Group Actions, Homogeneous Dynamics, and Number Theory

Wesleyan University, Middletown CT

Investigators

Abstract

Many questions in mathematics come with a set of symmetries that has a so-called group structure. Exploring the dynamics of underlying group actions, namely to study how points in spaces behave under symmetries, has recently played a central role in proving important results and resolving longstanding questions in mathematics. The overall aim of this research project is to advance the technique of using dynamics of group actions as a powerful tool to study questions that arise in algebra, geometry, and number theory, and to gain a deeper understanding of the connections between these mathematical fields. In this project special attention will be given to the following three directions: (1) Develop reduction theory of indefinite integral quadratic forms and consider its generalization to number fields using methods of dynamical systems. (2) Investigate random walks on Lie groups and equidistribution problems in homogenous spaces in light of recent developments in spectral theory and the theory of automorphic forms. (3) Study the height bounds of generators of arithmetic groups incorporating dynamical methods into classical algebraic and geometric theory of arithmetic groups. The project also aims to seek new interactions and develop the techniques in the study of groups, dynamics, and number theory.