学术科研

06072022
TsungJu Lee：Mirror symmetry and CalabiYau fractional complete intersections
Abstract: Recently, Hosono, Lian, Takagi, and Yau studied the family of K3 surfaces arising from double covers of \(\mathbf{P}^{2}\) branch over six lines in general position and proposed a singular version of mirror symmetry. In this talk, I will review their results on K3 surfaces and introduce a construction of mirror pairs of certain singular CalabiYau varieties based on Batyrev and Borisov's dual nefpartitions. I will discuss the topological test as well as the quantum test on these singular CalabiYau pairs. If time permits, I will also discuss some relevant results on the B side of these singular CalabiYau varieties. This is based on the joint works with S. Hosono, B. Lian and S.T. Yau.

05242022
Mathieu Ballandras：Intersection cohomology of character varieties for punctured Riemann surfaces
Abstract: Character varieties for punctured Riemann surfaces admit natural resolutions of singularities. Those resolutions are constructed thanks to Springer Theory and they carry a Weyl group action on their cohomology. This structure gives a relation between intersection cohomology of character varieties and cohomology of the resolutions. Thanks to a result from Mellit for smooth character varieties, previous relation allows to compute the Betti numbers for intersection cohomology.

05242022
Guangbo Xu：Integervalued GromovWitten type invariants for symplectic manifolds
Abstract: GromovWitten invariants for a general target are rationalvalued but not necessarily integervalued. This is due to the contribution of curves with nontrivial automorphism groups. In 1997 Fukaya and Ono proposed a new method in symplectic geometry which can count curves with a trivial automorphism group. While ordinary GromovWitten invariants only use the orientation on the moduli spaces, this integervalued counts are supposed to use also the (stable) complex structure on the moduli spaces. In this talk I will present the recent joint work with Shaoyun Bai in which we rigorously defined the integervalued GromovWitten type invariants in genus zero for a symplectic manifold and describe a conjecture relating the ordinary GromovWitten invariants with the integer counts. This talk is based on the preprint https://arxiv.org/abs/2201.02688.

05172022
Ruotao Yang: Untwisted Gaiotto equivalence for GL(MN)
Abstract: this is a joint work with Roman Travkin. A conjecture of Davide Gaiotto predicts that the category of representations of quantum supergroup U_q(gl(MN)) can be realized as a category of twisted Dmodules with certain equivariant condition on the affine Grassmannian Gr_N. The untwisted version of the above conjecture says that the category of representations of the degenerate supergroup is equivalent to the category of (nontwisted) Dmodules, with the same equivariant condition on Gr_N. In the case of M=N1 and M=N, the latter was proved by A. Braverman, M. Finkelberg, V. Ginzburg, and R. Travkin. In this paper, we proved all other cases.Also, we prove that we can realize the category of representations of the degenerate supergroup as a category of Dmodules on the mirabolic subgroup Mir_L(F) with certain equivariant conditions for any L bigger than N and M.

05102022
Tyler Kelly：Open FJRW theory and Mirror Symmetry
Abstract: A LandauGinzburg (LG) model is a triplet of data (X, W, G) consisting of a regular function W:X à C from a quasiprojective 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 GromovWitten invariants for LG models. We define a new open enumerative theory for certain LandauGinzburg 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 LandauGinzburg model to a LandauGinzburg model using these invariants. This allows us to prove a mirror symmetry result analogous to that established by ChoOh, FukayaOhOhtaOno, 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 wallcrossing phenomena). This is joint work with Mark Gross and Ran Tessler.

04262022
Yuchen Fu：KazhdanLusztig Equivalence at the Iwahori Level
Abstract：We construct an equivalence between Iwahoriintegrable 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 RiemannHilbert. 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) indcoherent 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 LandauGinzburg/CalabiYau 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：ladic local systems, Higgs bundles and Arthur trace formulas
Abstract: Deligne raised the question of counting ladic 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 semistable 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 ladic local systems and it proves some new cases of Deligne's conjectures.

04082022
Junping Jiao：Boundedness of polarised CalabiYau 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 CalabiYau pairs. As a corollary, we prove that smooth CalabiYau 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 GromovWitten theory of CalabiYau 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.

03112022
Eduardo Gonzalez：NilHecke 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 3d QFTs, in particular, its manifestation as duality between symplectic resolutions of singularities, and the representation theory of corresponding noncommutative algebras.