2010-11-12

Vivek Shende, Princeton University

Quantum Invariants of Plane Curve Singularities and Their Links

I study the generating function of Hilbert schemes of points on a locally planar curve. I will explain how this function encodes the numbers n_h of maximally nodal curves in a generic deformation of an h-parameter family of curves containing C. (For instance, n_1 is the Milnor number.) In the case that C has a unique singularity, I will discuss a conjecture that the homological analogue of a certain refinement of this function is the Poincare polynomial of the unreduced Khovanov-Rozansky triply graded homology of the link of the singularity.