Algebra, geometry, and combinatorics of link homology

Algebra, geometry, and combinatorics of link homology

This workshop, sponsored by AIM and the NSF, 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.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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