A Spectral Approach to Counting and Equidistribution

Abstract: Since the early 20th century, spectral methods have been used to obtain effective counting theorems for various objects of interest in number theory, geometry and group theory. In this talk I’ll start by introducing two classical problems: the Gauss circle problem, and the Apollonian counting problem. By surveying results on these problems (and some generalizations), I’ll demonstrate how to use spectral methods to obtain effective asymptotics for some very classical problems. Then I will try and explain how to generalize this method to apply to certain horospherical equidistribution theorems.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar