Online Seminar
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11032022
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.
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10272022
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.
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10132022
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.
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10102022
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.
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09222022
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.
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09132022
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.
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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.
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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 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.
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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 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.
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06142022
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.
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06072022
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.
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05312022
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.
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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.
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05172022
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.