Abstract: Hel Braun was a mathematician whose substantial work continues to impact research today.  Braun's research contributions lie in three areas: classical number theory problems about integers, modular and automorphic forms, and Jordan algebras.  I will introduce how each of these seemingly distinct topics led Braun naturally to the next, and I will highlight the ongoing impact of some of Braun's most significant contributions.  Reasons for interest today are largely different from those that motivated Braun and were likely unforeseen at the time.  I will also try to address apparent anomalies, like why you probably are unfamiliar with Braun despite the lasting impact of this work.  This colloquium-level talk is intended for a broad mathematical audience.  If your response to the title was "Who's Hel Braun?" this talk is for you.  If you already know who that is but asked "But why Hel Braun?" it is also for you.

Katy Woo, Princeton University

Abstract: We resolve Manin's conjecture for all Châtelet surfaces over Q (surfaces given by equations of the form x^2 + ay^2 = f(z)) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is estimating sums of Fourier coefficients of modular forms along polynomial values.

Langevin algorithms, integral to Markov Chain Monte Carlo methods, are crucial in machine learning, particularly for Bayesian inference in high-dimensional models and addressing challenges in stochastic non-convex optimization prevalent in deep learning. This talk delves into the practical aspects of stochastic Langevin algorithms through three illuminating examples. First, it explores their role in non-convex optimization, focusing on their efficacy in navigating complex landscapes. The discussion then extends to decentralized Langevin algorithms, emphasizing their relevance in distributed optimization scenarios, where data is dispersed across multiple sources. Lastly, the focus shifts to constrained sampling, aiming to sample from a target distribution subject to constraints. In each scenario, we introduce new algorithms with convergence guarantees and showcase their performance and scalability to large datasets through numerical examples.

Shira Gilat, UPenn

Abstract:The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras.  Supernatural matrices are a minimal solution to the equation of unital algebras M_n(X) ∼= X, which we compare to several similar conditions involving cancellation of matrices. This algebra has appeared under various names before, and it generalizes both McCrimmon's deep matrices algebra and m-petal Leavitt path algebra.

Mean-field games (MFGs) model non-cooperative games among large populations of agents and are widely applied in areas such as traffic flow, finance, and epidemic control. Inverse mean-field games address the challenge of inferring environmental factors from observed agent behavior. The coupled forward-backward structure of MFG equations makes solving these problems difficult and adds even greater complexity to their inverse problems. In this talk, I will introduce a policy iteration method for solving inverse MFGs. This method simplifies the problem by decoupling it into solving linear PDEs and linear inverse problems, leading to significant computational efficiency. The approach is flexible, accommodating a variety of numerical methods and machine learning tools. I will also present theoretical results that guarantee the convergence of our proposed method, along with numerical examples demonstrating its accuracy and efficiency.