Algebra, geometry, and combinatorics of link homology

July 31 to August 4, 2023

at the

American Institute of Mathematics, Pasadena, California

organized by

Nicolle Gonzalez, Eugene Gorsky, Matthew Hogancamp, Oscar Kivinen, and Alexei Oblomkov

Original Announcement

This workshop will be devoted to the recent developments in the intersection of Khovanov-Rozansky homology, affine Springer fibers, Hilbert schemes and link homology, and combinatorics.

The main topics for the workshop are:

  1. Computing Khovanov-Rozansky link homology for large families of links: Khovanov and Rozansky defined link homology categorifying the HOMFLY-PT link polynomial. This homology was computed explicitly for all torus links, but remains unknown for cables of torus knots and other algebraic links. Conjectures of Oblomkov, Rasmussen and Shende and their extensions give algebro-geometric description for this homology in terms of Hilbert schemes of points on singular curves, affine Springer fibers, Hitchin's integrable system, orbital integrals in algebraic number theory and even Coulomb branches of gauge theories. We plan to revisit these conjectures, relate all these various descriptions and attempt to match them with the knot homology side.
  2. Link homology and Hilbert schemes on the plane: another set of conjectures of Gorsky, Negut, Rasmussen, Oblomkov, Rozansky and others relates Khovanov-Rozansky homology to the Hilbert scheme of points on the plane. This variety plays a central role in modern geometric representation theory. We plan to find the analogues of geometric structures on Hilbert schemes (coordinate ring, Poisson bracket, vector bundles and their sections etc) in link homology.

  3. Combinatorics of Shuffle Conjecture and beyond: In algebraic combinatorics, the Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel and Ulyanov spurred a lot of activity and interest. This conjecture gives a combinatorial expression for matrix elements of a certain operator on symmetric functions, which turns out to be related to the action of the full twist braid in link homology. While the Shuffle conjecture was recently proven by Carlsson and Mellit, many of its cousins are closely related to other computations in link homology and remain open.
  4. Categorical skein theory: The skein theory which underlies the HOMFLY-PT polynomial is closely related to the theory of symmetric functions. Various topological operations such as adding a meridian or a full twist can be interpreted as certain operators on symmetric functions. A general, and more abstract direction to approach the above problems is related to the categorification of the skeins of the annulus and of the torus, and their relation to the categorifications of the Heisenberg and elliptic Hall algebras, as well as the categorification of the Carlsson-Mellit algebra which appeared in their proof of the Shuffle Conjecture.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Workshop videos