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 G-Higgs moduli and G^L-Higgs 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 Kazhdan-Lusztig 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 p-adic integration. This program was suggested by Prof. Ruan, and it is a joint work with Weiqiang He, Xiaoyu Su, Bin Wang and Yaoxiong Wen.

Yang Zhou：A generalization of mixed-spin-P fields

Abstract. The theory of Mixed-Spin fields was introduced by Chang-Li-Li-Liu for the quintic threefold. Chang-Guo-Li has successfully applied it to prove famous conjectures on the higher-genus Gromov-Witten 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 Fan-Jarvis-Ruan, removing their technique assumption about good lifitings.This is a joint work with Huai-Liang Chang, Shuai Guo, Jun Li and Wei-Ping Li.

Jagadish Pine：A universal moduli space of stable parabolic vector bundles over the marked stable curves

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.

Minxin Huang: Quantum Periods and TBA-like Equations for a Class of Calabi-Yau 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 Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

Changjian Su: Motivic Chern classes of Schubert cells and applications

Abstract: Motivic Chern classes in K-theory 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 p-adic dual groups and the K-theoretic stable envelopes. 

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 Mordeil-Weil lattice would also play an important role.

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 semi-orthogonal 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.

Yingchun Zhang: Seiberg Duality conjecture for finite type quivers and Gromov-Witten Theory

Abstract: This is the second work on Seiberg Duality. This work proves that the Seiberg duality conjecture holds for star-shaped quivers: the Gromov-Witten theories before and after a quiver mutation at the center node of a star-shaped 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  Gromov-Witten theories of the varieties obtained by those finite quiver mutations are equivalent with non-trivial transformations on their k\ahler variables.  Furthermore, Gromov-Witten 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.

Zhiyu Liu: Castelnuovo bound and Gromov-Witten invariants of the quintic 3-fold

Abstract：One of the most challenging problems in geometry and physics is to compute higher genus Gromov-Witten (GW) invariants of compact Calabi-Yau 3-folds, such as the famous quintic 3-fold. I will briefly describe how physicists compute GW-invariants of the quintic 3-fold 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.

Younghan Bae: Surfaces on Calabi-Yau fourfolds

Abstract: Consider a smooth projective Calabi-Yau 4-fold X(including hyperkahler and abelian 4-folds) over C. By the work of Borisov-Joyce/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 Kiem-Li/Kiem-Park 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.

Elba Garcia-Failde: The negative counterpart of Witten’s r-spin 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 Kazarian--Norbury. We also introduce a new collection of cohomology classes, which correspond to negative r-th roots (previously r=2) of the canonical bundle and form a cohomological field theory (CohFT), the negative analogue of Witten’s r-spin 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, W-constraints. The strategy draws inspiration from our proof, together with S. Charbonnier (https://arxiv.org/abs/2203.16523), of Witten’s r-spin conjecture from 1993 (Faber—Shadrin—Zvonkine’s theorem from 2010) that claims that (positive) r-spin intersection numbers satisfy the r-KdV hierarchy. We also obtain new (tautological) relations on the moduli space of curves in a (negative) analogous way to Pandharipande--Pixton--Zvonkine. The talk will be an overview of these four topics (r=2/>2; positive/negative) and their connections.

Yaoxiong Wen: Mirror symmetry for special nilpotent orbit closures

Abstract: Inspired by the work of Gukov-Witten, we investigate stringy E-polynomials 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.

Hu Zhao: Commutativity of quantization and reduction for quiver representations

Abstract: Given a finite quiver, its double may be viewed as its non-commutative “cotangent” space, and hence is a non-commutative symplectic space. Crawley-Boevey, Etingof and Ginzburg constructed the non-commutative reduction of this space while Schedler constructed its quantization. We show that the non-commutative 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.

Thomas Lam: Richardson varieties and positroid varieties

Abstract: Richardson varieties are intersections of opposite Schubert varieties in a (generalized) flag variety.  Positroid varieties are similar subvarieties of the Grassmannian.  I will survey various aspects of the geometry of these spaces, and mention some relations to category O, knot homology, cluster algebras, and particle physics.