Attempting to give a brief answer to the question "What is Number Theory?" inevitably involves a statement equivalent to it being the study of the integers. While this is certainly not wrong, Number Theory starts to feel astronomically far removed from just "studying integers" rather quickly, especially when studying graduate level material and beyond.

 

In this talk, we will (hopefully) bring Number Theory a bit back down to Earth by "studying integers" with some sophisticated mathematical machinery. In particular, we will examine how theta series and Eisenstein series may be used to answer the classical question of how many ways an integer can be represented as a sum of two squares. We will then generalize to higher dimensional theta functions attached to arbitrary lattices, Siegel-Eisenstein series, and how these fancy gizmos help us "study integers."