** Speaker:** Ben Brubaker, University of Minnesota

** Title:** Combinatorial Solutions to Automorphic Problems

** Introduction: ** These two talks will explain some interesting combinatorial
problems and solutions that arise when studying
automorphic forms and L-functions. The talks will present
two very different combinatorial approaches, so are meant
to stand alone, and will not assume any prior knowledge of
automorphic forms. Instead they are meant as an invitation
to a very rich dictionary between the two subjects, with
implications in both directions.

** Lecture 1:** Generating functions for Schur polynomials and
automorphic forms

**Abstract: ** Character values of highest weight representations
of Lie groups may be represented using the Weyl character
formula or as a generating function over a combinatorial basis
for the representation -- e.g. Young tableaux or Gelfand-Tsetlin patterns.
We'll explain a simultaneous deformation of these two expressions
for the character, and how it relates to certain matrix coefficients
used in the construction of automorphic L-functions. We'll also
briefly discuss several methods for proving the generating function
identity, including crystal bases, square ice, and via Hamiltonians on
fermions in a Dirac sea.

** Lecture 2:** Hecke algebra modules and automorphic forms

**Abstract:** The Iwahori-Hecke algebra arises naturally as a space
of functions on an algebraic group over a local field, so it is not so
surprising that it should play a starring role in automorphic forms.
We'll explain how, given one single critical fact, the Hecke algebra
allows one to easily compute spherical functions arising in automorphic
forms, a class of special functions closely related to symmetric and
non-symmetric Macdonald polynomials. At the end, we propose a recipe
for determining the one magic fact.