America/New_York) | MA

8/18/2020 Geometry and Physics Seminar
2020-08-18 09:30 - 11:30 | MA

8/19/2020 Quantum Matter Seminar
2020-08-19 09:30 - 11:00 | MA

8/20/2020 Quantum Matter
2020-08-20 09:30 - 11:00 | MA

8/25/2020 Geometry and Physics Seminar
2020-08-25 09:30 - 10:30 | MA

8/26/2020 Quantum Matter Seminar
2020-08-26 09:30 - 11:00 | MA

8/27/2020 Quantum Matter Seminar
2020-08-27 09:30 - 11:00 | MA

Graph Representation Learning: Recent Advances and Open Challenges
2020-09-16 15:00 - 16:00 | Virtual
Speaker: William Hamilton, McGill University and MILA
Title: Graph Representation Learning: Recent Advances and Open Challenges
Abstract: Graph-structured data is ubiquitous throughout the natural and social sciences, from telecommunication networks to quantum chemistry. Building relational inductive biases into deep learning architectures is crucial if we want systems that can learn, reason, and generalize from this kind of data. Recent years have seen a surge in research on graph representation learning, most prominently in the development of graph neural networks (GNNs). Advances in GNNs have led to state-of-the-art results in numerous domains, including chemical synthesis, 3D-vision, recommender systems, question answering, and social network analysis. In the first part of this talk I will provide an overview and summary of recent progress in this fast-growing area, highlighting foundational methods and theoretical motivations. In the second part of this talk I will discuss fundamental limitations of the current GNN paradigm and propose open challenges for the theoretical advancement of the field.

Self-induced regularization from linear regression to neural networks
2020-09-23 15:00 - 16:00 | Virtual
Speaker: Andrea Montanari, Departments of Electrical Engineering and Statistics, Stanford
Title: Self-induced regularization from linear regression to neural networks
Abstract: Modern machine learning methods –most noticeably multi-layer neural networks– require to fit highly non-linear models comprising tens of thousands to millions of parameters. Despite this, little attention is paid to the regularization mechanism to control model’s complexity. Indeed, the resulting models are often so complex as to achieve vanishing training error: they interpolate the data. Despite this, these models generalize well to unseen data : they have small test error. I will discuss several examples of this phenomenon, beginning with a simple linear regression model, and ending with two-layers neural networks in the so-called lazy regime. For these examples precise asymptotics could be determined mathematically, using tools from random matrix theory. I will try to extract a unifying picture. A common feature is the fact that a complex unregularized nonlinear model becomes essentially equivalent to a simpler model, which is however regularized in a non-trivial way. [Based on joint papers with: Behrooz Ghorbani, Song Mei, Theodor Misiakiewicz, Feng Ruan, Youngtak Sohn, Jun Yan, Yiqiao Zhong]

10/1/2020 Quantum Matter seminar
2020-10-01 10:30 - 12:00 | MA

4/15/2020 Quantum Matter seminar
2020-10-05 10:30 - 12:00 | MA

Statistical, mathematical, and computational aspects of noisy intermediate-scale quantum computers
2020-10-14 09:00 - 10:00 | MA
Speaker: Gil Kalai (Hebrew University and IDC Herzliya)
Title: Statistical, mathematical, and computational aspects of noisy intermediate-scale quantum computers
Abstract: Noisy intermediate-scale quantum (NISQ) Computers hold the key for important theoretical and experimental questions regarding quantum computers. In the lecture I will describe some questions about mathematics, statistics and computational complexity which arose in my study of NISQ systems and are related to

a) My general argument “against” quantum computers,
b) My analysis (with Yosi Rinott and Tomer Shoham) of the Google 2019 “quantum supremacy” experiment.
Relevant papers:
Yosef Rinott, Tomer Shoham and Gil Kalai, Statistical aspects of the quantum supremacy demonstration, https://gilkalai.files.wordpress.com/2019/11/stat-quantum2.pdf
Gil Kalai, The Argument against Quantum Computers, the Quantum Laws of Nature, and Google’s Supremacy Claims, https://gilkalai.files.wordpress.com/2020/08/laws-blog2.pdf
Gil Kalai, Three puzzles on mathematics, computations, and games, https://gilkalai.files.wordpress.com/2019/09/main-pr.pdf

