Gaussian multiplicative chaos (GMC) is a well-studied random measure appearing as a universal object in the study of Gaussian or approximately Gaussian log-correlated fields. On the other hand, no general framework exists for the study of multiplicative chaos associated to non-Gaussian log-correlated fields. In this talk, we examine a canonical model: the log-correlated random Fourier series, or random wave model, with i.i.d. random coefficients taken from a general class of distributions. The associated multiplicative chaos measure was shown to be non-degenerate when the inverse temperature is subcritical ($\gamma < \sqrt{2d}$) by Junnila. The resulting chaos is easily seen to not be a GMC in general, leaving open the question of what properties are shared between this non-Gaussian chaos and GMC. We answer this question through the lens of absolute continuity, showing that there exists a coupling between this chaos and a GMC such that the two are almost surely mutually absolutely continuous.

This talk will present new results on the optimal control of self-organization, motivated by a growing body of empirical work on biological pattern formation in dynamic environments. We pose a boundary control problem for the classic supercritical Turing pattern, asking the best way to reach a non-trivial steady state by controlling the boundary flux of a reactant species. Via the Pontryagin approach, first-order optimality conditions for a generic reaction-diffusion system with a suitable bifurcation structure are derived. Using formal asymptotics, we construct approximate closed-form optimal solutions in feedback law form that are valid for any Turing-unstable system near criticality, which are verified against numerical solutions for a representative reaction-diffusion model.

Algebraic curves and abelian varieties play a central role in modern algebraic geometry, with links to complex analysis, number theory, topology and others. Curves and abelian varieties are closely related: a fundamental example of an abelian variety is the Jacobian of an algebraic curve. In this talk, I will give a discussion of curves, abelian varieties and their moduli spaces. Time permitting, I will present some new tools aimed at studying geometric classes on the moduli space of abelian varieties, and conclude with a discussion of several open questions in this area. 

The study of p-adic Lie groups and their representations is a central piece of the p-adic Langlands program.  One tool which is used to study these is the notion of a saturated pro-p group, and the famous result of Lazard which states that every p-adic Lie group contains an open saturable subgroup.  In this talk, we will demonstrate a family of open saturated subgroups of G(F) for G a reductive group over a p-adic field F, which is indexed by the semisimple Bruhat-Tits building of G, given a mild assumption on G.  We will then review some group-theoretic consequences of this result.

The exchange property, introduced by Crawley and Jonsson in 1964 in the study of direct decompositions of algebraic systems and later extended to modules and rings by Warfield, plays a central role in modern decomposition theory. One of the main open problems in the area is whether the finite exchange property implies the full exchange property. This talk surveys the development of exchange theory from its module-theoretic origins to its ring-theoretic formulation via exchange rings. The last part of the talk is based on joint work with A. Cigdem Ozcan and focuses on lifting theory, including idempotent, regular, and unit lifting ideals and morphisms, and their interaction with local morphisms.

We discuss construction of a derived Lagrangian intersection theory of moduli spaces of perfect complexes, with support on divisors on compact Calabi-Yau threefolds. Our goal is to compute deformation invariants associated to a fixed linear system of divisors in CY3. We apply a Tyurin degeneration of the CY3 into a normal-crossing singular variety composed of Fano threefolds meeting along their anti-canonical divisor. We show that the moduli space over the Fano 4 fold given by total space of the degeneration family satisfies a relative Lagrangian foliation structure which leads to realizing the moduli space as derived critical locus of a global (-1)-shifted potential function. We construct a flat Gauss-Manin connection to relate the periodic cyclic homology induced by matrix factorization category of such function to the derived Lagrangian intersection of the corresponding “Fano moduli spaces”. The latter provides one with categorification of DT invariants over the special fiber (of degenerating family). The alternating sum of dimensions of the categorical DT invariants of the special fiber induces numerical DT invariants. If there is time, we show how in terms of “non-derived” virtual intersection theory, these numerical DT invariants relate to counts of D4-D2-D0 branes which are expected to have modularity property by the S-duality conjecture. This talk is based on joint work with Jacob Krykzca.

The mathematical core of deep learning is function approximation by neural networks trained on data using stochastic gradient descent. I will explain an emerging geometric framework for the analysis of this process. This includes a collection of rigorous results on training dynamics for the deep linear network (DLN) as well as general principles for arbitrary neural networks. The mathematics ranges over a surprisingly broad range, including geometric invariant theory, random matrix theory, and minimal surfaces. However, little background in these areas will be assumed and the talk will be accessible to a broad audience. The talk is based on joint work with several co-authors: Yotam Alexander, Nadav Cohen (Tel Aviv), Kathryn Lindsey (Boston College), Alan Chen, Zsolt Veraszto and Tianmin Yu (Brown).

TBA

In this talk, we explore how data scientists in industry are using modern AI tools such as ChatGPT to write code and perform mathematical reasoning. Chris Deotte is a Senior Data Scientist at NVIDIA, a seven-time Kaggle Grandmaster, and holds a PhD in mathematics.

In recent years, data scientists and mathematicians have increasingly shifted from writing all code and derivations by hand to collaborating with AI assistants such as ChatGPT, Claude, and Gemini. These tools are now capable of generating high-quality code, solving mathematical problems, and accelerating research and development workflows.

We will examine concrete examples of how these AI tools perform on real-world coding and mathematical tasks. In particular, we will demonstrate how ChatGPT recently wrote over 99% of the code for a gold-medal-winning solution in an online competition focused on predicting mouse behavior from keypoint time-series data.