Lecture Series 

Speaker: Tomoyuki Arakawa

Title: 4D/3D QFT and representation theory

Abstract: 4D/3D quantum field theory in theoretical physics is conceptually rich and gives rise to many interesting mathematical structures, even though a fully rigorous mathematical formulation of the theories themselves is still lacking. A relatively recent discovery by Beem et al. shows that to every 4D N=2 superconformal field theory one can associate a representation-theoretic object called a vertex algebra, which serves as an invariant (or observable) of the theory. Although vertex algebras are inherently algebraic, those arising as invariants of 4D QFT display striking connections with certain geometric objects that also appear as invariants of the same physical theories. Similarly, to each 3D N=4 gauge theory one can associate two vertex algebras—the A-twisted and B-twisted boundary VOAs—by the work of Gaiotto and his collaborators, which may be viewed as refinements of the Higgs and Coulomb branches.  In this lecture, I will discuss some representation-theoretic aspects of these phenomena.

Speaker: Andrew Neitzke

Title: Physics and Mathematics of Class S

Abstract: I will give an overview of a family of quantum field theories known as "Class S", and some of their applications in physics and mathematics. A rough plan of the lectures:

Lecture 1: Quantum field theories from surfaces

We will explain that certain four-dimensional supersymmetric quantum field theories can be parameterized by Riemann surfaces with marked points and decorations. Different ways of cutting the surface into pairs of pants correspond to different but equivalent descriptions of the same physics.

Lecture 2: Geometry of vacua

We will explain how the low-energy behavior of class S field theories is controlled by the geometry of Hitchin's integrable system. Thus physical questions about effective couplings, particles and charges get related to geometry of Higgs bundles and spectral curves.

Lecture 3: Connections to other subjects

We will discuss links between class S theories and various subjects in physics and mathematics, including two-dimensional conformal field theory, cluster coordinates, and wall-crossing.

Research Talk

Speaker: Andrea Appel

Title: Quantum affine symmetric pairs and boundary integrability

Abstract: The Yang–Baxter equation (YBE) and the reflection equation (RE), also known as the boundary Yang–Baxter equation, encode the fundamental consistency conditions for quantum integrable systems on a line and on a half-line, respectively. While the YBE is naturally governed by quantum groups, solutions of the RE—K-matrices—are closely tied to quantum symmetric pairs (QSPs). In this talk, I will survey recent developments, many from joint work with B. Vlaar, focusing on how QSPs provide a structural framework for constructing and classifying integrable boundary conditions.

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: Mikhail Khovanov 

Title: Diagrammatics of the Delannoy Category 

Abstract: We review the construction of the Delannoy category, as defined by N.Harman, A.Snowden and N.Snyder. This category has a "positive" subcategory which categories the ring of integer-valued polynomials. We explain the diagrammatics of the Delannoy category (work in progress with N.Snyder).

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: Whittaker functions, antispherical modules, and formulas of Kato-Lusztig type

Abstract: The antispherical module for an affine Hecke algebra plays a central role in many different aspects of algebraic representation theory.  This module also has a p-adic incarnation in terms of certain spaces of Whittaker functions.   After explaining this connection and also its relation to the so-called geometric Casselman-Shalika or Kato-Lusztig formulas, we will explain how to lift this story to the setting of p-adic loop groups where the affine Hecke algebra is replaced by a variant of the DAHA. We will also explain the relation to the double affine geometric Satake program and deformations of this story that are naturally related to the quantum geometric Langlands correspondence. Joint work with Valentin Buciumas and Yanze Chen.

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.