Eduardo Gonzalez：Nil-Hecke algebras and Quantum Cohomology

Abstract: Let $M$ be a (monotone) symplectic manifold endowed with a Hamiltonian action of a compact Lie group $G$ with maximal torus $T$. We construct an action of the cohomology of the affine flag manifold $LG/T$ onto the quantum cohomology of $M$. Our construction generalises that of Okounkov et. all of shift operators and Seidel's representation. This is joint work with Cheuk Yu Mak and Dan Pomerleano.

Binyong Sun: Lie group representations and coadjoint orbits

Abstract: Kirillov philosophy predicts that there is a correspondence between Lie group representations and coadjoint orbits of the Lie group. We will explain some examples of  the correspondence. Basic notations from representation theory of real reductive groups  will also be explained.

Ben Webster：3d mirror symmetry and symplectic singularities

Abstract: I'll give a short induction to the mathematical aspects of mirror symmetry for 3-d QFTs, in particular, its manifestation as duality between symplectic resolutions of singularities, and the representation theory of corresponding non-commutative algebras.

Peng Zhao: Cluster Configuration Spaces and Hypersurface Arrangements

Abstract: Cluster algebras have inspired many recent developments in physics. In the other direction, the study of scattering amplitudes motivates us to study certain moduli spaces associated with finite-type cluster algebras. For A-type it is M_{0,n}, the moduli space of n marked points on CP^1. By introducing a gluing construction for other types, we realize such spaces as hypersurface arrangements. We make contact with Fomin-Zelevinsky's work on Y-systems and Fomin-Shapiro-Thurston's work on surface cluster algebras. We study various topological properties using a finite-field method and propose conjectures about quasi-polynomial point count, dimensions of cohomology, and Euler characteristics for the D_n space up to n=10. Based on 2109.13900 with Song He, Yihong Wang, Yong Zhang, and upcoming work with Yihong Wang.

Adrien Sauvaget ：Integral of Psi-classes on DR cycles

Abstract: I will present a closed formula expressing the integral of a single psi-class on generalized DR-cycles as well as an analogue conjectural formula for a spin refinement of DR cycle. Some applications and open problem will be discussed depending on the time and the topics of interest of the participants.

Song Yu：Open/closed correspondence via relative/local correspondence

Abstract: We discuss a mathematical approach to the open/closed correspondence proposed by Mayr, which is a correspondence between the disk invariants of toric Calabi-Yau threefolds and genus-zero closed Gromov-Witten invariants of toric Calabi-Yau fourfolds. We establish the correspondence in two steps: First, a correspondence between the disk invariants and the genus-zero maximally-tangent relative Gromov-Witten invariants of relative Calabi-Yau threefolds, which follows from the topological vertex (Li-Liu-Liu-Zhou, Fang-Liu). Second, a correspondence between the maximally-tangent relative invariants and the closed invariants, which can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting. Our correspondences are based on localization. We also discuss generalizations and implications of our correspondences. Joint work with Chiu-Chu Melissa Liu.

Denis Nesterov：Quasimaps to moduli spaces of sheaves

Abstract:  We will discuss two different compactifications of the moduli space of maps from smooth curves to punctorial Hilbert schemes of a surface. Namely, the moduli space of stable maps (Gromov-Witten theory) and the relative Hilbert scheme of one-dimensional subschemes (Donaldson-Thomas theory) on threefolds of the type Surface x Curve for a varying nodal curve. The (virtual) intersection theories of these moduli spaces are related by a certain wall-crossing, which is provided by the theory of quasimaps to moduli spaces of sheaves.

