Online Seminar

12312022
Gregory W. Moore: Supersymmetric QFT & Invariants Of Smooth FourManifolds
Abstract. After describing briefly some aspects of nonabelian YangMills theory and Donaldson invariants the physicists' derivation of the "Witten conjecture" expressing Donaldson invariants in terms of SeibergWitten invariants will be sketched. Time permitting, more recent developments related to this physical approach will be sketched.

12192022
Xueqing Wen：Topological mirror symmetry for parabolic Higgs moduli of type B/C—From local to global
Abstract. The moduli of Higgs bundles on curve was firstly studied by Hitchin in 1980’s and has been vastly studied during the last over 30 years. One of the most remarkable observation about Higgs moduli was proposed by Hausel and Thaddeus in 2003 that there is a mirror relation between the GHiggs moduli and G^LHiggs moduli, here G^L is the Langlands dual group of G. They conjectured that there is a topological mirror symmetry between SL_n/PGL_n Higgs moduli and this conjecture was proved by Groechenig, Wyss and Ziegler also by Maulik and Shen using different methods. The mirror phenomenon of Higgs moduli was also noticed by Gokov, Kapustin and Witten by the viewpoint of physics. Especially, Gokov and Witten proposed in physics that there should be a mirror relation of parabolic Higgs bundles for Langlands dual groups and nilpotent orbits inserted at the marked points. If one considers Higgs bundles with nilpotent orbits inserted, the most interesting case is type B/C. In this talk, we will show that how to relate nilpotent orbits in type B/C using KazhdanLusztig map and loop Lie algebra, and then use these local computations to prove a topological mirror symmetry statement for parabolic Higgs moduli of type B/C using padic integration. This program was suggested by Prof. Ruan, and it is a joint work with Weiqiang He, Xiaoyu Su, Bin Wang and Yaoxiong Wen.

12142022
Yang Zhou：A generalization of mixedspinP fields
Abstract. The theory of MixedSpin fields was introduced by ChangLiLiLiu for the quintic threefold. ChangGuoLi has successfully applied it to prove famous conjectures on the highergenus GromovWitten invariants. In this talk I will explain a generalization of the construction to more spaces. The key is the stability condition which guarantees the separatedness and properness of certain moduli spaces. It also generalizes the construction of the mathematical Gauged Linear Sigma Model by FanJarvisRuan, removing their technique assumption about good lifitings.This is a joint work with HuaiLiang Chang, Shuai Guo, Jun Li and WeiPing Li.

12072022
Abstract. In this talk we will discuss a construction of moduli space of stable parabolic vector bundles $\overline{\mathfrak{U}}_{_{g, n, r}}$ over $\overline{M}_{_{g, n}}$. The objects that appear over the boundary of $\overline{M}_{_{g, n}}$ i.e., over singular curves will remain vector bundles. We will also discuss about the singularity of the total space and the fiber over a marked stable curve which are "nicer" than their torsion free counterpart.

12012022
Minxin Huang: Quantum Periods and TBAlike Equations for a Class of CalabiYau Geometries
Abstract. We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)like difference equations, generalize previous works to a large class of CalabiYau geometries described by threeterm quantum operators. We give two methods to derive the TBAlike equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBAlike equations which are nevertheless related to the same quantum period.

11252022
Changjian Su: Motivic Chern classes of Schubert cells and applications
Abstract: Motivic Chern classes in Ktheory are generalizations of the MacPherson classes in homology. I will talk about some recent developments about motivic Chern classes of the Schubert cells in the flag varieties, and their applications to representations of padic dual groups and the Ktheoretic stable envelopes.

11182022
Dan Xie: One classification of theories with eight supercharges
Abstract. We will discuss the classification of theories with eight supercharges by using the associated Coulomb branch geometry,which is described by a mixed Hodge module over Coulomb branch. The classification of rank one case is related to classification of rational elliptic surface,and the rank two is related to genus two Lefchetz fibration over P^1. Arithmetic aspects such as MordeilWeil lattice would also play an important role.

11102022
Shizhuo Zhang：Categorical Torelli problem for Fano varieties
Abstract: It is well known that for a smooth Fano variety, bounded derived category of coherent sheaves determine their isomorphism classes. It is natural to ask whether it is possible to reconstruct them with less information, say a semiorthogonal component, known as categorical Torelli problem. I will talk about recent progress on this problem, in particular for Fano threefolds and fourfolds.It is based on the work myself and the joint work with Zhiyu Liu, Augustinas Jacovskis and Soheyla Feyzbakhsh.

