CMSA Colloquium: Knot homology and sheaves on the Hilbert scheme of points on the plane

The knot homology (defined by Khovavov,Rozansky) provide us with a refinements of the knot polynomial knot invariant defined by Jones. However, the knot homology are much harder to compute compare to the polynomial invariant of Jones. In my talk I present recent developments that allow us to use tools of algebraic geometry to compute the homology of torus knots and prove long-standing conjecture on the Poncare duality the knot homology. In more details, using physics ideas of Kapustin-Rozansky-Saulina, in the joint work with Rozansky, we provide a mathematical construction that associates to a braid on n strands a complex of sheaves on the Hilbert scheme of n points on the plane. The knot homology of the closure of the braid is a space of sections of this sheaf. The sheaf is also invariant with respect to the natural symmetry of the plane, the symmetry is the geometric counter-part of the mentioned Poincare duality.

Zoom link: https://harvard.zoom.us/j/95767170359 (Password: cmsa)