The abelian/nonabelian correspondence and quantum cohomology

Given a “good” action of a reductive complex algebraic
group G on a projective manifold X, the abelian/nonabelian correspondence refers to a precise relation that exists between the cohomology rings of the Geometric Invariant Theory quotients X//G and X//T, where T is a maximal abelian subgroup in G. In this talk, I will explain how to extend this relation to the (big) quantum cohomology
rings of the two quotients. This is joint work with Bertram, Kim, and Sabbah.