Yingchun Zhang：Local Seiberg Duality to flag variety and tautological bundles over flag variety

Abstract：Seiberg Duality conjecture is proposed by Yongbin Ruan and relates GLSM of two different quivers related via quiver mutation. In this talk, I will introduce the result of the conjecture applied to $A_n$ type quiver and we prove that small I functions of flag variety (before duality) and a complete intersection in another quiver (after mutation) are equal under variable change. Moreover, we can extend our result to the tautological bundle over flag variety. 

Xiaohan Yan: Quantum K-theory of flag varieties via non-abelian localization

Abstract: Quantum K-theory studies the K-theoretic analogue of Gromov-Witten invariants defined as holomorphic Euler characteristics of sheaves on the moduli space of stable maps. Generating functions of such invariants, which are called the (K-theoretic) big J-functions, play a crucial role in the theory. In this talk, we provide a reconstruction theorem of the permutation-invariant big J-function of partial flag varieties (regarded as GIT quotients of vector spaces) using a family of finite-difference operators, from the quantum K-theory of their associated abelian quotients which is well-understood. The idea is to use a recursive characterization of the big J-functions based on the geometry of isolated fixed points and connecting 1-dimensional orbits on the flag varieties in toric-equivariant settings, but along the way we will need to address the issue of possibly non-isolated fixed points on the abelian quotient. A portion of this talk is based on a joint work with Alexander Givental.

Wei Gu: On phases of 3d N=2 Chern-Simons-matter theories.

Abstract: In this talk, we will first review some aspects of 2d gauged linear sigma models (GLSMs). Then, we investigate phases of 3d N = 2 Chern-Simons-matter theories, extending to three dimensions the celebrated correspondence between 2d gauged Wess-Zumino-Witten (GWZW) models and non-linear sigma models (NLSMs) with geometric targets. We find that although the correspondence in 3d and 2d are closely related by circle compactification, an important subtlety arises in this process, changing the phase structure of the 3d theory. Namely, the effective theory obtained from the circle compactification of a phase of a 3d N = 2 gauge theory is, in general, different from the phase of the 3d N = 2 theory on R ^2 × S^1, which means taking phases of a 3d gauge theory do not necessarily commute with compactification.   Finally, if time permits, we will also talk about Givental-Lee's and KWRZ's (Kapustin, Willet, Ruan, Zhang) quantum K theories from 3d gauge theories.This is a joint work w/ Du Pei and Ming Zhang.

Peter Koroteev: 3d Mirror Symmetry for Instanton Moduli Spaces

3d Mirror Symmetry for Instanton Moduli Spaces

Abstract: We prove that the Hilbert scheme of $k$ points on $\mathbb{C}^2$ (Hilb$^k[\mathbb{C}^2]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its K\ahler and equivariant parameters as well as inverting the weight of the $\mathbb{C}^\times_\hbar$-action. First, we find a two-parameter family $X_{k,l}$ of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of Hilb$^k[\mathbb{C}^2]$ is obtained via direct limit $l\to\infty$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted $\hbar$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-$N$ sheaves on $\mathbb{P}^2$ with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

Junliang Shen: Cohomology of the moduli of Higgs bundles and the Hausel-Thaddeus conjecture

Cohomology of the moduli of Higgs bundles and the Hausel-Thaddeus conjecture

Abstract: In this talk, I will discuss some structural results for the cohomology of the moduli of semi-stable SL_n Higgs bundles on a curve. One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration. If time permits, we will also discuss the case where the rank of the Higgs bundle is not coprime to the degree, so that the moduli spaces are singular due to the presence of the strictly semi-stable loci. We will explain how intersection cohomology comes into play naturally. Based on joint work with Davesh Maulik.

Baohua Fu: An introduction to nilpotent orbits and their birational geometry

Abstract: I'll first review the symplectic geometrical part of coadjoint orbits and the connection to symplectic singularities.Then I'll move to the classification of nilpotent orbits and their crepant resolutions given by Springer maps. Different crepant resolutions are connected by stratified Mukai flops. Although we have a clear understanding of the geometry of these flops, the derived equivalence between them is still mysterious.If time permits, I'll also describe a construction of a new family of 4-dimensional isolated symplectic singularities with trivial local fundamental groups, which answers a question of Beauville raised in 2000.

