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.

This talk will be aimed at a general audience and will assume no prior dynamics knowledge.

Graduate students: there will be no refreshments; despite that, you should attend the talk