The Peterson Isomorphism: Moduli of Curves and Alcove Walks

In this talk, I will explain the combinatorial tool of folded alcove walks, in addition to surveying a wide range of applications in combinatorics, representation theory, and algebraic geometry.  As a concrete example, I will describe a labeling of the points of the moduli space of genus 0 curves in the complete complex flag variety using the combinatorial machinery of alcove walks.  Following Peterson, this geometric labeling explains the "quantum equals affine" phenomenon which relates the quantum cohomology of this flag variety to the homology of the affine Grassmannian. This is joint work with Arun Ram.