A. Okounkov

Quantum cohomology of symplectic resolutions

Certain holomorphic symplectic varieties of interest, such as Hilbert scheme of points on C^2 and more general moduli of sheaves on surfaces have a singular affine Poisson blowdown X_0 (which for Hilbert schemes is the symmetric power of the surface). Even more classically, such blowdowns exist for cotangent bundles to homogeneous varieties G/P. Equivariant quantum cohomology of such symplectic resolutions show a particularly close connections to classical structures in representation theory. There is an ongoing project to better understand them pursued by several groups from several directions. I will explain the general shape of this program and some of the more interesting examples worked out so far.