CMSA

Global Anomalies on the Hilbert Space
2021-02-17 10:30 - 12:00 (America/New_York) | Virtual
Speaker: Jaume Gomis (Perimeter PI)
Title: Global Anomalies on the Hilbert Space
Abstract: We will discuss an elementary way of detecting some global anomalies from the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory, give a physical description of the imprint of the “layers” that enter in the cobordism classification of anomalies and discuss applications, including how anomalies can imply a supersymmetric spectrum in strongly coupled (nonsupersymmetric) gauge theories.

2/16/2021 Computer Science for Mathematicians
2021-02-16 13:30 - 14:30 | Virtual
Speaker: Michael P. Kim (UC Berkeley)
Title: Outcome Indistinguishability
Abstract: Prediction algorithms assign numbers to individuals that are popularly understood as individual “probabilities” — e.g., what is the probability of 5-year survival after cancer diagnosis? — and which increasingly form the basis for life-altering decisions. The understanding of individual probabilities in the context of such unrepeatable events has been the focus of intense study for decades within probability theory, statistics, and philosophy. Building off of notions developed in complexity theory and cryptography, we introduce and study Outcome Indistinguishability (OI). OI predictors yield a model of probabilities that cannot be efficiently refuted on the basis of the real-life observations produced by Nature.
We investigate a hierarchy of OI definitions, whose stringency increases with the degree to which distinguishers may access the predictor in question. Our findings reveal that OI behaves qualitatively differently than previously studied notions of indistinguishability. First, we provide constructions at all levels of the hierarchy. Then, leveraging recently-developed machinery for proving average-case fine-grained hardness, we obtain lower bounds on the complexity of the more stringent forms of OI. The hardness result provides scientific grounds for the political argument that, when inspecting algorithmic risk prediction instruments, auditors should be granted oracle access to the algorithm, not simply historical predictions.
Joint work with Cynthia Dwork, Omer Reingold, Guy N. Rothblum, Gal Yona; to appear at STOC 2021.

2/15/2021 Math Physics Seminar
2021-02-15 10:00 - 11:00 | MA

2/11/2021 Quantum Matter Seminar
2021-02-11 10:30 - 12:00 | MA

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.

2/10/2021 Strongly Correlated Quantum Materials
2021-02-10 10:00 - 12:00 | Virtual

2/9/2021 Computer Science for Math
2021-02-09 11:30 - 12:30 | MA

2/4/2021 Quantum Matter Seminar
2021-02-04 10:30 - 12:00 | MA

2/3/2021 Strongly Correlated Quantum Materials
2021-02-03 10:30 - 12:00 | MA

2/2/2021 Computer Science for Mathematicians
2021-02-02 11:30 - 12:30 | MA

2/1/2021 Math Physics
2021-02-01 10:00 - 11:00 | MA

1/28/2021 Quantum Matter Seminar
2021-01-28 10:30 - 2021-01-29 00:00 | Virtual

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.

1/27/2021 Quantum Matter Seminar
2021-01-27 10:30 - 12:00 | MA

CMSA Math-Science Literature Lecture: Discrepancy Theory and Randomized Controlled Trials
2021-01-27 09:00 - 10:30 | Virtual
Dan Spielman (Yale University)
Title: Discrepancy Theory and Randomized Controlled Trials
Abstract: Discrepancy theory tells us that it is possible to partition vectors into sets so that each set looks surprisingly similar to every other. By “surprisingly similar” we mean much more similar than a random partition. I will begin by surveying fundamental results in discrepancy theory, including Spencer’s famous existence proofs and Bansal’s recent algorithmic realizations of them. Randomized Controlled Trials are used to test the effectiveness of interventions, like medical treatments. Randomization is used to ensure that the test and control groups are probably similar. When we know nothing about the experimental subjects, uniform random assignment is the best we can do. When we know information about the experimental subjects, called covariates, we can combine the strengths of randomization with the promises of discrepancy theory. This should allow us to obtain more accurate estimates of the effectiveness of treatments, or to conduct trials with fewer experimental subjects. I will introduce the Gram-Schmidt Walk algorithm of Bansal, Dadush, Garg, and Lovett, which produces random solutions to discrepancy problems. I will then explain how Chris Harshaw, Fredrik Sävje, Peng Zhang, and I use this algorithm to improve the design of randomized controlled trials. Our Gram-Schmidt Walk Designs have increased accuracy when the experimental outcomes are correlated with linear functions of the covariates, and are comparable to uniform random assignments in the worst case.
Talk chair: Salil Vadhan
Video

Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants for algebraic surfaces
2021-01-25 08:00 - 09:00 | Virtual
During the Spring 2021 Semester Artan Sheshmani (CMSA/ I.M. A.U.) will be teaching a CMSA special lecture series on Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants for algebraic surfaces.
In order to attend this series, please fill out this form.
The lectures will be held Mondays from 8:00 – 9:30 AM ET and Wednesdays from 8:00 – 9:00 AM ET beginning January 25 on Zoom.
You can watch Prof. Sheshmani describe the series here.

