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.

Yunfeng Jiang： The virtual fundamental class for the moduli space of general type surfaces

Abstract:  Sir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class on the moduli stack of  lci (locally complete intersection) covers over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the virtual fundamental class. This can be taken as a generalization of the tautological invariants on the moduli space of stable  curves to the moduli space of stable surfaces.  The method presented  can also be applied to the moduli space of stable map spaces from semi-log-canonical surfaces to projective varieties.

Sheldon Katz：Enumerative Geometry of the Mirror Quintic

Abstract: Motivated by a prediction of Hayashi, Jefferson, Kim, Ohmori, and Vafa based on a proposed generalization of the topological vertex formalism, the Gromov-Witten or Gopakumar-Vafa invariants of low degree are computed for the mirror quintic, involving all 101 Kahler parameters.  A conjectural description of the Mori cone of the mirror quintic is also proposed, with evidence; however the validity of the computed invariants are not dependent on this conjecture.  

Bohan Fang: Homological mirror symmetry for toric Calabi-Yau 3-orbifolds via constructible sheaves

The mirror of a toric Calabi-Yau 3-orbifold is an affine curve usually referred as mirror curve. I will describe a constructible sheaf model for the Fukaya category on this curve by Dyckerhoff-Kapranov. We first identify the B-model category of matrix factorization with this category in the case of affine orbifolds, where the mirror curve is a multiple cover of a pair of pants. By the descent properties on both sides, one can proceed to the general homological mirror symmetry statement by gluing together affine pieces. This talk is based on the joint work with Qingyuan Bai.

Tsung-Ju Lee：Mirror symmetry and Calabi--Yau 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 Calabi--Yau varieties based on Batyrev and Borisov's dual nef-partitions. I will discuss the topological test as well as the quantum test on these singular Calabi--Yau pairs. If time permits, I will also discuss some relevant results on the B side of these singular Calabi--Yau varieties. This is based on the joint works with S. Hosono, B. Lian and S.-T. Yau.

Guangbo Xu：Integer-valued Gromov-Witten type invariants for symplectic manifolds

Abstract:  Gromov-Witten invariants for a general target are rational-valued but not necessarily integer-valued. 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 Gromov-Witten invariants only use the orientation on the moduli spaces, this integer-valued 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 integer-valued Gromov-Witten type invariants in genus zero for a symplectic manifold and describe a conjecture relating the ordinary Gromov-Witten invariants with the integer counts. This talk is based on the preprint https://arxiv.org/abs/2201.02688.

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.

Ruotao Yang: Untwisted Gaiotto equivalence for GL(M|N)

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(M|N)) can be realized as a category of twisted D-modules 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 (non-twisted) D-modules, with the same equivariant condition on Gr_N. In the case of M=N-1 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 D-modules on the mirabolic subgroup Mir_L(F) with certain equivariant conditions for any L bigger than N and M.

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.

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.