Triple Descent and a Fine-Grained Bias-Variance Decomposition
2020-10-14 15:00 - 16:00 | MA
Speaker: Jeffrey Pennington, Google Brain
Title: Triple Descent and a Fine-Grained Bias-Variance Decomposition
Abstract: Classical learning theory suggests that the optimal generalization performance of a machine learning model should occur at an intermediate model complexity, striking a balance between simpler models that exhibit high bias and more complex models that exhibit high variance of the predictive function. However, such a simple trade-off does not adequately describe the behavior of many modern deep learning models, which simultaneously attain low bias and low variance in the heavily overparameterized regime. Recent efforts to explain this phenomenon theoretically have focused on simple settings, such as linear regression or kernel regression with unstructured random features, which are too coarse to reveal important nuances of actual neural networks. In this talk, I will describe a precise high-dimensional asymptotic analysis of Neural Tangent Kernel regression that reveals some of these nuances, including non-monotonic behavior deep in the overparameterized regime. I will also present a novel bias-variance decomposition that unambiguously attributes these surprising observations to particular sources of randomness in the training procedure.

Generalization bounds for rational self-supervised learning algorithms, or "Understanding generalizations requires rethinking deep learning"
2020-10-28 15:00 - 16:00 | MA
Speakers: Boaz Barak and Yamini Bansal, Harvard University Dept. of Computer Science
Title: Generalization bounds for rational self-supervised learning algorithms, or “Understanding generalizations requires rethinking deep learning”
Abstract: The generalization gap of a learning algorithm is the expected difference between its performance on the training data and its performance on fresh unseen test samples. Modern deep learning algorithms typically have large generalization gaps, as they use more parameters than the size of their training set. Moreover the best known rigorous bounds on their generalization gap are often vacuous. In this talk we will see a new upper bound on the generalization gap of classifiers that are obtained by first using self-supervision to learn a complex representation of the (label free) training data, and then fitting a simple (e.g., linear) classifier to the labels. Such classifiers have become increasingly popular in recent years, as they offer several practical advantages and have been shown to approach state-of-art results. We show that (under the assumptions described below) the generalization gap of such classifiers tends to zero as long as the complexity of the simple classifier is asymptotically smaller than the number of training samples. We stress that our bound is independent of the complexity of the representation that can use an arbitrarily large number of parameters. Our bound assuming that the learning algorithm satisfies certain noise-robustness (adding small amount of label noise causes small degradation in performance) and rationality (getting the wrong label is not better than getting no label at all) conditions that widely (and sometimes provably) hold across many standard architectures. We complement this result with an empirical study, demonstrating that our bound is non-vacuous for many popular representation-learning based classifiers on CIFAR-10 and ImageNet, including SimCLR, AMDIM and BigBiGAN. The talk will not assume any specific background in machine learning, and should be accessible to a general mathematical audience. Joint work with Gal Kaplun.

Some exactly solvable models for machine learning via Statistical physics
2020-11-04 15:00 - 16:00 | Virtual
Speaker: Florent Krzakala, EPFL
Title: Some exactly solvable models for machine learning via Statistical physics
Abstract: The increasing dimensionality of data in the modern machine learning age presents new challenges and opportunities. The high dimensional settings allow one to use powerful asymptotic methods from probability theory and statistical physics to obtain precise characterizations and develop new algorithmic approaches. Statistical mechanics approaches, in particular, are very well suited for such problems. Will give examples of recent works in our group that build on powerful methods of statistical physics of disordered systems to analyze some relevant questions in machine learning and neural networks, including overparameterization, kernel methods, and the behavior gradient descent algorithm in a high dimensional non-convex landscape.