1/21/2021 Quantum Matter
2021-01-21 08:30 - 10:00 | MA

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.

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.

CMSA Math-Science Literature Lecture: Quantum topology and new types of modularity
2021-01-13 09:00 - 10:30 | Virtual
Don Zagier (Max Planck Institute for Mathematics and International Centre for Theoretical Physics)
Title: Quantum topology and new types of modularity
Abstract: The talk concerns two fundamental themes of modern 3-dimensional topology and their unexpected connection with a theme coming from number theory. A deep insight of William Thurston in the mid-1970s is that the vast majority of complements of knots in the 3-sphere, or more generally of 3-manifolds, have a unique metric structure as hyperbolic manifolds of constant curvature -1, so that 3-dimensional topology is in some sense not really a branch of topology at all, but of differential geometry. In a different direction, the work of Vaughan Jones and Ed Witten in the late 1980s gave rise to the field of Quantum Topology, in which new types of invariants of knot complements and 3-manifolds are introduced that have their origins in ideas coming from quantum field theory. These two themes then became linked by Kashaev’s famous Volume Conjecture, now some 25 years old, which says that the Kashaev invariant _N of a hyperbolic knot K (this is a quantum invariant defined for each positive integer N and whose values are algebraic numbers) grows exponentially as N tends to infinity with an exponent proportional to the hyperbolic volume of the knot complement. About 10 years ago, I was led by numerical experiments to the discovery that Kashaev’s invariant could be upgraded to an invariant having rational numbers as its argument (with the original invariant being the value at 1/N) and that the Volume Conjecture then became part of a bigger story saying that the new invariant has some sort of strange transformation property under the action x -> (ax+b)/(cx+d) of the modular group SL(2,Z) on the argument. This turned out to be only the beginning of a fascinating and multi-faceted story relating quantum invariants, q-series, modularity, and many other topics. In the talk, which is intended for a general mathematical audience, I would like to recount some parts of this story, which is joint work with Stavros Garoufalidis (and of course involving contributions from many other authors). The “new types of modularity” in the title refer to a specific byproduct of these investigations, namely that there is a generalization of the classical notion of holomorphic modular form – which plays an absolutely central role in modern number theory – to a new class of holomorphic functions in the upper half-plane that no longer satisfy a transformation law under the action of the modular group, but a weaker extendability property instead. This new class, called “holomorphic quantum modular forms”, turns out to contain many other functions of a more number-theoretical nature as well as the original examples coming from quantum invariants.
Talk chair: Mark Kisin
Video

A universal triangulation for flat tori
2021-01-13 09:00 - 10:00 | Virtual
Speaker:Francis Lazarus, CNRS / Grenoble University
Title: A universal triangulation for flat tori
Abstract: A celebrated theorem of Nash completed by Kuiper implies that every smooth Riemannian surface has a C¹ isometric embedding in the Euclidean 3-space E³. An analogous result, due to Burago and Zalgaller, states that every polyhedral surface, obtained by gluing Euclidean triangles, has an isometric PL embedding in E³. In particular, this provides PL isometric embeddings for every flat torus (a quotient of E² by a rank 2 lattice). However, the proof of Burago and Zalgaller is partially constructive, relying on the Nash-Kuiper theorem. In practice, it produces PL embeddings with a huge number of vertices, moreover distinct for every flat torus. Based on a construction of Zalgaller and on recent works by Arnoux et al. we exhibit a universal triangulation with less than 10.000 vertices, admitting for any flat torus an isometric embedding that is linear on each triangle. Based on joint work with Florent Tallerie.

12/23/2020 Quantum Matter Seminar
2020-12-22 10:30 - 12:00 | MA

12/17/2020 Strongly Correlated Quantum Materials
2020-12-17 10:30 - 12:00 | MA

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

12/16/2020 Strongly Correlated Quantum Materials
2020-12-16 10:30 - 12:00 | MA

12/15/2020 Computer Science for Math
2020-12-15 11:30 - 12:30 | MA

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

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.

12/9/2020 Strongly Correlated Quantum Materials
2020-12-09 10:30 - 12:00 | MA

12/7/2020 Math Physics Seminar
2020-12-07 10:30 - 11:30 | MA
