Online Seminar

12242021
Nawaz Sultani：GromovWitten invariants of some nonconvex complete intersections
Abstract: For convex complete intersections, the GromovWitten (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).

12142021
Cheng Shu：Mixed Hodge polynomial of character variety
Abstract: Hausel, Letellier and RodriguezVillegas computed the Epolynomial of character varieties with generic semisimple 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.

12082021
Hans Jockers：Remarks on Modularity in Quantum KTheory
Abstract: In this talk we present some observations about the modular properties of 3d BPS halfindices of particular N=2 3d gauge theories. These indices connect to quantum Ktheory via the 3d gauge theory quantum Ktheory correspondence. They are solutions to certain qdifference equations, which — for particular classes of N=2 3d gauge theories — relate to the theory of bilateral qseries and modular qcharacters of two dimensional conformal field theories in a certain massless limit.

11302021
Jie Zhou: Twisted Sectors in QuasiHomogeneous Polynomial Singularities and Automorphic Forms
Abstract: We study oneparameter deformations of CalabiYau type Fermat polynomial singularities along degreeone directions. We show that twisted sectors in the vanishing cohomology are automorphic forms for certain triangular groups. We prove consequentially that genus zero GromovWitten generating series of the corresponding Fermat CalabiYau varieties are components of automorphic forms. The main tools we use are period mappings for quasihomogeneous polynomial singularities, RiemannHilbert correspondence, and genus zero mirror symmetry.This is joint work with Yongbin Ruan and Dingxin Zhang.

11232021
Zijun Zhou: Virtual Coulomb branch and quantum Ktheory
Abstract: In this talk, I will introduce a virtual variant of the quantized Coulomb branch by BravermanFinkelbergNakajima, 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 Ktheoretic 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 qdifference module. As an application, this gives a proof of the invariance of the quantum qdifference module under variation of GIT.

11162021
Yang Zhou：Quasimap wallcrossing and applications
Abstract: The theory of GromovWitten 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 epsilonstable quasimaps, defined for a large class of GIT quotients. When epsilon tends to infinity, one recovers GromovWitten invariants. When epsilon tends to zero, the invariants are closely related to the Bmodel in physics. The space of epsilons has a wallandchamber structure, and a wallcrossing formula was conjectured by CiocanFontanine and Kim. In this talk, I will explain the wallcrossing phenomenon, sketch a proof of the wallcrossing formula and discuss its variants and applications.

11092021
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 Ginvariant 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 GromovWitten 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 GromovWitten invariants of Y.

10262021
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.

06252021
Xiaohan Yan: Quantum Ktheory of flag varieties via nonabelian localization
Abstract: Quantum Ktheory studies the Ktheoretic analogue of GromovWitten invariants defined as holomorphic Euler characteristics of sheaves on the moduli space of stable maps. Generating functions of such invariants, which are called the (Ktheoretic) big Jfunctions, play a crucial role in the theory. In this talk, we provide a reconstruction theorem of the permutationinvariant big Jfunction of partial flag varieties (regarded as GIT quotients of vector spaces) using a family of finitedifference operators, from the quantum Ktheory of their associated abelian quotients which is wellunderstood. The idea is to use a recursive characterization of the big Jfunctions based on the geometry of isolated fixed points and connecting 1dimensional orbits on the flag varieties in toricequivariant settings, but along the way we will need to address the issue of possibly nonisolated fixed points on the abelian quotient. A portion of this talk is based on a joint work with Alexander Givental.

06152021
Wei Gu: On phases of 3d N=2 ChernSimonsmatter 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 ChernSimonsmatter theories, extending to three dimensions the celebrated correspondence between 2d gauged WessZuminoWitten (GWZW) models and nonlinear 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 GiventalLee'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.

06082021
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 selfdual under threedimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant Ktheory 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 twoparameter family $X_{k,l}$ of selfmirror quiver varieties of type A and study their quantum Ktheory algebras. The desired quantum Ktheory 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 (qLanglands) 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 torsionfree rank$N$ sheaves on $\mathbb{P}^2$ with the help of a different (threeparametric) family of type A quiver varieties with known mirror dual.

06022021
Junliang Shen: Cohomology of the moduli of Higgs bundles and the HauselThaddeus conjecture
Cohomology of the moduli of Higgs bundles and the HauselThaddeus conjecture
Abstract: In this talk, I will discuss some structural results for the cohomology of the moduli of semistable SL_n Higgs bundles on a curve. One consequence is a new proof of the HauselThaddeus conjecture proven previously by GroechenigWyssZiegler via padic 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 semistable loci. We will explain how intersection cohomology comes into play naturally. Based on joint work with Davesh Maulik.

05132021
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 4dimensional isolated symplectic singularities with trivial local fundamental groups, which answers a question of Beauville raised in 2000.

04282021
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 Kmoduli spaces, and its fundamental properties. As a byproduct, it also completes the solution of YauTianDonaldson Conjecture to all Fano varieties case (including singular ones).Lecture 1: we will discuss the background of Kstability and algebraic geometer’s gradually evolving understanding of the concept.Lecture 2: we will discuss the construction of Kmoduli 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 Kstability, including the compactedness of the Kmoduli as well as the YauTianDonaldson Conjecture for general Fano varieties.