Consider a finite modular tensor category $\mathcal A$. In [DGGPR] the authors exhibit a 3-dimensional topological field theory  $Z_{\mathcal A}: \operatorname{Bord}_{\mathcal A} \to \operatorname{Vect}$, which, in the case where $\mathcal A$ is semisimple, recovers the usual Reshetikhin-Turaev TQFT. In the present work we show that this extends naturally to a TQFT $Z_{\operatorname{Ch}(\mathcal A)}$, which takes values in the symmetric tensor category $\operatorname{Ch(Vect)}$ of linear cochains. This cochain valued theory furthermore respects (certain classes of) homotopies.

[DGGPR] M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, and I. Runkel. 3-dimensional TQFTs from non-semisimple modular categories. Sel. Math. New Ser., 28(2):42, 2022.

In 1944, von Neumann and Morgenstern raised a question in their famous book Theory of Games and Economic Behavior: For a rational agent with preferences over all probabilistic mixtures of finitely many deterministic outcomes, is there always a unique utility function on deterministic outcomes whose expected value on probabilistic mixtures represents the preferences? Under natural assumptions on the preference order, they answered the question positively. We attempt to generalize this result to the case when the outcomes are infinite. We first identify the outcomes with the extreme points of a Choquet simplex, a natural generalization of the classical simplex to infinite-dimensional spaces. We then prove a similar result in the setting of Choquet simplex.

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.

We will discuss recent progress on the corners problem from additive combinatorics and its connection to communication complexity. In particular, we will sketch a proof that every set $A \subseteq [N]^2$ of density $\gg exp(- polylog N)$ must contain three points of the form $(x,y)$, $(x+d,y)$, $(x,y+d)$ for some nonzero $d$. We will also show how this result implies strong lower bounds for the multiparty communication complexity of the Exactly-$N$ function.

Based on joint work with Michael Jaber, Yang P. Liu, Shachar Lovett, and Mehtaab Sawhney.

An IRS is a conjugacy-invariant probability measure on the space of subgroups of a locally compact group. We are interested in those IRS that give full measure to the set of compact subgroups. This talk is about what we know about those, how it connects to the structure theory of locally compact groups, and what we would still like to figure out.

Joint with Tal Cohen, Helge Glöckner and Gil Goffer.

This study considers simultaneous inference of economic time series models for potential policy implications. The process of interest is an unknown function of either time or an observable random vector in macro-finance. To overcome the well-known slow convergence issue with the traditional asymptotic-based approach, we utilize a Gaussian approximation for our time-dependent processes. Relevant theories and finite-sample simulations justify our approach. The empirical applications include the U.S. Phillips curve in macroeconomics and the forward premium puzzle in international finance.