Online Seminar
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12242021
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).
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12142021
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.
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12082021
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.
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11302021
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.
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11232021
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.
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11162021
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.
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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 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.
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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.
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06252021
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.
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06152021
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.
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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 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.
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06022021
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.
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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 4-dimensional isolated symplectic singularities with trivial local fundamental groups, which answers a question of Beauville raised in 2000.
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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 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.