Nawaz Sultani：Gromov--Witten invariants of some non-convex complete intersections

Abstract: For convex complete intersections, the Gromov-Witten (GW) invariants are often computed using the Quantum Lefshetz Hyperplane theorem, which relates the invariants to those of the ambient space. However, even in the genus 0 theory, the convexity condition often fails when the target is an orbifold, and so Quantum Lefshetz is no longer guaranteed. In this talk, I will showcase a method to compute these invariants, despite the failure of Quantum Lefshetz, for orbifold complete intersections in stack quotients of the form [V // G]. This talk will be based on joint work with Felix Janda (Notre Dame) and Yang Zhou (Harvard), and upcoming work with Rachel Webb (Berkeley).

Cheng Shu：Mixed Hodge polynomial of character variety

Abstract: Hausel, Letellier and Rodriguez-Villegas computed the E-polynomial of character varieties with generic semi-simple conjugacy classes. Their computation led to a conjectural formula for the mixed Hodge polynomial of character varieties. We will recall their results and introduce a new family of character varieties that are unitary in the global sense. The same method gives a conjectural formula for the mixed Hodge polynomial, which is built of Macdonald polynomials and wreath Macdonald polynomials.

Hans Jockers：Remarks on Modularity in Quantum K-Theory

Abstract: In this talk we present some observations about the modular properties of 3d BPS half-indices of particular N=2 3d gauge theories. These indices connect to quantum K-theory via the 3d gauge theory quantum K-theory correspondence. They are solutions to certain q-difference equations, which — for particular classes of N=2 3d gauge theories — relate to the theory of bilateral q-series and modular q-characters of two dimensional conformal field theories in a certain massless limit.

Jie Zhou: Twisted Sectors in Quasi-Homogeneous Polynomial Singularities and Automorphic Forms

Abstract: We study one-parameter deformations of Calabi-Yau type Fermat polynomial sin-gularities along degree-one directions. We show that twisted sectors in the vanishing cohomology are automorphic forms for certain triangular groups. We prove consequen-tially that genus zero Gromov-Witten generating series of the corresponding Fermat Calabi-Yau varieties are components of automorphic forms. The main tools we use are period mappings for quasi-homogeneous polynomial singularities, Riemann-Hilbert correspondence, and genus zero mirror symmetry.This is joint work with Yongbin Ruan and Dingxin Zhang.

Zijun Zhou: Virtual Coulomb branch and quantum K-theory

Abstract: In this talk, I will introduce a virtual variant of the quantized Coulomb branch by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of quasimaps into the holomorphic symplectic quotient T^*N///G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under variation of GIT.

Yang Zhou：Quasimap wall-crossing and applications

Abstract: The theory of Gromov-Witten invariants is a curve counting theory defined by integration on the moduli of stable maps. Varying the stability condition gives alternative compactifications of the moduli space and defines similar invariants. One example is epsilon-stable quasimaps, defined for a large class of GIT quotients. When epsilon tends to infinity, one recovers Gromov-Witten invariants. When epsilon tends to zero, the invariants are closely related to the B-model in physics. The space of epsilons has a wall-and-chamber structure, and a wall-crossing formula was conjectured by Ciocan-Fontanine and Kim. In this talk, I will explain the wall-crossing phenomenon, sketch a proof of the wall-crossing formula and discuss its variants and applications.

Mark Shoemaker：Towards a mirror theorem for GLSMs

Abstract: A gauged linear sigma model (GLSM) consists roughly of a complex vector space V, a group G acting on V, a character \theta of G, and a G-invariant function w on V.  This data defines a GIT quotient Y = [V //_\theta G] and a function on that quotient.  GLSMs arise naturally in a number of contexts, for instance as the mirrors to Fano manifolds and as examples of noncommutative crepant resolutions. GLSMs provide a broad setting in which it is possible to define an enumerative curve counting theory, simultaneously generalizing FJRW theory and the Gromov-Witten theory of hypersurfaces. Despite a significant effort to rigorously define the enumerative invariants of a GLSM, very few computations of these invariants have been carried out.  In this talk I will describe a new method for computing generating functions of GLSM invariants.  I will explain how these generating functions arise as derivatives of generating functions of Gromov-Witten invariants of Y.