Positivity and integrality from higher representation theory

Abstract: 

One of the important consequences of the geometric methods introduced in representation theory over the last thirty years has been the discovery that representations of important algebras in Lie theory often have natural - or "canonical" bases; when written in these bases, the representation theory of the algebra can be formulated over the positive integers, instead of, say, over the complex numbers.  One byproduct of such positive integrality is that a wealth of combinatorics emerges.  Of course, proving these various positivity and integrality statements is sometimes hard work.  In this talk we will explain how ideas from "higher representation theory" can be used to prove positivity and integrality statements.  In particular, we will explain some of the mathematics behind the recent proof by Elias-Williamson of the positivity of Kazdhan-Lusztig polynomials in the representation theory of the Hecke algebra.