Dynamical Methods in Counting Questions and Diophantine Approximation

Dynamical systems model such varied examples as planetary motion, spread of disease and the flow of electric currents in conductive material. The area of mathematical dynamical systems consists of the study of the evolution over time of a system under a transformation rule that governs the behavior of the system. Methods from the study of dynamical systems, when applied to systems of algebraic origin and admitting a lot of symmetries, shed light, perhaps strikingly, on some of the oldest and most well-studied questions in mathematics, namely solutions in whole numbers, or integers, of polynomial equations, and approximations of arbitrary real numbers by fractions. In turn, developments in number theory, driven by dynamical methods, have had interesting and surprising applications, reaching recently as far as impacting wireless communication technologies. This project aims to deepen these fruitful connections between dynamics and number theory by, in particular, developing new methods for counting the number of integer solutions to certain types of highly symmetric polynomial equations. An important component of this project is geared towards training graduate students in this area, at the intersection of dynamical systems and number theory, through research and professional mentoring. The goal of this project is fourfold: 1) develop methods in homogeneous dynamics to resolve outstanding questions regarding the distribution of rational points near manifolds and self-similar sets; 2) develop techniques in the theory of random walks and linear representations of algebraic groups aimed at studying counting problems of integral points on affine homogeneous varieties; 3) develop spectral tools for the study of the dynamics of the Kontsevich-Zorich cocycle over the Teichmueller geodesic flow, along with applications to rigidity problems for horocycle flows on moduli spaces of Abelian differentials; 4) develop new methods for the study of the mixing properties of the geodesic flow on infinite volume locally symmetric spaces of negative curvature. 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.