Alexei Oblomkov

Title: Matrix factorizations and knot homology

Abstract:

Recently, several groups of researchers proposed relations between the HOMFLY-PT homology and sheaves on the Hilbert schemes of points in the plane, see the work of Aganagic, Cherednik, Gorsky, Negut, Hogancamp, Oblomkov, Rasmussen, Rozansky, Shende. In my talk, I will discuss an approach due to Oblomkov and Rozansky. For a braid $b$, we construct a two-periodic complex of coherent sheaves $S_b$ such that $H^*(S_b)$ is the HOMFLYPT homology of the closure of $b$.

In the heart of our construction is a realization of the braid group inside some specific category of matrix factorizations. The main goal of the talk is to give an introduction to the theory of matrix factorization and to construct the above mentioned braid group action. As an application we will compute the knot homology of the torus knots.