November 18: Ben Wormleighton

Title: McKay correspondence and walls for G-Hilb

Abstract:

The McKay correspondence takes many guises but at its core connects the geometry of minimal resolutions for quotient singularities C^n / G to the representation theory of the group G. I will introduce the classical situation for SL(2), along with its categorification and extension to three dimensions. When G is an abelian subgroup of SL(3), Craw-Ishii showed that every minimal resolution can be realised as a moduli space of stable quiver representations, although the chamber structure for the stability parameter and associated wall-crossing behaviour is in general poorly understood. I will describe my recent work computing the walls and wall-crossing behaviour for the chamber corresponding to a particular minimal resolution called the G-Hilbert scheme. Time permitting, I will also discuss ongoing work with Yukari Ito (IPMU) and Tom Ducat (Bristol) to better understand the geometry, chambers, and corresponding representation theory for other minimal resolutions.