Meeting Time: Friday, 3:00-5:00 p.m. | Location: 2-135
Zoom Link: https://mit.zoom.us/j/94469771032
944 6977 1032
For the Passcode, please contact Pavel Etingof at firstname.lastname@example.org.
Spin link homology and webs in type B
In their study of GL(N)-GL(m) Howe duality, Cautis-Kamnitzer-Morrison observed that the GL(N) Reshetikhin-Turaev link invariant can be computed in terms of quantum gl(m). This idea inspired Cautis and Lauda-Queffelec-Rose to give a construction of GL(N) link homology in terms of Khovanov-Lauda's categorified quantum gl(m). There is a Spin(2n+1)-Spin(m) Howe duality, and a quantum analogue which was first studied by Wenzl. In the first half of the talk I will explain how to use this duality to compute the Spin(2n+1) link polynomial, and present calculations which suggest that the Spin(2n+1) link invariant is obtained from the GL(2n) link invariant by folding. In the second part of the talk, I will introduce the parallel categorified constructions and explain how to use them to define Spin(2n+1) link homology.
This is based on joint work in progress with Ben Elias and David Rose.
| Yasuyuki Kawahigashi
(University of Tokyo)
Quantum 6j-symbols and braiding
I will explain certain 4-tensors appearing in studies of two-dimensional topological order from a viewpoint of subfactor theory of Jones and alpha-induction there, which is a tensor functor arising from a modular tensor category and a Frobenius algebra in it. They are understood with quantum 6j-symbols and braiding.
Affine Kazhdan-Lusztig polynomials on the subregular cell: with an application to character formulae
(joint with Vasily Krylov) I will explain the computation of special values of parabolic affine inverse Kazhdan-Lusztig polynomials, which give explicit formulas for certain irreducible representations of affine Lie algebras that generalize Kac and Wakimoto's results. By Bezrukavnikov's equivalence, the canonical basis in the subregular part of the anti-spherical module can be identified with irreducible objects in the exotic t-structure on the equivariant derived category of the subregular Springer fiber. We describe the irreducible objects explicitly using an equivariant derived McKay correspondence. In doing so, we identify the module with a module Lusztig defines, which compatibly extends to the regular cell.
(University of North Carolina at Chapel Hill)
From DAHA superpolynomials for algebraic links to motivic ones
The focus will be on a recent construction of the motivic superpolynomials for arbitrary singularities (multi-branch and non-square-free). They will be introduced from scratch, which includes the definition of varieties of torsion-free sheaves of any rank over curve singularities. Our motivic superpolynomials are q,t,a-generalizations of orbital integrals associated with Affine Springer Fibers of type A in the case of the most general characteristic polynomials. I will not use the theory of ASF. The key conjecture is their coincidence with the DAHA superpolynomials of the corresponding (colored) algebraic links. The latter (due to Cherednik-Danilenko) will be defined. This coincidence can be seen as a high-level Shuffle Conjecture. As an application, the DAHA vertex will be considered and its relation to the q-theory of Riemann’s zeta. Also, q,t,a-deformations of the modified rho-invariants of algebraic knots will be discussed; classically, rho is defined via the Atiyah-Patodi-Singer eta invariant, but I will need only some formulas in this talk. See https://arxiv.org/abs/2304.02200.
|Do Kien Hoang