Chenyang Xu: Moduli spaces of Fano varieties

Abstract: We will describe the purely algebraic construction of moduli spaces parametrizing Fano varieties with K-(semi,poly)stability, called K-moduli spaces, and its fundamental properties. As a by-product, it also completes the solution of Yau-Tian-Donaldson Conjecture to all Fano varieties case (including singular ones).Lecture 1: we will discuss the background of K-stability and algebraic geometer’s gradually evolving understanding of the concept.Lecture 2: we will discuss the construction of K-moduli spaces.Lecture 3: we will focus on a new finite generation theorem we proved recently (joint with Yuchen Liu and Ziquan Zhuang), which completes the solution to several main questions in K-stability, including the compactedness of the K-moduli as well as the Yau-Tian-Donaldson Conjecture for general Fano varieties.

Xueqing Wen: Parabolic Hitchin maps and their generic fiber

Abstract: I will talk about the parabolic version of BNR correspondence, which relates certain parabolic Higgs bundles to the line bundles on the normalized spectral curve. As a consequence, we can give a modular interpretation of generic fiber of the parabolic Hitchin map, which can be applied to study the Langlands duality between parabolic Hitchin systems If time permits, I will also talk about some properties of parabolic global nilpotent cone, which is the most special fiber of the parabolic Hitchin map. This is a joint work with Xiaoyu Su and Bin Wang.

Jeongseok Oh: Counting sheaves on Calabi-Yau 4-folds

Abstract: We define a localised Euler class for isotropic sections, and isotropic cones, in SO(N) bundles. We use this to give an algebraic definition of Borisov-Joyce sheaf counting invariants on Calabi-Yau 4-folds. When a torus acts, we prove a localisation result. This talk is based on the joint work with Richard P. Thomas.

Si Li: Regularized integral and chiral elliptic index

Abstract: We explain a geometric formulation of the low energy effective theory of sigma models in terms of bundles of factorization algebras. As a case study in dimension two,  we introduce a geometric renormalization method for regularized integrals over configuration spaces of Riemann surfaces. It provides a mathematical formulation of correlation functions of non-local operators in a chiral CFT that will lead to an algebraic analogue of elliptic index theory.

Heeyeon Kim: Moduli space of vortices and 3d supersymmetric gauge theories

Abstract: I will discuss the geometric interpretation of the twisted index of 3d supersymmetric gauge theories on a closed Riemann surface. I will show that the twisted index reproduces the virtual Euler characteristic of the moduli space of solutions to vortex equations on the Riemann surface. I will also discuss 3d N = 4 mirror symmetry in this context, which implies non-trivial relations between enumerative invariants associated to the moduli space of vortices. Finally, I will comment on level structures and a wall-crossing formula of the twisted indices derived from the gauge theory point of view.

Philsang Yoo: Topological String Theory and S-duality

Abstract: S-duality is a non-trivial equivalence of two physical theories. The main aim of the talk is to explain how to mathematically understand S-duality in the context of twisted version of string theory, or more precisely, type IIB supergravity theory. Consequences of the new framework include a derivation of the geometric Langlands conjecture in a way different from the famous proposal of Kapustin and Witten. This talk is based on a joint work with Surya Raghavendran.

Huazhong Ke: Gamma conjecture II for quadrics

Abstract: Gamma conjecture II was proposed by Galkin, Golyshev and Iritani for quantum cohomology of Fano manifolds. The conjecture concerns the asymptotic behavior near irregular singularities of flat sections of Dubrovin connection of a Fano manifold X, and tries to describe the asymptotic behavior in terms of full exceptional collections of the bounded derived category of coherent sheaves of X, via the Gamma-integral structure of the quantum cohomology of X. Recently, we have obtained a sufficient condition for Gamma conjecture II, and used it to prove the conjecture for smooth quadric hypersurfaces. In this talk, we will give a brief introduction to Gamma conjecture II, and report our work. This is joint work with Xiaowen Hu.

Yang-Hui He：Universes as Bigdata: from Geometry, to Physics, to Machine-Learning

Abstract: We briefly overview how historically string theory led theoretical physics first to algebraic/differential geometry, and then to computational geometry, and now to data science. Using the Calabi-Yau landscape - accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades - as a starting-point and concrete playground, we then launch to review our recent programme in machine-learning mathematical structures and address the tantalizing question of how AI helps doing mathematics, ranging from geometry, to representation theory, to combinatorics, to number theory.