Towards AI for mathematical modeling of complex biological systems: Machine-learned model reduction, spatial graph dynamics, and symbolic mathematics
2020-11-11 15:00 - 16:00 | Virtual
Speaker: Eric Mjolsness, Departments of Computer Science and Mathematics, UC Irvine
Title: Towards AI for mathematical modeling of complex biological systems: Machine-learned model reduction, spatial graph dynamics, and symbolic mathematics
Abstract: The complexity of biological systems (among others) makes demands on the complexity of the mathematical modeling enterprise that could be satisfied with mathematical artificially intelligence of both symbolic and numerical flavors. Technologies that I think will be fruitful in this regard include (1) the use of machine learning to bridge spatiotemporal scales, which I will illustrate with the “Dynamic Boltzmann Distribution” method for learning model reduction of stochastic spatial biochemical networks and the “Graph Prolongation Convolutional Network” approach to course-graining the biophysics of microtubules; (2) a meta-language for stochastic spatial graph dynamics, “Dynamical Graph Grammars”, that can represent structure-changing processes including microtubule dynamics and that has an underlying combinatorial theory related to operator algebras; and (3) an integrative conceptual architecture of typed symbolic modeling languages and structure-preserving maps between them, including model reduction and implementation maps.

11/12/2020 Condensed Matter Seminar
2020-11-12 10:30 - 12:00 | MA

Re-pricing avalanches
2020-11-18 09:45 - 11:00 | Virtual
Speaker: Jose A. Scheinkman (Columbia)

Title: Re-pricing avalanches

Abstract: Monthly aggregate price changes exhibit chronic fluctuations but the aggregate shocks that drive these fluctuations are often elusive. Macroeconomic models often add stochastic macro-level shocks such as technology shocks or monetary policy shocks to produce these aggregate fluctuations. In this paper, we show that a state-dependent pricing model with a large but finite number of firms is capable of generating large fluctuations in the number of firms that adjust prices in response to an idiosyncratic shock to a firm’s cost of price adjustment. These fluctuations, in turn, cause fluctuations in aggregate price changes even in the absence of aggregate shocks. (Joint work with Makoto Nirei.)

Universes as Big Data, or Machine-Learning Mathematical Structures
2020-11-18 15:00 - 16:00 | Virtual
Speaker: Yang-Hui He, Oxford University, City University of London and Nankai University
Title: Universes as Big Data, or Machine-Learning Mathematical Structures
Abstract: We review how historically the problem of string phenomenology lead theoretical physics first to algebraic/differetial geometry, and then to computational geometry, and now to data science and AI. With the concrete playground of the Calabi-Yau landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of physical and mathematical interest, from geometry, to group theory, to combinatorics and number theory.

Members’ Seminar
2020-11-20 15:02 - 2021-01-01 15:02 | MA
The CMSA Members’ Seminar will occur every Friday at 9:30am ET on Zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. Please email the seminar organizers to obtain a link. This year’s seminar is organized by Tianqi Wu. The Schedule will be updated below.
Previous seminars can be found here.
Spring 2021:

Date
Speaker
Title/Abstract

1/29/2021
Cancelled

2/5/2021
Itamar Shamir
Title: Boundary CFT and conformal anomalies
Abstract: Boundary and defects in quantum field theory play an important role in many recent developments in theoretical physics. I will discuss such objects in the setting of conformal field theories, focusing mainly on conformal anomalies. Boundaries or defects can support various kinds of conformal anomalies on their world volume. Perhaps the one which is of greatest theoretical importance is associated with the Euler density in even dimensions. I will show how this anomaly is related to the one point function of exactly marginal deformations and how it arises explicitly from various correlation functions.

2/12/2021
Louis Fan
Title: Joint distribution of Busemann functions in corner growth models
Abstract: The 1+1 dimensional corner growth model with exponential weights is a centrally important exactly solvable model in the Kardar-Parisi-Zhang class of statistical mechanical models. While significant progress has been made on the fluctuations of the growing random shape, understanding of the optimal paths, or geodesics, is less developed. The Busemann function is a useful analytical tool for studying geodesics. We present the joint distribution of the Busemann functions, simultaneously in all directions of growth, in terms of mappings that represent FIFO (first-in-first-out) queues. As applications of this description we derive a marked point process representation for the Busemann function across a single lattice edge and point out its implication on structure of semi-infinite geodesics. This is joint work with Timo Seppäläinen.

