Mikhail Mazin, Kansas State University

Title: Bijectivity of zeta map on rational parking functions via Brouwer fixed point theorem

Abstract: Let (m,n) be a pair of relatively prime positive integers. A function f:{1,...,n}->{0,...,m-1} is called an n/m-parking function if for every i from 1 to m the number of j such that f(i) is less than j is greater than or equal to in/m. Zeta is a map from the set of n/m-parking functions to itself. The bijectivety of zeta was first conjectured by myself, Eugene Gorsky, and Monica Vazirani in 2014. It was proved by Jon Mccamond, Hugh Thomas, and Nathan Williams in 2019 via a beautiful argument involving Brouwer fixed point theorem.

In this talk I will explain how zeta map is motivated by the topology of Affine Springer Fibers and combinatorics of Affine Symmetric group, and then present the proof of bijectivity due to Mccamond, Thomas, and Williams.