Hilbert schemes, Torus Knots, and Khovanov Homology

Khovanov homology is an invariant of knots in S^3 which categorifies the Jones polynomial. Let C be a singular plane curve. I'll describe some conjectures relating the geometry of the Hilbert scheme of points on C to a variant of Khovanov homology which categorifies the HOMFLY-PT polynomial. These conjectures suggest a relation between HOMFLY-PT homology of torus knots and the representation theory of the rational Cherednik algebra. As a consequence, we get some easily testable predictions about the Khovanov homology of torus knots.