Braids and sheaves: a geometric incarnation of Khovanov-Rozansky homology

In this talk, I will survey recent work on the connection between various categorical knot invariants (e.g. Khovanov’s construction via Soergel bimodules) to categories of coherent sheaves on Hilbert schemes. I will emphasize the conjectural framework proposed by Gorsky, Rasmussen and myself, as well as the matrix factorization version due to Oblomkov and Rozansky. I will end the talk by describing relations between the two approaches.