Compactified Jacobians, Cherednik algebras and knot homology

Tuesday, January 22, 2013 - 15:00 to 15:50
Speaker Information
Eugene Gorsky
Stony Brook University

Notes

Recruitment talk

Abstract or Additional Information

A complex plane curve singularity intersects a small three–dimensional sphere by a link. It turns out that many topological invariants of this link can be obtained from the algebraic geometry of this singularity. In particular, D. Maulik recently proved a conjecture of A. Oblomkov and V. Shende showing that the HOMFLY-PT polynomial of the link can be read off the Euler characteristics of the Hilbert schemes of points on the curve.

I will describe the structure of the homology of the Hilbert schemes on a curve xm= yn which is expected to match the Khovanov-Rozansky homology of the corresponding torus knot. In particular, I will show its relation to the combinatorics of the bigraded Catalan numbers of A. Garsia and M. Haiman and representation theory of rational Cherednik algebras. The talk is based on the joint works with M. Mazin, A. Oblomkov, J. Rasmussen and V. Shende.

Research Area