Karen Smith: Geometry and Combinatorics via Finite Fields

Schedule

14:00-15:00 Precolloquium lecture by Alessandro De Stefani (Room FB54 , AlbaNova)

15:15-16:15 Colloquium lecture by Karen Smith (Room Oskar Klein, AlbaNova)

16:15-17:00 SMC social get together with refreshments

Abstract

Many natural problems in mathematics can be framed in terms of solutions to polynomial equations. These solution sets, called algebraic varieties, have many interesting geometric properties which have been  studied for centuries for their own intrinsic beauty, as well as  used to solve seemingly unrelated problems in many different contexts. In this talk, I will explain how certain geometric properties of  varieties can be understood  by shifting focus to the algebraic features of their corresponding coordinate rings. We will see that by ``reducing modulo p" we can introduce effective new tools for understanding the singularities of a complex variety. As a recent application, we discuss an application to an important new type of algebra arising in combinatorial representation theory called a cluster algebra due to Fomin and Zelevinsky. Specifically, we  prove that certain cluster algebras always have rational singularities over the real or  complex numbers, by studying the corresponding cluster algebras over finite fields.