Osama Khalil
University of Illinois Chicago
Title
Exponential mixing via additive combinatorics
Abstract
A guiding principle in additive combinatorics is that failures of randomness are typically accounted for by rigid algebraic structures. This perspective has proved powerful in examining the distribution of sets in a wide range of settings, such as integer points near varieties, prime numbers, or orbits of dynamical systems. In this context, randomness is reflected in the smoothness of the natural measures these sets support. The resulting dichotomy for a given measure takes the following form: either typical Fourier coefficients decay polynomially with the frequency, or a substantial portion of the mass aligns with linear subspaces across many scales. In this talk, I will explain how this viewpoint has recently led to a new approach to studying exponential mixing for geodesic flows and to related problems at the interface of dynamics, geometry, and harmonic analysis.