CAREER: Mixing and Equidistribution in Number Theory and Geometry

Dynamics is the study of the evolution of a system under a transformation rule governing its behavior over time. It encompasses such varied examples as planetary motion, the spread of disease and the flow of electric currents in conductive material. It turns out that many fundamental problems in number theory and geometry can also be understood in terms of the long-time behavior of certain dynamical systems of algebraic origin. Furthermore, the algebraic nature of these systems makes it possible to employ tools from a wide array of mathematical disciplines for their investigation. This project aims to develop new methods in the theory of algebraic dynamical systems with the goal of resolving central questions in the fields of Diophantine approximation and surface geometry. The educational component aims at training early-career mathematicians and providing mentorship to students at all levels. This includes a workshop geared towards training graduate students and postdocs on active research directions in dynamics, as well as outreach workshops aimed at encouraging students to pursue careers in the mathematical sciences. The research program has three interrelated goals. One goal is to study the distribution of rational points near self-similar sets from the perspective of homogeneous dynamics. The PI will also investigate limits of horocycle-invariant measures on moduli spaces of Abelian differentials, A third aim is to establish rates of mixing of geodesic flows on negatively curved manifolds. Progress on these questions will involve development of dynamical methods at the intersection of representation theory, geometry, and additive combinatorics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.