Georgia Institute of Technology College of Sciences

An Anosov representation of a hyperbolic group $\Gamma$ is a representation which quasi-isometrically embeds $\Gamma$ into a semisimple Lie group - say, SL(d, R) - in a way which imitates and generalizes the behavior of a convex cocompact group acting on a hyperbolic metric space. It is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, I will discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu, which shows that every hyperbolic group that acts geometrically on a CAT(0) cube complex admits a 1-Anosov representation into SL(d, R) for some d. Mainly, the proof exploits the relationship between the combinatorial/CAT(0) geometry of right-angled Coxeter groups and the projective geometry of a convex domain in real projective space on which a Coxeter group acts by reflections.

The aim of this talk is to discuss recent advances around the convergence problem in mean field control theory and the study of associated nonlinear PDEs.

We are interested in optimal control problems involving a large number of interacting particles and subject to independent Brownian noises. As the number of particles tends to infinity, the problem simplifies into a McKean-Vlasov type optimal control problem for a typical particle. I will present recent results concerning the quantitative analysis of this convergence. More precisely, I will discuss an approach based on the analysis of associated value functions. These functions are solutions of Hamilton-Jacobi equations in high dimension and the convergence problem translates into a stability problem for the limit equation which is posed on a space of probability measures.

I will also discuss the well-posedness of this limiting equation, the study of which seems to escape the usual techniques for Hamilton-Jacobi equations in infinite dimension.

Equation learning has been a holy grail of scientific research for decades. Only recently has the capability of learning equations directly from data become a computationally feasible task, due to the availability of high-resolution data and fast algorithms capable of surpassing the inherent combinatorial complexity of most model classes. Weak form equation learning has arisen as an advantageous framework for efficiently selecting models from data with noise and nonsmoothness, qualities inherent to observed data. By viewing the dynamics through the guise of test functions, the weak form affords a flexible representation of the governing equations that naturally incorporates these data maladies. More generally, the weak form has been shown to reveal alternative dynamical descriptions, such as coarse-grained and reduced-order models, opening the door to hierarchical model discovery. In this talk I will give a broad overview of historical advances in weak form equation learning and parameter inference, from the 1950s to WSINDy and more recent algorithms. I will then give an outlook for future research directions in this field, in light of now-known computational limitations and recently demonstrated successes, both theoretical and applied, with applications to molecular dynamics, plasma physics, cell biology, and weather forecasting.

It is a fundamental question in Ramsey theory to determine the smallest possible independence number of an H-free hypergraph on n vertices. In the case of graphs, the problem was famously solved for H=K3 by Kim and for H=K4 (up to a logarithmic factor) by Mattheus-Verstraete in 2023. Even C4 and K5 remain wide open. We study the problem for 3-uniform hypergraphs and conjecture a full classification: the minimum independence number is poly(n) if and only if H is contained in the iterated blowup of the single-edge hypergraph. We prove this conjecture for all H with at most two tightly connected components. Based on joint work with Conlon, Fox, Gunby, Mubayi, Suk, Verstraete, and Yu.