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Speaker: Jie Zhou, Tsinghua UniversityTitle: Regularized integrals on Riemann surfaces and correlations functions in 2d chiral CFTsVenue: virtualAbstract: I will report a recent approach of regularizing divergent integrals on configuration spaces of Riemann surfaces, introduced by Si Li and myself in arXiv:2008.07503, with an emphasis on genus one cases where modular forms arise naturally. I will then talk about some applications in studying correlation functions in 2d chiral CFTs, holomorphic anomaly equations, etc. If time permits, I will also mention a more algebraic formulation of this notion of regularized integrals in terms of mixed Hodge structures. The talk is partially based on joint works with Si Li. |
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Speaker: Benjamin Gammage, Harvard UniversityTitle: 2-categorical 3d mirror symmetryVenue: virtualAbstract: It is by now well-known that mirror symmetry may be expressed as an equivalence between categories associated to dual Kahler manifolds. Following a proposal of Teleman, we inaugurate a program to understand 3d mirror symmetry as an equivalence between 2-categories associated to dual holomorphic symplectic stacks. We consider here the abelian case, where our theorem expresses the 2-category of spherical functors as a 2-category of coherent sheaves of categories. Applications include categorifications of hypertoric category O and of many related constructions in representation theory. This is joint work with Justin Hilburn and Aaron Mazel-Gee. |
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Speaker: Dhyan Vas Aranha, SISSATitle: Virtual localization for Artin stacksVenue: virtualAbstract: This is a report about work in progress with: Adeel Khan, Aloysha Latyntsev, Hyeonjun Park and Charanya Ravi. We will describe a virtual Atiyah-Bott formula for Artin stacks. In the Deligne-Mumford case our methods allow us to remove the global resolution hypothesis for the virtual normal bundle. |
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Speaker: Konstantin Aleshkin, Columbia UniversityTitle: Higgs-Coulomb correspondence in abelian GLSMVenue: virtualAbstract: We construct a certain type of Gauged Linear Sigma Model quasimap invariants that generalize the original ones and are easier to compute. Higgs-Coulomb correspondence provides identification of generating functions of our invariants with certain analytic functions that can be represented as generalized inverse Mellin transforms. Analytic continuation of these functions provides wall-crossing results for GLSM and generalizes Landau- Ginzburg/Calabi-Yau correspondence. The talk is based on a joint work in progress with Melissa Liu. |
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Speaker: Zijun Zhou, Kavli IPMUTitle: Virtual Coulomb branch and quantum K-theoryVenue: virtualAbstract: In this talk, I will introduce a virtual variant of the quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of quasimaps into the holomorphic symplectic quotient T^*N//G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under variation of GIT. |
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Speaker: Carl LianTitle: Curve-counting with fixed domain (“Tevelev degrees”)Venue: VirtualAbstract: We will consider the following problem: if (C,x_1,…,x_n) is a fixed general pointed curve, and X is a fixed target variety with general points y_1,…,y_n, then how many maps f:C -> X in a given homology class are there, such that f(x_i)=y_i? When considered virtually in Gromov-Witten theory, the answer may be expressed in terms of the quantum cohomology of X, leading to explicit formulas in some cases (Buch-Pandharipande). The geometric question is more subtle, though in the presence of sufficient positivity, it is expected that the virtual answers are enumerative. I will give an overview of recent progress on various aspects of this problem, including joint work with Farkas, Pandharipande, and Cela, as well as work of other authors. |
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Speaker: Yu-Shen Lin, Boston UniversityTitle: SYZ Conjecture beyond Mirror SymmetryVenue: virtualAbstract: Strominger-Yau-Zaslow conjecture is one of the guiding principles in mirror symmetry, which not only predicts the geometric structures of Calabi-Yau manifolds but also provides a recipe for mirror construction. Besides mirror symmetry, the SYZ conjecture itself is the holy grail in geometrical analysis and closely related to the behavior of the Ricci-flat metrics. In this talk, we will explain how SYZ fibrations on log Calabi-Yau surfaces detect the non-standard semi-flat metric which generalized the semi-flat metrics of Greene-Shapere-Vafa-Yau. Furthermore, we will use the SYZ fibration on log Calabi-Yau surfaces to prove the Torelli theorem of gravitational instantons of type ALH^*. This is based on the joint works with T. Collins and A. Jacob. |
