The many mirrors of the Grassmannian (and beyond)

Abstract: 

The mirror of a smooth Fano toric variety is a Laurent polynomial, and this Laurent polynomial encodes interesting enumerative data of the toric variety.  For example, the Jacobian ring of the Laurent polynomial is isomorphic to the small quantum cohomology ring of the toric variety. In this talk, I’ll describe this aspect of the (now basically classical) toric mirror theorem, and explain how this picture is expected to extend to other smooth Fano varieties. The Grassmannian’s strong combinatorial structure  makes it one of the nicest examples of non-toric Fano varieties to study in this context. I’ll present several mirror constructions for the Grassmannian that each reflect a different perspective on its geometry. Finally, I’ll discuss some extensions of these constructions to generalizations of the Grassmannian.