Online Seminar
-
05102022
Tyler Kelly:Open FJRW theory and Mirror Symmetry
Abstract: A Landau-Ginzburg (LG) model is a triplet of data (X, W, G) consisting of a regular function W:X à C from a quasi-projective variety X with a group G acting on X leaving W invariant. An enumerative theory developed by Fan, Jarvis, and Ruan gives FJRW invariants, the analogue of Gromov-Witten invariants for LG models. We define a new open enumerative theory for certain Landau-Ginzburg LG models. Roughly speaking, this involves computing specific integrals on certain moduli of disks with boundary and interior marked points. One can then construct a mirror Landau-Ginzburg model to a Landau-Ginzburg model using these invariants. This allows us to prove a mirror symmetry result analogous to that established by Cho-Oh, Fukaya-Oh-Ohta-Ono, and Gross for mirror symmetry for toric Fano manifolds. If time permits, I will explain some key features that this enumerative geometry enjoys (e.g., open topological recursion relations and wall-crossing phenomena). This is joint work with Mark Gross and Ran Tessler.
-
04262022
Yuchen Fu:Kazhdan-Lusztig Equivalence at the Iwahori Level
Abstract:We construct an equivalence between Iwahori-integrable representations of affine Lie algebras and representations of the mixed quantum group, thus confirming a conjecture by Gaitsgory. Our proof utilizes factorization methods: we show that both sides are equivalent to algebraic/topological factorization modules over a certain factorization algebra, which can then be compared via Riemann-Hilbert. On the quantum group side this is achieved via general machinery of homotopical algebra, whereas the affine side requires inputs from the theory of (renormalized) ind-coherent sheaves as well as compatibility with global Langlands over P1. This is joint work with Lin Chen.
-
04192022
Abstract: The moduli spaces of pointed stable curves have played a major role in enumerative algebraic geometry. Much is known about their cohomology but we still don't have a complete understanding of the symmetric group actions by permuting the marked points. I will talk about a new construction of the moduli spaces of pointed stable curves of genus 0, by an investigation on wall crossings of moduli spaces of quasimaps, which was motivated by the Landau-Ginzburg/Calabi-Yau correspondence. Using this construction, we give a closed formula for the characters of the symmetric group actions on the cohomology. Motivated by Manin and Orlov's question about the existence of an equivariant full exceptional sequece in the derived category of the moduli spaces, it is natural to ask if the cohomology groups are permutation representations or not. Using our closed formula, we provide partial answers to this question. Based on a joint work - arXiv:2203.05883 - with Jinwon Choi and Donggun Lee.
-
04122022
Hongjie Yu:l-adic local systems, Higgs bundles and Arthur trace formulas
Abstract: Deligne raised the question of counting l-adic local systems on a curve and made some conjectures about it. The global Langlands correspondence proved by L. Lafforgue offers the possibility to do the counting by enumerating some cuspidal automorphic representations which can be done using Arthur's trace formula. From another perspective, Ngô, Chaudoaurd and Laumon observed a relation between the moduli of semi-stable Higgs bundles and a Lie algebra analogue of the trace formula. We relate Arthur's trace formula to its Lie algebra analogue, which implies a numerical relation between Higgs bundles and l-adic local systems and it proves some new cases of Deligne's conjectures.
-
04082022
Junping Jiao:Boundedness of polarised Calabi-Yau fibrations
Abstract: In this talk, we investigate the boundedness of good minimal models with intermediate Kodaira dimensions. We prove that good minimal models are bounded modulo crepant birational when the base (canonical models) are bounded and the general fibers of the Iitaka fibration are in a bounded family of polarized Calabi-Yau pairs. As a corollary, we prove that smooth Calabi-Yau varieties with a polarised fibration structure are bounded modulo flop.
-
03292022
Georg Oberdieck:Holomorphic anomaly equations for the Hilbert schemes of points of a K3 surface
Abstract: Holomorphic anomaly equations are structural properties predicted by physics for the Gromov-Witten theory of Calabi-Yau manifolds. In this talk I will explain the conjectural form of these equations for the Hilbert scheme of points of a K3 surface, and explain how to prove them for genus 0 and up to three markings. As a corollary, for fixed n, the (reduced) quantum cohomology of Hilb^n K3 is determined up to finitely many coefficients.
-
03172022
Du Pei: On Quantization of Coulomb Branches
Abstract: Quantum field theories can often be used to uncover hidden algebraic structures in geometry and hidden geometric structures in algebra. In this talk, I will demonstrate how such a phase transition can relate the moduli space of Higgs bundles with the moduli space of vortices.
-
03072022
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.
-
02182022
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.
-
01252022
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.
-
01182022
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.
-
01122022
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.
-
01042022
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.
-
12282021
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.