CMSA

CMSA Math-Science Literature Lecture: Homotopy spectra and Diophantine equations
2020-11-20 08:00 - 09:30 (America/New_York) | Virtual
Yuri Manin (Max Planck Institute for Mathematics)
Title: Homotopy spectra and Diophantine equations
Abstract: For a long stretch of time in the history of mathematics, Number Theory and Topology formed vast, but disjoint domains of mathematical knowledge. Origins of number theory can be traced back to the Babylonian clay tablet Plimpton 322 (about 1800 BC) that contained a list of integer solutions of the “Diophantine” equation $a^2+b^2=c^2$: archetypal theme of number theory, named after Diophantus of Alexandria (about 250 BC). Topology was born much later, but arguably, its cousin — modern measure theory, — goes back to Archimedes, author of Psammites (“Sand Reckoner”), who was approximately a contemporary of Diophantus. In modern language, Archimedes measures the volume of observable universe by counting the number of small grains of sand necessary to fill this volume. Of course, many qualitative geometric models and quantitative estimates of the relevant distances precede his calculations. Moreover, since the estimated numbers of grains of sand are quite large (about $10^{64}$), Archimedes had to invent and describe a system of notation for large numbers going far outside the possibilities of any of the standard ancient systems. The construction of the first bridge between number theory and topology was accomplished only about fifty years ago: it is the theory of spectra in stable homotopy theory. In particular, it connects $Z$, the initial object in the theory of commutative rings, with the sphere spectrum $S$. This connection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory. In this talk based upon the authors’ (Yu. Manin and M. Marcolli) joint research project, I suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds.
Talk chair: Michael Hopkins
Slides | Video

11/19/20 Strongly Correlated Quantum Materials
2020-11-19 11:30 - 12:00 | MA

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.

11/18/2020 RMPT
2020-11-18 14:00 - 15:00 | MA

11/18/2020 Strongly Correlated Quantum Materials
2020-11-18 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.)

CMSA Math-Science Literature Lecture: Log Calabi-Yau fibrations
2020-11-18 08:00 - 09:30 | Virtual
Caucher Birkar (University of Cambridge)
Title: Log Calabi-Yau fibrations
Abstract: Fano and Calabi-Yau varieties play a fundamental role in algebraic geometry, differential geometry, arithmetic geometry, mathematical physics, etc. The notion of log Calabi-Yau fibration unifies Fano and Calabi-Yau varieties, their fibrations, as well as their local birational counterparts such as flips and singularities. Such fibrations can be examined from many different perspectives. The purpose of this talk is to introduce the theory of log Calabi-Yau fibrations, to remind some known results, and to state some open problems.
Video

11/17/2020 Computer Science for Mathematicians
2020-11-17 11:30 - 12:30 | MA

11/16/2020 Mathematical Physics Seminar
2020-11-16 10:00 - 11:00 | MA

CMSA Math-Science Literature Lecture: Classical and quantum integrable systems in enumerative geometry
2020-11-16 08:00 - 09:30 | Virtual
Andrei Okounkov (Columbia University)
Title: Classical and quantum integrable systems in enumerative geometry
Abstract: For more than a quarter of a century, thanks to the ideas and questions originating in modern high-energy physics, there has been a very fruitful interplay between enumerative geometry and integrable system, both classical and quantum. While it is impossible to summarize even the most important aspects of this interplay in one talk, I will try to highlight a few logical points with the goal to explain the place and the role of certain more recent developments.
Talk chair: Cumrun Vafa
Video

CMSA Math-Science Literature Lecture: Knot Invariants From Gauge Theory in Three, Four, and Five Dimensions
2020-11-13 08:00 - 09:30 | Virtual
Edward Witten (IAS)
Title: Knot Invariants From Gauge Theory in Three, Four, and Five Dimensions
Abstract: I will explain connections between a sequence of theories in two, three, four, and five dimensions and describe how these theories are related to the Jones polynomial of a knot and its categorification.
Talk chair: Cliff Taubes
Video

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

11/11/2020 RM&PT Seminar
2020-11-11 15:00 - 16:00 | MA

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.

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

11/10/2020 Computer Science for Mathematicians
2020-11-10 11:30 - 12:30 | MA

11/6/2020 Strongly Correlated Quantum Materials Lecture
2020-11-06 12:30 - 15:00 | MA

11/05/2020 Quantum Matter Seminar
2020-11-05 10:30 - 12:00 | MA

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.

11/04/2020 RMPT Seminar
2020-11-04 14:00 - 15:00 | MA

11/3/2020 Computer Science for Mathematicians
2020-11-03 11:30 - 12:30 | MA

11/2/2020 Math-Physics Seminar
2020-11-02 10:30 - 11:30 | MA

11/9/2020 Math-Physics Seminar
2020-11-02 10:30 - 11:30 | MA

10/29/2020 Quantum Matter Seminar
2020-10-29 10:30 - 12:00 | MA

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.

10/28/2020 RM&PT seminar
2020-10-28 14:00 - 15:00 | MA

10/28/2020 Strongly Correlated Quantum Materials
2020-10-28 10:30 - 12:00 | MA

10/27/2020 Computer Science for Mathematicians
2020-10-27 11:30 - 12:30 | MA

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

10/21/2020 Quantum Matter Seminar
2020-10-21 10:30 - 12:00 | MA
