I will report on recent progress on influential conjectures from the 1970s and 1980s (Berry-Tabor, Bohigas-Giannoni-Schmit), which suggest that the spectral statistics of the Laplace-Beltrami operator on a given compact Riemannian manifold should be described either by a Poisson point process or by a random matrix ensemble, depending on whether the  geodesic flow is integrable or “chaotic”. This talk will straddle aspects of analysis, geometry, probability, number theory and ergodic theory, and should be accessible to a broad audience. The two most recent results presented in this lecture were obtained in collaboration with Laura Monk and with Wooyeon Kim and Matthew Welsh. 

The universal infinitesimal symmetry of a holomorphic field theory is the Lie algebra of holomorphic vector fields. We introduce the higher-dimensional Virasoro algebra and prove a local, universal, form of the Riemann–Roch theorem using Feynman diagrams. We use the concept of a (Jouanoulou) higher-dimensional chiral algebra as developed recently with Gui and Wang. We will remark on applications to superconformal field theory. This project is joint work with Zhengping Gui.

We consider particles following a diffusion process in a multi-well potential and attracted by their barycenter (corresponding to the particle approximation of the Wasserstein flow of a suitable free energy). It is well-known that this process exhibits phase transitions: at high temperature, the mean-field limit has a single stationary solution, the N-particle system converges to equilibrium at a rate independent from N and propagation of chaos is uniform in time. At low temperature, there are several stationary solutions for the non-linear PDE, and the limit of the particle system as N and t go to infinity do not commute. We show that, in the presence of multiple stationary solutions, it is still possible to establish local convergence rates for initial conditions starting in some Wasserstein balls (this is a joint work with Julien Reygner). In terms of metastability for the particle system, we also show that for these initial conditions, the exit time of the empirical distribution from some neighborhood of a stationary solution is exponentially large with N and approximately follows an exponential distribution, and that propagation of chaos holds uniformly over times up to this expected exit time (hence, up to times which are exponentially large with N). Exactly at the critical temperature below which multiple equilibria appear, the situation is somewhat degenerate and we can get uniform in N convergence estimates, but polynomial instead of exponential.

Pfaffian functions, and by extension Pfaffian and semi-Pfaffian sets, play a crucial role in various areas of mathematics, including o-minimal theory. Incidence combinatorics has recently experienced a surge of activity, fuelled by the introduction of the polynomial partitioning method of Guth and Katz. While traditionally restricted to simple geometric objects such as points and lines, focus has shifted towards incidence questions involving higher dimensional algebraic or semi-algebraic sets. We present a generalization of the polynomial partitioning method to semi-Pfaffian sets and illustrate how this leads to Pfaffian generalizations of classic results in incidence geometry, such as the Szemerédi-Trotter Theorem. Finally, we outline an application of semi-Pfaffian geometry and Khovanskii's bound to the robustness of neural networks.

The theory of characters for infinite groups, initiated by Thoma, is a natural generalization of the representation theory of finite groups. More precisely, a character on a discrete group is a normalised positive definite function which is conjugation invariant and extremal. Connes conjectured a rigidity result for characters of an important family of discrete groups, namely, irreducible lattices in higher-rank semisimple Lie groups. The conjecture states that every character is either the trace of a finite-dimensional representation, or vanishes off the center. This rigidity property implies the Stuck-Zimmer conjecture for such lattices, namely, ergodic actions are either essentially transitive or essentially free. I will present a recent joint result with Michael Glasner, Yuval Gorfine, Liam Hanany and Arie Levit in which we prove that non-uniform irreducible lattices in higher-rank semisimple groups are character rigid. As a result, we also obtain a resolution of the Stuck-Zimmer conjecture for all non-uniform lattices.

The problem of integration in finite terms is the problem of finding exact closed forms for antiderivatives of functions, within a given class of functions. Liouville introduced his elementary functions (built from polynomials, exponentials, logarithms and trigonometric functions) and gave a solution to the problem for that class, nearly 200 years ago. The same problem was shown to be decidable and an algorithm given by Risch in 1969.

We introduce the class of exponentially-algebraic functions, generalising the elementary functions and much more robust than them, and give characterisations of them both in terms of o-minimal local definability and in terms of their types in a reduct of the theory of differentially closed fields.

We then prove the analogue of Liouville's theorem for these exponentially-algebraic functions and give some new decidability results.

This is joint work with Rémi Jaoui, Lyon