2/19/2021
Daniel Junghans
Title: Control issues of the KKLT scenario in string theory
Abstract: The simplest explanation for the observed accelerated expansion of the universe is that we live in a 4-dimensional de Sitter space. We analyze to which extent the KKLT proposal for the construction of such de Sitter vacua in string theory is quantitatively controlled. As our main finding, we uncover and quantify an issue which one may want to call the “singular-bulk problem”. In particular, we show that, generically, a significant part of the manifold on which string theory is compactified in the KKLT scenario becomes singular. This implies a loss of control over the supergravity approximation on which the construction relies.

2/26/2021
Tsung-Ju Lee
Title: SYZ fibrations and complex affine structures
Abstract: Strominger–Yau–Zaslow conjecture has been a guiding principle in mirror symmetry. The conjecture predicts the existence of special Lagrangian torus fibrations of a Calabi–Yau manifold near a large complex structure limit point. Moreover, the mirror is given by the dual fibrations and the Ricci-flat metric is obtained from the semi-flat metric with corrections from holomorphic discs whose boundaries lie in a special Lagrangian fiber. By a result of Collins–Jacob–Lin, the complement of a smooth elliptic curve in the projective plane admits a SYZ fibration. In this talk, I will explain how to compute the complex affine structure induced from this SYZ fibration and show that it agrees with the affine structure used in Carl–Pumperla–Siebert. This is based on a joint work with Siu-Cheong Lau and Yu-Shen Lin.

3/5/2021
Cancelled

3/11/2021
9:00pm ET
Ryan Thorngren
Title: Symmetry protected topological phases, anomalies, and their classification
Abstract: I will give an overview of some mathematical aspects of the subject of symmetry protected topological phases (SPTs), especially as their theory relates to index theorems in geometry, cobordism of manifolds, and group cohomology.

3/18/2021
9:00pm ET
Ryan Thorngren
Title: Symmetry protected topological phases, anomalies, and their classification
Abstract: I will give an overview of some mathematical aspects of the subject of symmetry protected topological phases (SPTs), especially as their theory relates to index theorems in geometry, cobordism of manifolds, and group cohomology.

3/26/2021
8:30am ET
Aghil Alaee
Title: Rich extra dimensions are hidden inside black holes
Abstract: In this talk, I present an argument that shows why it is difficult to see rich extra dimensions in the Universe.

4/2/2021
8:30am ET
Enno Keßler
Title: Super Stable Maps of Genus Zero
Abstract: I will report on a supergeometric generalization of J-holomorphic curves. Supergeometry is a mathematical theory of geometric spaces with anti-commuting coordinates and functions which is motivated by the concept of supersymmetry from theoretical physics. Super J-holomorphic curves and super stable maps couple the equations of classical J-holomorphic curves with a Dirac equation for spinors and might, in the future, lead to a supergeometric generalization of Gromov-Witten invariants.

4/9/2021
Juven Wang
Video
Title: Ultra Unification
Abstract: Strong, electromagnetic, and weak forces were unified in the Standard Model (SM) with spontaneous gauge symmetry breaking. These forces were further conjectured to be unified in a simple Lie group gauge interaction in the Grand Unification (GUT). In this work, we propose a theory beyond the SM and GUT by adding new gapped Topological Phase Sectors consistent with the nonperturbative global anomaly matching and cobordism constraints (especially from the baryon minus lepton number B − L and the mixed gauge-gravitational anomaly). Gapped Topological Phase Sectors are constructed via symmetry extension, whose low energy contains unitary topological quantum field theories (TQFTs): either 3+1d non-invertible TQFT (long-range entangled gapped phase), or 4+1d invertible or non-invertible TQFT (short-range or long-range entangled gapped phase), or right-handed neutrinos, or their combinations. We propose that a new high-energy physics frontier beyond the conventional 0d particle physics relies on the new Topological Force and Topological Matter including gapped extended objects (gapped 1d line and 2d surface operators or defects, etc., whose open ends carry deconfined fractionalized particle or anyonic string excitations). I will also fill in the dictionary between math, QFT, and condensed matter terminology, and elaborate more on the nonperturbative global anomalies of Z2, Z4, Z16 classes useful for beyond SM. Work is based on arXiv:2012.15860, arXiv:2008.06499, arXiv:2006.16996, arXiv:1910.14668.

