CMSA

Exact symmetries and threshold states in two-dimensional models for QCD
2021-03-17 10:30 - 2021-03-18 12:00 (America/New_York) | 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.

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

3/10/2021 Quantum Matter Seminar
2021-03-10 19:30 - 21:00 | MA

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.

3/9/2021 Computer Science for Mathematicians
2021-03-09 11:30 - 2021-03-10 12:30 | MA

3/8/2021 Math Physics Seminar
2021-03-08 10:00 - 11:00 | MA

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

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%.

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

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

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.

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

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.

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

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

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

Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces
2021-02-22 10:00 - 11:00 | MA
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.

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.

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.

2/16/2021 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