11032022
Yingchun Zhang: Seiberg Duality conjecture for finite type quivers and GromovWitten Theory
Abstract: This is the second work on Seiberg Duality. This work proves that the Seiberg duality conjecture holds for starshaped quivers: the GromovWitten theories before and after a quiver mutation at the center node of a starshaped quiver are equivalent.In particular, it is known that a $D_4$type quiver goes back to itself after finite times quiver mutations. We prove that GromovWitten theories of the varieties obtained by those finite quiver mutations are equivalent with nontrivial transformations on their k\ahler variables. Furthermore, GromovWitten theory and k\ahler variables of $D_4$ go back to the original ones after finite times quiver mutations. This is a joint work with Weiqiang He.

10272022
Zhiyu Liu: Castelnuovo bound and GromovWitten invariants of the quintic 3fold
Abstract：One of the most challenging problems in geometry and physics is to compute higher genus GromovWitten (GW) invariants of compact CalabiYau 3folds, such as the famous quintic 3fold. I will briefly describe how physicists compute GWinvariants of the quintic 3fold up to genus 53, using five mathematical conjectures. Three of them were proved and one of the remaining two conjectures was solved in some genus. I will explain how to prove the last open one, referred to as Castelnuovo bound in the literature. This talk is based on the joint work with Yongbin Ruan.

10132022
Younghan Bae: Surfaces on CalabiYau fourfolds
Abstract: Consider a smooth projective CalabiYau 4fold X(including hyperkahler and abelian 4folds) over C. By the work of BorisovJoyce/Oh Thomas, the moduli space of stable sheaves on X with fixed Chern character has a virtual fundamental class. This cycle vanishes when the (2,2)class of the Chern character is Hodge theoretically `nontrivial'. I will fix this issue by using KiemLi/KiemPark cosection technique. Also, I will try to explain why the surface counting theory has its own unique feature compared to curve counting theories. This talk is based on joint projects with Martijn Kool and Hyeonjun Park.

10102022
Elba GarciaFailde: The negative counterpart of Witten’s rspin conjecture
Abstract: In 1990, Witten conjectured that the generating series of intersection numbers of psi classes is a tau function of the KdV hierarchy. This was first proved by Kontsevich. In 2017, Norbury conjectured that the generating series of intersection numbers of psi classes times a negative square root of the canonical bundle is also a tau function of the KdV hierarchy. In joint work with N. Chidambaram and A. Giacchetto (https://arxiv.org/abs/2205.15621), we prove Norbury’s conjecture and obtain polynomial relations among kappa classes which were recently conjectured by KazarianNorbury. We also introduce a new collection of cohomology classes, which correspond to negative rth roots (previously r=2) of the canonical bundle and form a cohomological field theory (CohFT), the negative analogue of Witten’s rspin CohFT, which turns out to be geometrically much simpler. We prove that the corresponding intersection numbers can be computed recursively using topological recursion (which I will briefly introduce) and, equivalently, Wconstraints. The strategy draws inspiration from our proof, together with S. Charbonnier (https://arxiv.org/abs/2203.16523), of Witten’s rspin conjecture from 1993 (Faber—Shadrin—Zvonkine’s theorem from 2010) that claims that (positive) rspin intersection numbers satisfy the rKdV hierarchy. We also obtain new (tautological) relations on the moduli space of curves in a (negative) analogous way to PandharipandePixtonZvonkine. The talk will be an overview of these four topics (r=2/>2; positive/negative) and their connections.

09222022
Yaoxiong Wen: Mirror symmetry for special nilpotent orbit closures
Abstract: Inspired by the work of GukovWitten, we investigate stringy Epolynomials of nilpotent orbit closures of type $B_n$ and $C_n$. Classically, there is a famous Springer duality between special orbits. Therefore, it is natural to speculate that the mirror symmetry we seek may coincide with Springer duality in the context of special orbits. Unfortunately, such a naive statement fails. To remedy the situation, we propose a conjecture which asserts the mirror symmetry for certain parabolic/induced covers of special orbits. Then, we prove the conjecture for Richardson orbits and obtain certain partial results in general. In the mirror symmetry, we find an interesting seesaw phenonem where Lusztig's canonical quotient group plays an important role. This talk is based on the joint work with Baohua Fu and Yongbin Ruan, https://arxiv.org/abs/2207.10533.

09132022
Hu Zhao: Commutativity of quantization and reduction for quiver representations
Abstract: Given a finite quiver, its double may be viewed as its noncommutative “cotangent” space, and hence is a noncommutative symplectic space. CrawleyBoevey, Etingof and Ginzburg constructed the noncommutative reduction of this space while Schedler constructed its quantization. We show that the noncommutative quantization and reduction commute with each other. Via the quantum and classical trace maps, such a commutativity induces the commutativity of the quantization and reduction on the space of quiver representations.