** Speaker:** Thomas Lam, University of Michigan

** Title:** Affine Schubert polynomials

** Abstract: **
I will discuss some new developments in affine Schubert
calculus, focusing on the "Schubert polynomial problem" for affine
Grassmannians and flag varieties. The talks will be based on joint
work with Seungjin Lee and Mark Shimozono.

In the first lecture, I will recall the basic relation between Schur polynomials, Schubert polynomials and the cohomology of Grassmannians and flag varieties. I will then discuss the analogous story for affine Grassmannians and affine flag varieties, focusing on the features special to the affine case: (1) a Hopf algebra structure, (2) a map from the affine Grassmannian to the affine flag variety, and (3) additional loop rotation equivariance.

In the second lecture, I will talk about presentations for the cohomology ring of the affine flag variety, and the Schubert polynomial problem. In particular, I will discuss Seungjin Lee's affine Schubert polynomials and new formulae for them, together with variations in different kinds of equivariant cohomology.