Roman Bezrukavnikov (MIT), Elliptic pairs and Hilbert schemes

I will describe a joint project with Galyna Dobrovolska and Ivan Losev. The goal is to prove an equivalence between certain noncommutative resolutions of the symmetric power of the complex plane and a category of perverse sheaves on the moduli space of stable pairs on an elliptic curve. Various ingredients come from geometric Langlands duality, from the theory of elliptic Hall algebras, from representation theory of rational DAHA in positive characteristic and from the theory of stabilities a la Bridgeland. The conjectural equivalence relates that type of stability on the coherent side to the classical GIT stability on the constructible side