Title: Monodromy of the Casimir connection at numerical quantum parameters

Abstract: 

The monodromy theorem for the Casimir connection of a symmetrisable Kac-Moody algebra was proved in earlier joint work with Valerio Toledano Laredo for a formal quantum parameter. That proof relied on deformation-theoretic methods and a strong rigidity result, which imply the statement for generic complex values of the quantum parameter.

In this talk, I will present an alternative and more direct proof, valid for every numerical parameter away from roots of unity. More precisely, I will introduce a canonical braided tensor functor from category O for the Kac-Moody algebra to the corresponding category for the quantum group, giving in particular another proof of the Drinfeld–Kohno theorem. On integrable modules, this functor readily identifies the monodromy of the Casimir connection with the quantum Weyl group action. This is based on forthcoming joint work with Valerio Toledano Laredo.

Speaker: Ivan Cherednik

Title: Motivic superpolynomials and instanton slices

Abstract:
The key will be my recent theorem that establishes a solid connection between motivic superpolynomials for modules of any rank over plane curve singularities and certain slices of Quot-schemes over isolated surface singularities, torsion-free modules there with fixed "conductors". This includes some new formulas for instanton sums for modules over A^2 supported at 0.

Our definition is of general nature and, generally, does not require plane curve singularities. The first examples resulted in surprising motivic formulas for superpolynomials of certain hyperbolic knots, including K12n242 and K12n725. It is not clear by now which non-algebraic knots can be obtained, but I will state a conjecture linking them to those from a recent work by Galashin-Lam and prior ones. If time permits, I will discuss the modification of ASF, Affine Springer Fibers, incorporating this new development, which indicates that ASF and several related theories (local LP is one of them) may exist beyond "curves".

Speaker: Saebyeok Jeong 

Title: Miura operators as R-matrices from M-brane Intersections

Abstract: In this talk, I will discuss how M2-M5 intersections in a twisted M-theory background yield the R-matrices of the quantum toroidal algebra of gl(1). These R-matrices are identified with the Miura operators for the q-deformed W- and Y-algebras. Additionally, I will show how the M2-M5 intersection (or equivalently, the Miura operator) generates the qq-characters of the 5d N=1 gauge theory, offering new insight into the algebraic meaning of the latter.

Speaker: Michael McBreen
Title: The small quantum group via microlocal sheaves

Abstract: I will describe joint work with Roman Bezrukavnikov, Pablo Boixeda Alvarez and Zhiwei Yun, which realizes a block of representations of the small quantum group as microlocal sheaves on a moduli space of wildly ramified Higgs bundles over the complex plane. Conjecturally, this is in turn equivalent to the Fukaya category of a related moduli of irregular local systems.

Speaker: Sunghyuk Park 

Title: Skein trace and applications

Abstract: 

The moduli space of rank n local systems on a Riemann surface S famously admits "cluster coordinates," which are now part of the "higher Teichmuller theory" of Fock and Goncharov. It was later discovered by Gaiotto, Moore, and Neitzke that these coordinate charts can be identified with the moduli of rank 1 local systems on a spectral curve Σ, a degree n branched cover over S. Fock and Goncharov showed that cluster varieties in general admit a q-deformation, while Turaev had previously established that the moduli space of rank n local systems on S admits a q-deformation to the gl(n) skein algebra. For the two deformations to agree, there must exist corresponding maps from the gl(n) skein of S to the gl(1) skein of Σ. Such maps were indeed constructed algebraically by Bonahon and Wong and others under the name "quantum trace," but their geometric origin -- and why such maps should exist a priori -- remained unclear. In this talk, I will give a geometric construction of these maps and their generalization, named "skein trace," by counting holomorphic curves. Time permitting, I will discuss two applications of this construction: a skein-valued lift of the Kontsevich-Soibelman wall-crossing formula, obtained by deforming Σ in the space of branched covers, and a definition of BPS q-series for fibered 3-manifolds via the skein trace map. 

Speaker: Manish Patnaik

Title: Harder—Narasimhan theory for Loop Groups

Abstract: Eisenstein series on loop groups have two distinct sources: Kapranov’s work on modularity in N=4 SUSY Yang-Mills theory and Garland’s much earlier work of reduction theory for loop groups. After briefly recalling the dictionary between these two pictures, we introduce a notion of semi-stability (Harder—Narasimhan theory) for loop groups, explain some of its peculiarities and illustrate how this leads to certain partial compactifications.  Following this, we make some speculations on how this work is connected back to theory of Eisenstein series. Joint work with Punya Satpathy.

Speaker: Huafeng Zhang

Title: Uni-triangular R-matrices of quantum affine algebras via Theta series

Abstract: The universal R-matrix of the quantum affine algebra associated to a finite-dimensional simple complex Lie algebra admits a Gauss decomposition into a lower uni-triangular part, an abelian part, and an upper uni-triangular part. In this talk, we explain a simple conjugation formula for the uni-triangular R-matrices with one tensor factor evaluated at an arbitrary finite-dimensional representation of the quantum affine algebra. Our formula involves the T-series of Frenkel--Hernandez and the Theta series introduced in a previous work.