Language Modeling for Mathematical Reasoning
2021-01-20 15:00 - 16:00 (America/New_York) | 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.

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

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.

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

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

Knowledge graph representation: From recent models towards a theoretical understanding
2021-01-27 15:00 - 16:00 | Massachusetts
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.

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

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

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

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

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

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

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

A Mathematical Exploration of Why Language Models Help Solve Downstream Tasks
2021-02-10 15:00 - 16:00 | Massachusetts
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.

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

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

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.

Global Anomalies on the Hilbert Space
2021-02-17 10:30 - 12:00 | 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.

2/18/2021 Quantum Matter Seminar
2021-02-18 10:30 - 12:00 | Virtual
Speaker: Xiao-Gang Wen (MIT)
Title: A solution to the chiral fermion problem
Abstract: Motivated by the relation between anomaly and topological/SPT order in one higher dimension, we propose a solution to the chiral fermion problem. In particular, we find several sufficient conditions, such that a chiral fermion field theory can be regularized by an interacting lattice model in the same dimension. We also discuss some related issues, such as mass without mass term, and why ‘topological’ phase transitions are usually not “topological” phase transitions.

Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces
2021-02-22 10:00 - 11:00 | Massachusetts
Speaker: Yu-Shen Lin (Boston University)
Title: Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces
Abstract: Strominger-Yau-Zaslow conjecture predicts the existence of special Lagrangian fibrations on Calabi-Yau manifolds. The conjecture inspires the development of mirror symmetry while the original conjecture has little progress. In this talk, I will confirm the conjecture for the complement of a smooth anti-canonical divisor in del Pezzo surfaces. Moreover, I will also construct the dual torus fibration on its mirror. As a consequence, the special Lagrangian fibrations detect a non-standard semi-flat metric and some Ricci-flat metrics that don’t obviously appear in the literature. This is based on a joint work with T. Collins and A. Jacob.

CMSA Math-Science Literature Lecture: Homological (homotopical) algebra and moduli spaces in Topological Field theories
2021-02-23 09:00 - 10:30 | Virtual
Kenji Fukaya (Simons Center for Geometry and Physics)
Title: Homological (homotopical) algebra and moduli spaces in Topological Field theories
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.
Talk chair: Peter Kronheimer
Slides | Video

2/24/2021 Quantum Matter Seminar
2021-02-24 10:30 - 12:00 | Massachusetts

2/23/2021 Computer Science for Mathematicians
2021-02-24 11:30 - 12:30 | Virtual

A Mathematical Language
2021-02-24 15:00 - 16:00 | Massachusetts
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.

2/25/2021 Quantum Matter Seminar
2021-02-25 10:30 - 12:00 | Massachusetts

Mathematical supergravity and its applications to differential geometry
2021-03-01 10:00 - 11:00 | Hybrid
Speaker: Carlos S. Shahbazi (Hamburg University)
Title: Mathematical supergravity and its applications to differential geometry
Abstract: I will discuss the recent developments in the mathematical theory of supergravity that lay the mathematical foundations of the universal bosonic sector of four-dimensional ungauged supergravity and its Killing spinor equations in a differential-geometric framework. I will provide the necessary context and background. explaining the results pedagogically from scratch and highlighting several open mathematical problems which arise in the mathematical theory of supergravity, as well as some of its potential mathematical applications. Work in collaboration with Vicente Cortés and Calin Lazaroiu.

3/2/2021 Computer Science for Mathematicians
2021-03-02 11:30 - 12:30 | Massachusetts

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

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%.

3/4/2021 Quantum Matter Seminar
2021-03-04 10:30 - 12:00 | Massachusetts
