Online Seminar
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04132021
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.
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04062021
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.
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03092021
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.
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03022021
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.
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01192021
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.
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01122021
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.
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01052021
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.
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12222020
Murad Alim: Difference equations for Gromov-Witten potentials
Abstract: I will discuss a difference equation for the Gromov-Witten potential of the resolved conifold. Using the Gopakumar-Vafa resummation of the Gromov-Witten invariants of any Calabi-Yau threefold, I will further show that similar difference equations are satisfied by the part of the resummed potential containing the contribution of the genus zero GV invariants.
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12152020
Todor Milanov: Matrix model for the total descendent potential of a simple singularity of type D
Abstract: I am planning to talk about my recent paper, joint with Alexander Alexandrov, with the same title. It was conjectured by Witten that the intersection theory on the moduli space of curves is governed by the KdV hierarchy. The conjecture was first proved by Kontsevich. He was able to express the intersection numbers in terms of the asymptotic expansion of a certain Hermitian matrix integral and to prove that the asymptotic expansion coincides with a tau-function of KdV given in the so-called Miwa parametrization. We were able to construct a Hermitian matrix integral whose asymptotic expansion coincides with the Miwa parametrization of the total descendent potential of a simple singularity of type D. Recalling a LG/LG mirror symmetry result of Fan--Jarvis--Ruan, we can also conclude that the asymptotic expansion of our integral gives the FJRW-invariants of the Berglund--Hubsch dual singularity D^T.
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12082020
Wei Gu: Nonabelian mirrors and Gromov-Witten invariants
Abstract: We propose Picard-Fuchs equations for periods of nonabelian mirrors [1] in this paper. Thenumber of parameters in our Picard-Fuchs equations is the rank of the gauge group in the nonabelian GLSM, although we reduce them to the actual number of Kähler parameters ultimately. These Picard-Fuchs equations are concise but novel, so we reproduce existing mathematical Picard-Fuchs equations of Gr(k,N) and Calabi-Yau manifolds as complete intersections in Grassmannians [2] to justify our proposal. Furthermore, our approach can be applied to other nonabelian GLSMs, so we compute mathematical Picard-Fuchs equations of some other Fano-spaces, which were not calculated in the literature before. Finally, the cohomology-valued generating functions of mirrors can be read off from our Picard-Fuchs equations. Using these generating functions, we compute Gromov-Witten invariants of various Calabi-Yau manifolds including complete intersection Calabi-Yau manifolds in Grassmannians and non-complete intersection Calabi-Yau examples such as Pfaffian Calabi- Yau threefold and Gulliksen-Negård Calabi-Yau threefold, and find agreement with existing results in the literature. However, the generating functions we propose for the non-complete intersection Calabi-Yau manifolds are genuinely new.
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12012020
Jeremy Guere: Congruences on K-theoretic Gromov-Witten invariants
Abstract: K-theoretic Gromov-Witten invariants of smooth projective varieties have been introduced by YP Lee, using the Euler characteristic of a virtual structure sheaf. In particular, they are integers. In this talk, I look at these invariants for the quintic threefold and I will explain how to compute them modulo 41, using the virtual localization formula under a finite group action, up to genus 19 and degree 40.
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11242020
Yaoxiong Wen: 3d Toric Mirror Symmetry
Abstract: In this talk, I will introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d N=2 abelian mirror symmetry construction in physics introduced by Dorey-Tong. More precisely, for a short exact sequence 0 -> Z^k -> Z^n -> Z^{n-k} -> 0,we consider the toric Artin stack $[C^n/(C^*)^k]$, and its mirror is given by the Gale dual of the above exact sequence, i.e., $[C^n/(C^*)^{n-k}]$. We introduce the modified equivariant K-theoretic I-functions for the mirror pair; they are defined by the contribution of fixed points. Under the mirror map, which switches the Kälher parameters and equivariant parameters and maps $q$ to $q^{-1}$, we see that modified I-functions with the effective level structure of mirror pair coincide. This talk is based on the joint work with Yongbin Ruan and Zijun Zhou.
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11172020
Ming Zhang: Verlinde/Grassmannian Correspondence
Abstract: In the 90s', Witten gave a physical derivation of an isomorphism between the Verlinde algebra of $GL(n)$ of level $l$ and the quantum cohomology ring of the Grassmannian $\text{Gr}(n,n+l)$. In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten's work by relating the $\text{GL}_{n}$ Verlinde numbers to the level $l$ quantum K-invariants of the Grassmannian $\text{Gr}(n,n+l)$, and refer to it as the Verlinde/Grassmannian correspondence. The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there. In this talk, I will first explain the background of this correspondence and its interpretation in physics. Then I will discuss the main idea of the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner.
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11102020
Will Donovan: Windows on the Pfaffian-Grassmannian correspondence
Abstract: The Pfaffian-Grassmannian correspondence relates certain pairs of non-birational Calabi-Yau threefolds which can be proved to be derived equivalent. I construct a family of derived equivalences using mutations of an exceptional collection on the relevant Grassmannian, and explain a mirror symmetry interpretation. This follows a physical analysis of Landau-Ginzburg B-models by Eager, Hori, Knapp, and Romo, and builds on work with Addington and Segal.