4/16/2021
Sergiy Verstyuk
Title: Deep learning methods for economics
Abstract: The talk discusses some recent developments in neural network models and their applicability to problems in international economics as well as macro-via-micro economics. Along the way, interpretability of neural networks features prominently.

4/23/2021
Yifan Wang
Title: Virtues of Defects in Quantum Field Theories
Abstract: Defects appear ubiquitously in many-body quantum systems as boundaries and impurities. They participate inextricably in the quantum dynamics and give rise to novel phase transitions and critical phenomena. Quantum field theory provides the natural framework to tackle these problems, where defects define extended operators over sub-manifolds of the spacetime and enrich the usual operator algebra. Much of the recent progress in quantum field theory has been driven by the exploration of general structures in this extended operator algebra, precision studies of defect observables, and the implications thereof for strongly coupled dynamics. In this talk, I will review selected developments along this line that enhance our understanding of concrete models in condensed matter and particle physics, and that open new windows to nonperturbative effects in quantum gravity.

4/30/2021
Yun Shi
Title: D-critical locus structure for local toric Calabi-Yau 3-fold
Abstract: Donaldson-Thomas (DT) theory is an enumerative theory which produces a count of ideal sheaves of 1-dimensional subschemes on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will give a brief introduction to motivic DT theory following the definition of Bussi-Joyce-Meinhardt, in particular the role of d-critical locus structure in the definition of motivic DT invariant. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint work in progress with Sheldon Katz.

5/7/2021
Thérèse Yingying Wu
Title: Topological aspects of Z/2Z eigenfunctions for the Laplacian on S^2
Abstract: In this talk, I will present recent work with C. Taubes on an eigenvalue problem for the Laplacian on the round 2-sphere associated with a configuration of an even number of distinct points on that sphere, denoted as C_2n. I will report our preliminary findings on how eigenvalues and eigenfunctions change as a function of the configuration space. I will also discuss how the compactification of C_2n is connected to the moduli space of algebraic curves (joint work with S.-T. Yau). There is a supergeometry tie-in too.

5/14/2021
Du Pei
Title: Three applications of TQFTs
Abstract: Topological quantum field theories (TQFTs) often serve as a bridge between physics and mathematics. In this talk, I will illustrate how TQFTs that arise in physics can help to shed light on 1) the quantization of moduli spaces 2) quantum invariants of 3-manifolds, and 3) smooth structures on 4-manifolds.

5/21/2021
Farzan Vafa
Title: Active nematic defects and epithelial morphogenesis
Abstract: Inspired by recent experiments that highlight the role of topological defects in morphogenesis, we develop a minimal framework to study the dynamics of an active curved surface driven by its nematic texture (a rank 2 symmetric traceless tensor). Allowing the surface to evolve via relaxational dynamics (gradient flow) leads to a theory linking nematic defect dynamics, cellular division rates, and Gaussian curvature. Regions of large positive (negative) curvature and positive (negative) growth are colocalized with the presence of positive (negative) defects, and cells accumulate at positive defects and are depleted at negative defects. We also show that activity stabilizes a bound $+1$ defect state by creating an incipient tentacle, while a bound $+1$ defect state surrounded by two $-1/2$ defects can create a stationary ring configuration of tentacles, consistent with experimental observations. The talk is based on a recent paper with L Mahadevan [arXiv:2105.0106].

Fall 2020:

Date
Speaker
Title/Abstract

9/11/2020
Moran Koren
Title: Observational Learning and Inefficiencies in Waitlists
Abstract: Many scarce resources are allocated through waitlists without monetary transfers. We consider a model, in which objects with heterogeneous qualities are offered to strategic agents through a waitlist in a first-come-first-serve manner. Agents, upon receiving an offer, accept or reject it based on both a private signal about the quality of the object and the decisions of agents ahead of them on the list. This model combines observational learning and dynamic incentives, two features that have been studied separately. We characterize the equilibrium and quantify the inefficiency that arises due to herding and selectivity. We find that objects with intermediate expected quality are discarded while objects with a lower expected quality may be accepted. These findings help in understanding the reasons for the substantial discard rate of transplant organs of various qualities despite the large shortage of organ supply.

