This is a six hour lecture course to be given at Northeastern in the week of September 12-16.

Monday, Tuesday, 4.30-6.30pm, Churchill (building 54 here) 103, Thursday 3-5pm, Lake 509.

Title: Khovanov-Rozansky homology and Hilbert schemes of points

Abstract: Khovanov and Rozansky introduced a knot homology theory which categorifies the HOMFLY polynomial. This homology has a lot of interesting properties, but it is notoriously hard to compute. I will introduce HOMFLY homology and discuss its conjectural relation to algebraic geometry of the Hilbert scheme of points on the plane. I will also outline a possible strategy of the proof using the recent work of Elias and Hogancamp on categorical diagonalization. All notions will be introduced in lectures, no preliminary knowledge is assumed.

Lecture 1: Hecke algebra and HOMFLY polynomial

Lecture 2: Soergel bimodules and HOMFLY homology

Lecture 3: Hilbert schemes and flag Hilbert schemes

Lecture 4: From braids to sheaves on Hilbert schemes

Lecture 5: Categorical diagonalization I

Lecture 6: Categorical diagonalization II

Notes for lectures 1 and 2 by Kostya Tolmachov.

Notes for lectures 3 and 4 by Jose Simental Rodriguez.

Notes for lectures 5 and 6 by Galyna Dobrovolska.