Mirror Symmetry Week

Hypertoric varieties are hyperkahler analogues of toric varieties. They arise as the simplest examples of symplectic resolutions, a class of spaces important in representation theory. I will discuss aspects of mirror symmetry for hypertoric varieties. This includes joint work with Daniel Shenfeld on their quantum cohomology, and work in progress on homological mirror symmetry.

The K-theory of Nakajima quiver varieties contains the classes of the so-called tautological bundles. We will show how to compute operators in terms of these classes, with focus on the example of the moduli space of sheaves on a surface, where we will encounter explicit formulas via the shuffle algebra.

There are two natural circle actions in Floer theory, one on symplectic cohomology and one on the Hochschild homology of the Fukaya category. We show the open-closed map between these two complexes is S^1-equivariant, and thus induces maps between equivariant homology theories, which include for the Fukaya category cyclic, negative cyclic, and periodic cyclic homology groups. Thus, in nice cases, we can describe the periodic cyclic homology of the Fukaya category as ordinary (De Rham) cohomology. Finally, we say a few words about the resulting equivariant homology theories when additionally equipped with a dilation, (the mirror to) an infinitesimal C^* action.

I will present a conjectural formula for the Poincare polynomial of the Hilbert scheme of points on a planar curve (joint with Rasmussen and Shende). The formula is written in terms of the Khovanov-Rozansky invariants of the links of the singularities of the curve. In the case of toric curve $\{ x^m=y^n\}$ the Poincare polynomial also could be written in terms of equivariant Euler characteristic of some sheaf of some particular equivariant sheaf on $Hilb^n(\CC^2)$ (joint with Yun). If time permits I will also discuss cohomology ring of the compactified Jacobian of the toric curve and the conjectural description of the ring for general curve singularity (joint with Yun).

Geometric Langlands duality is a conjectural equivalence of a derived category of coherent sheaves with a derived category of constructible sheaves, while homological mirror symmetry is an equivalence of a derived category of coherent sheaves with a Fukaya category. According to Kapustin, Witten et al. the two stories have a significant overlap. I will discuss examples of results and conjectures describing how these dualities interact with abelian hearts and with automorphism groups of the categories.

Motivated, on the one hand, by R. Bezrukavnikov's conjectures in the representation theory of the rational Cherednik algebra in characteristic p, and, on the other hand, by geometric Langlands duality for the trivial local system, we calculate explicitly the result of the Drinfeld-Laumon construction applied to the direct summands in the Springer-Laumon sheaf for the trivial local system on a curve which correspond to irreducible representations of the symmetric group (the proof is completed for a curve of genus 0 and in progress for curves of positive genus).