9/18/2020
Michael Douglas
Title: A talk in two parts, on strings and on computers and math
Abstract: I am dividing my time between two broad topics. The first is string theory, mostly topics in geometry and compactification. I will describe my current work on numerical Ricci flat metrics, and list many open research questions. The second is computation and artificial intelligence. I will introduce transformer models (Bert,GPT) which have led to breakthroughs on natural language processing, describe their potential for helping us do math, and sketch some related theoretical problems.

9/25/2020
Cancelled – Math Science Lecture

10/2/2020
Cancelled – Math Science Lecture

10/9/2020
Wai Tong (Louis) Fan
Title: Stochastic PDE as scaling limits of interacting particle systems
Abstract: Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.
In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics. Joint work with Rick Durrett.

10/16/2020
Tianqi Wu
Title: Koebe circle domain conjecture and the Weyl problem in hyperbolic 3-space
Abstract: In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. The Weyl problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 into the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture, the Weyl problem and the work of Alexandrov and Thurston on convex surfaces. This is a joint work with Feng Luo.

10/23/2020
Changji Xu
Title: Random Walk Among Bernoulli Obstacles
Abstract: Place an obstacle with probability $1 – p$ independently at each vertex of $mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. This is called random walk among Bernoulli obstacles. The most prominent feature of this model is a strong localization effect: the random walk will be localized in a very small region conditional on the event that it survives for a long time. In this talk, we will discuss some recent results about the behaviors of the conditional random walk, in quenched, annealed, and biased settings.

10/30/2020
Michael Simkin
Title: The differential equation method in Banach spaces and the $n$-queens problem
Abstract: The differential equation method is a powerful tool used to study the evolution of random combinatorial processes. By showing that the process is likely to follow the trajectory of an ODE, one can study the deterministic ODE rather than the random process directly. We extend this method to ODEs in infinite-dimensional Banach spaces.
We apply this tool to the classical $n$-queens problem: Let $Q(n)$ be the number of placements of $n$ non-attacking chess queens on an $n times n$ board. Consider the following random process: Begin with an empty board. For as long as possible choose, uniformly at random, a space with no queens in its row, column, or either diagonal, and place on it a queen. We associate the process with an abstract ODE. By analyzing the ODE we conclude that the process almost succeeds in placing $n$ queens on the board. Furthermore, we can obtain a complete $n$-queens placement by making only a few changes to the board. By counting the number of choices available at each step we conclude that $Q(n) geq (n/C)^n$, for a constant $C>0$ associated with the ODE. This is optimal up to the value of $C$.

11/6/2020
Kenji Kawaguchi
Title: Deep learning: theoretical results on optimization and mixup
Abstract: Deep neural networks have achieved significant empirical success in many fields, including the fields of computer vision, machine learning, and artificial intelligence. Along with its empirical success, deep learning has been theoretically shown to be attractive in terms of its expressive power. However, the theory of the expressive power does not ensure that we can efficiently find an optimal solution in terms of optimization, robustness, and generalization, during the optimization process of a neural network. In this talk, I will discuss some theoretical results on optimization and the effect of mixup on robustness and generalization.

11/13/2020
Omri Ben-Eliezer
Title: Sampling in an adversarial environment
Abstract: How many samples does one need to take from a large population in order to truthfully “represent” the population? While this cornerstone question in statistics is very well understood when the population is fixed in advance, many situations in modern data analysis exhibit a very different behavior: the population interacts with and is affected by the sampling process. In such situations, the existing statistical literature does not apply.
We propose a new sequential adversarial model capturing these situations, where future data might depend on previously sampled elements; we then prove uniform laws of large numbers in this adversarial model. The results, techniques, and applications reveal close connections to various areas in mathematics and computer science, including VC theory, discrepancy theory, online learning, streaming algorithms, and computational geometry.
Based on joint works with Noga Alon, Yuval Dagan, Shay Moran, Moni Naor, and Eylon Yogev.

