Online Seminar

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.

09062022
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.

09022022
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 semilogcanonical 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 semilogcanonical surfaces to projective varieties.

06202022
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 GromovWitten or GopakumarVafa 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.

06142022
Bohan Fang: Homological mirror symmetry for toric CalabiYau 3orbifolds via constructible sheaves
The mirror of a toric CalabiYau 3orbifold is an affine curve usually referred as mirror curve. I will describe a constructible sheaf model for the Fukaya category on this curve by DyckerhoffKapranov. We first identify the Bmodel 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.

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.

05312022
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.

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.

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.