Algebraic Combinatorics Seminar - Gilyoung Cheong

Title: P\'olya enumeration theorems in algebraic geometry

Abstract:

We will start by comparing Macdonald's formula of the generating function for the symmetric powers of a compact complex manifold and Grothendieck's formula of the zeta series of a projective variety over a finite field, an explicit version of Dwork's rationality result. After seeing a common generalization of the two formulas, we will see how it is related to a classical theorem in combinatorics called the P\'olya enumeration theorem, which has to do with counting colorings of a graph modulo symmetries. If time permits, we will discuss another version of this enumeration theorem with distinct vertices, a geometric analogue of which is a joint work with Yifeng Huang.