String-Math Seminar: Monodromic deformation of Khovanov-Rozansky homology and Hilbert schemes

Seminar | October 15 | 2-3 p.m. | 402 LeConte Hall

 Matthew Hogancamp, University of Southern California

 Department of Mathematics

A conjecture of Gorsky-Negut-Rasmussen asserts the existence of a pair of adjoint functors relating the Hecke category for symmetric groups and the Hilbert scheme of points in the plane. One topological consequence of this conjecture is the prediction of a deformation of the triply graded Khovanov-Rozansky link homology which restores the missing \(q\rightarrow tq^{-1}\) symmetry of KR homology for links. In this talk I will discuss a candidate for such a deformation, constructed in joint work with Eugene Gorsky, which indeed facilitates connections with Hilbert schemes. For instance our main result explicitly computes the homologies (both deformed and undeformed) of the \((n,nk)\) torus links, summed over all \(n\geq 0\), as a graded algebra. Combining with work of Haiman this gives a functor from the Hecke category to sheaves on the relevant Hilbert scheme.