Categorified Hecke algebras, link homology, and Hilbert schemes

October 1 to October 5, 2018

at the

American Institute of Mathematics, San Jose, California

organized by

Eugene Gorsky, Andrei Negut, and Alexei Oblomkov

Original Announcement

This workshop will be devoted to the representation theory of the type A Hecke algebras, their categorifications and connections to the Hilbert scheme of points and link invariants.

The Hecke algebra and its generalizations are central objects in modern representation theory. It can be also used to define a topological invariant of links known as HOMFLY-PT polynomial. Soergel defined a categorification of the Hecke algebra using a certain category of bimodules, now known as Soergel bimodules. Elias and Williamson used Soergel bimodules to resolve the long-standing Kazhdan-Lusztig conjecture. Rouquier used complexes of Soergel bimodules to categorify the braid group. Khovanov and Rozansky used Soergel bimodules and Rouquier complexes to categorify the construction of the HOMFLY-PT polynomial. The resulting invariant is known as Khovanov-Rozansky homology, it is a powerful link invariant, which, however, is notoriously hard to compute from the definition. By drawing parallels between link homology and other fields of mathematics, we hope to uncover new ways in which the former may be effectively computed.

The Hilbert scheme of points on the plane is a central object of study in modern algebraic geometry and geometric representation theory. Furthermore, the work of Haiman related it to algebraic combinatorics and Macdonald polynomials. In recent years, several results and conjectures relating the Hilbert scheme to knot invariants were put forward. Specifically, the organizers (together with Rasmussen, Rozansky and Shende) proposed an extensive research program aimed at understanding the algebraic structure of the Khovanov-Rozansky homology and related theories. They associate to a braid a coherent sheaf on the Hilbert scheme, and conjecture that its sheaf cohomology is isomorphic to the HOMFLY-PT homology of the braid we started with. Recently, a significant progress towards the proof of these conjectures was achieved by Elias, Hogancamp and Mellit.

The main topics for the workshop are

To prove the precise relation between the categories of (type A) Soergel bimodules, coherent sheaves on the Hilbert scheme of points, and various categories of matrix factorizations;
To formulate, develop and prove the analogues of the above relation for extended affine Soergel bimodules;
To describe the Drinfeld center of the homotopy category of Soergel bimodules;
To give an algebro-geometric construction of Khovanov-Rozansky link homology;
To understand the combinatorial structure of link homology.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Workshop Videos

Papers arising from the workshop:

Evaluations of annular Khovanov-Rozansky homology

by Eugene Gorsky, Paul Wedrich

Serre duality for Khovanov-Rozansky homology

by Eugene Gorsky, Matthew Hogancamp, Anton Mellit, Keita Nakagane

Generalized q,t-Catalan numbers

by Eugene Gorsky, Graham Hawkes, Anne Schilling, Julianne Rainbolt