Title: Quantum invariants of plane curve singularities
Speaker: Vivek Shende
Speaker Info: MIT
Consider a critical point of a function f(x,y) in two complex variables. Milnor showed that the number of non-degenerate critical points into which it splits under a general perturbation is encoded by the Alexander polynomial of the link of the singular fibre of f:C^2 --> C passing through the critical point.Date: Thursday, January 10, 2013
Analogously, one can consider a (k-1)-parameter family of such functions f, and ask: into how many k-nodal curves does the fiber containing a critical point split under a general perturbation? As we will explain, they are counted by the coefficients of the HOMFLY polynomial of the link. The relation goes through the Hilbert schemes of points, which parameterize subschemes of the singular curve.
These latter spaces and their generalizations in fact contain contain enough information to recover all the coefficients of the HOMFLY polynomial, and, conjecturally, their cohomologies are the Khovanov-Rozansky homology of the link.