11/20/2020
Charles Doran
Title: The Calabi-Yau Geometry of Feynman Integrals
Abstract: Over the past 30 years Calabi-Yau manifolds have proven to be the key geometric structures behind string theory and its variants. In this talk, I will show how the geometry and moduli of Calabi-Yau manifolds provide a new framework for understanding and computing Feynman integrals. An important organizational principle is provided by mirror symmetry, and specifically the DHT mirror correspondence. This is joint work with Andrey Novoseltsev and Pierre Vanhove.

Colloquia & Seminars,Seminars

Machine learning and su(3) structures on six manifolds
2020-12-09 15:00 - 16:00 | Virtual
Speaker: James Gray – Virginia Tech
Title: Machine learning and su(3) structures on six manifolds
Abstract: In this talk we will discuss the application of Machine Learning techniques to obtain numerical approximations to various metrics of SU(3) structure on six manifolds. More precisely, we will be interested in SU(3) structures whose torsion classes make them suitable backgrounds for various string compactifications. A variety of aspects of this topic will be covered. These will include learning moduli dependent Ricci-Flat metrics on Calabi-Yau threefolds and obtaining numerical approximations to torsional SU(3) structures.

The Inside View: Raymarching and the Thurston Geometries
2020-12-16 12:00 - 13:00 | MA
On Wednesday, December 16 at 12:00 p.m. EST, WAM and CMSA will host a holiday seminar featuring Sabetta Matsumoto, Georgia Institute of Technology who will present The Inside View: Raymarching and the Thurston Geometries.
The properties of euclidean space seem natural and obvious to us, to the point that it took mathematicians over two thousand years to see an alternative to Euclid’s parallel postulate. The eventual discovery of hyperbolic geometry in the 19th century shook our assumptions, revealing just how strongly our native experience of the world blinded us from consistent alternatives, even in a field that many see as purely theoretical. Non-euclidean spaces are still seen as unintuitive and exotic, but with direct immersive experiences we can get a better intuitive feel for them. The latest wave of virtual reality hardware, in particular the HTC Vive, tracks both the orientation and the position of the headset within a room-sized volume, allowing for such an experience. We create realtime rendering to explore the three-dimensional geometries of the Thurston/Perelman geometrization theorem. In this talk, we use the “inside view” of each manifold to try to understand its geometry and what life might be like on the inside. Joint work with Rémi Coulon, Henry Segerman and Steve Trettel.
Visit the event page

AI and Theorem Proving
2021-01-13 15:00 - 16:00 | Virtual
Speaker: Josef Urban, Czech Technical University
Title: AI and Theorem Proving
Abstract: The talk will discuss the main approaches that combine machine learning with automated theorem proving and automated formalization. This includes learning to choose relevant facts for “hammer” systems, guiding the proof search of tableaux and superposition automated provers by interleaving learning and proving (reinforcement learning) over large ITP libraries, guiding the application of tactics in interactive tactical systems, and various forms of lemmatization and conjecturing. I will also show some demos of the systems, and discuss autoformalization approaches such as learning probabilistic grammars from aligned informal/formal corpora, combining them with semantic pruning, and using neural methods to learn direct translation from Latex to formal mathematics.

Language Modeling for Mathematical Reasoning
2021-01-20 15:00 - 16:00 | Virtual
Speaker: Christian Szegedy
Title: Language Modeling for Mathematical Reasoning
Abstract: In this talk, I will summarize the current state of the art of transformer based language models and give examples on non-trivial reasoning task language models can solve in higher order logic reasoning. I will also discuss how to inject injective bias into transformer networks via pretraining on very simple synthetic tasks and representing graph structures for transformer networks.

