From Khovanov homology to Hilbert schemes of points

In this talk, we will present a framework that takes in a monoidal category C with some extra data, and outputs a pair of adjoint functors from C to the derived category of a certain algebraic space. Our main application is when C is the category of type A Soergel bimodules, in which case we conjecture that the resulting algebraic space is the flag Hilbert scheme of points on the plane. This would allow us to associate to any braid a sheaf on the flag Hilbert scheme, whose equivariant Euler characteristic (conjecturally) matches the triply graded Khovanov homology of the closure of the braid. We show how this leads to a geometrization of the Jones-Ocneanu trace using Hilbert schemes. Joint work with Eugene Gorsky and Jacob Rasmussen.