for this workshop
Categorified Hecke algebras, link homology, and Hilbert schemes
American Institute of Mathematics, San Jose, California
Eugene Gorsky, Andrei Negut, and Alexei Oblomkov
This workshop, sponsored by AIM and the NSF, 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.
The workshop will differ from typical conferences in some regards. 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.
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