Knowledge graph representation: From recent models towards a theoretical understanding
2021-01-27 15:00 - 16:00 | MA
Speaker: Carl Allen and Ivana Balažević – University of Edinburgh School of Informatics
Title: Knowledge graph representation: From recent models towards a theoretical understanding
Abstract: Knowledge graphs (KGs), or knowledge bases, are large repositories of facts in the form of triples (subject, relation, object), e.g. (Edinburgh, capital_of, Scotland). Many models have been developed to succinctly represent KGs such that known facts can be recalled (question answering) and, more impressively, previously unknown facts can be inferred (link prediction). Subject and object entities are typically represented as vectors in R^d and relations as mappings (e.g. linear transformations) between them. Such representation can be interpreted as positioning entities in a space such that relations are implied by their relative locations. In this talk we give an overview of knowledge graph representation including select recent models; and, by drawing a connection to word embeddings, explain a theoretical model for how semantic relationships can correspond to geometric structure.

A Mathematical Exploration of Why Language Models Help Solve Downstream Tasks
2021-02-10 15:00 - 16:00 | MA
Speaker: Nikunj Saunshi, Dept. of Computer Science, Princeton University
Title: A Mathematical Exploration of Why Language Models Help Solve Downstream Tasks
Abstract: Autoregressive language models pretrained on large corpora have been successful at solving downstream tasks, even with zero-shot usage. However, there is little theoretical justification for their success. This paper considers the following questions: (1) Why should learning the distribution of natural language help with downstream classification tasks? (2) Why do features learned using language modeling help solve downstream tasks with linear classifiers? For (1), we hypothesize, and verify empirically, that classification tasks of interest can be reformulated as next word prediction tasks, thus making language modeling a meaningful pretraining task. For (2), we analyze properties of the cross-entropy objective to show that eps-optimal language models in cross-entropy (log-perplexity) learn features that are O(sqrt{eps}) good on such linear classification tasks, thus demonstrating mathematically that doing well on language modeling can be beneficial for downstream tasks. We perform experiments to verify assumptions and validate our theoretical results. Our theoretical insights motivate a simple alternative to the cross-entropy objective that performs well on some linear classification tasks.

A Mathematical Language
2021-02-24 15:00 - 16:00 | MA
Speaker: Thomas Hales, Univ. of Pittsburgh Dept. of Mathematics
Title: A Mathematical Language
Abstract: A controlled natural language for mathematics is an artificial language that is designed in an explicit way with precise computer-readable syntax and semantics. It is based on a single natural language (which for us is English) and can be broadly understood by mathematically literate English speakers. This talk will describe the design of a controlled natural language for mathematics that has been influenced by the Lean theorem prover, by TeX, and by earlier controlled natural languages. The semantics are provided by dependent type theory.

Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models
2021-03-03 15:00 - 16:00 | Virtual
Speaker: Jason Rute, CIBO Technologies
Title: Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models
Abstract: Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT ({bf P}roof {bf A}rtifact {bf C}o-{bf T}raining), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32% to 48%.

The Ramanujan Machine: Using Algorithms for the Discovery of Conjectures on Mathematical Constants
2021-03-10 15:00 - 16:00 | Virtual
Speaker: Ido Kaminer, Technion – Israel Institute of Technology, Faculty of Electrical Engineering
Title: The Ramanujan Machine: Using Algorithms for the Discovery of Conjectures on Mathematical Constants
Abstract: In the past, new conjectures about fundamental constants were discovered sporadically by famous mathematicians such as Newton, Euler, Gauss, and Ramanujan. The talk will present a different approach – a systematic algorithmic approach that discovers new mathematical conjectures on fundamental constants. We call this approach “the Ramanujan Machine”. The algorithms found dozens of well-known formulas as well as previously unknown ones, such as continued fraction representations of π, e, Catalan’s constant, and values of the Riemann zeta function. Part of the conjectures were in retrospect simple to prove, whereas others remained so far unproved. We will discuss these puzzles and wider open questions that arose from this algorithmic investigation – specifically, a newly-discovered algebraic structure that seems to generalize all the known formulas and connect between fundamental constants. We will also discuss two algorithms that proved useful in finding conjectures: a variant of the meet-in-the-middle algorithm and a gradient descent algorithm tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values; consequently, they conjecture formulas without providing proofs or requiring prior knowledge of the underlying mathematical structure. This way, our approach reverses the conventional usage of sequential logic in formal proofs; instead, using numerical data to unveil mathematical structures and provide leads to further mathematical research.

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.
