Richárd Rimányi: 3d mirror symmetry in (elliptic) enumerative geometry

Abstract:

A surprising number of problems in enumerative geometry can be approached through the method of assigning characteristic classes to singularities. We will explore these characteristic classes in different settings, encountering a wide spectrum of mathematical concepts, from geometric theorems of the ancient Greeks to Schubert Calculus, quivers, and quantum groups. After examining characteristic classes of singularities in general, we will focus on a sophisticated version: Aganagic–Okounkov’s elliptic stable envelope. Fresh understanding of d=3, N=4 quantum field theories predicts a surprising new structure theorem for stable envelopes, called 3D mirror symmetry. The class of spaces where we state and prove 3D mirror symmetry (called Cherkis bow varieties) have intriguing geometry and combinatorics.