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Speaker: Xingyang YuTitle: 2d N=(0,1) gauge theories, Spin(7) orientifolds and trialityVenue: Virtual |
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Speaker: Dori BejleriTitle: Wall crossing for moduli of stable log varietiesVenue: Virtual |
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Speaker: Pierrick BousseauTitle: Gromov-Witten theory of complete intersectionsVenue: VirtualAbstract: I will describe an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. This is joint work with Hülya Argüz, Rahul Pandharipande, and Dimitri Zvonkine (arxiv:2109.13323). |
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Speaker:Title: Cosection localization for virtual fundamental classes of d-manifolds and Donaldson-Thomas invariants of Calabi-Yau fourfoldsVenue: VirtualSpeaker: Michail Savvas, UT Austin Title: Cosection localization for virtual fundamental classes of d-manifolds and Donaldson-Thomas invariants of Calabi-Yau fourfolds Abstract: Localization by cosection, first introduced by Kiem-Li in 2010, is one of the fundamental techniques used to study invariants in complex enumerative geometry. Donaldson-Thomas (DT) invariants counting sheaves on Calabi-Yau fourfolds were first defined by Borisov-Joyce in 2015 by combining derived algebraic and differential geometry. In this talk, we develop the theory of cosection localization for derived manifolds in the context of derived differential geometry of Joyce. As a consequence, we also obtain cosection localization results for (-2)-shifted symplectic derived schemes. This provides a cosection localization formalism for the Borisov-Joyce DT invariant. As an immediate application, the stable pair… |
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Speaker: Michail SavvasTitle: Cosection localization for virtual fundamental classes of d-manifolds and Donaldson-Thomas invariants of Calabi-Yau fourfoldsVenue: VirtualAbstract: Localization by cosection, first introduced by Kiem-Li in 2010, is one of the fundamental techniques used to study invariants in complex enumerative geometry. Donaldson-Thomas (DT) invariants counting sheaves on Calabi-Yau fourfolds were first defined by Borisov-Joyce in 2015 by combining derived algebraic and differential geometry. In this talk, we develop the theory of cosection localization for derived manifolds in the context of derived differential geometry of Joyce. As a consequence, we also obtain cosection localization results for (-2)-shifted symplectic derived schemes. This provides a cosection localization formalism for the Borisov-Joyce DT invariant. As an immediate application, the stable pair invariants of hyperkähler fourfolds, constructed by Maulik-Cao-Toda, vanish, as expected. |
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Speaker: Hossein MovasatiTitle: Gauss-Manin connection in disguise: Quasi Jacobi forms of index zeroVenue: VirtualAbstract: We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields. This is a joint work with J. Cao and R. Villaflor. (arXiv:2109.00587) |
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Speaker: Xiaowen HuTitle: On singular Hilbert schemes of pointsVenue: VirtualAbstract: It is well known that the Hilbert schemes of points on smooth surfaces are smooth. In higher dimensions the Hilbert schemes of points are in general singular. In this talk we will present some examples and conjectures on the local structures of the Hilbert scheme of points on $\mathbb{P}^3$. As an application we study a conjecture of Wang-Zhou on the Euler characteristics of the tautological sheaves on Hilbert schemes of points. |
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Speaker: Andrea T. RicolfiTitle: D-critical structure(s) on Quot schemes of points of Calabi-Yau 3-foldsVenue: VirtualAbstract: D-critical schemes and Artin stacks were introduced by Joyce in 2015, and play a central role in Donaldson-Thomas theory. They typically occur as truncations of (-1)-shifted symplectic derived schemes, but the problem of constructing the d-critical structure on a “DT moduli space” without passing through derived geometry is wide open. We discuss this problem, and new results in this direction, when the moduli space is the Hilbert (or Quot) scheme of points on a Calabi-Yau 3-fold. Joint work with Michail Savvas. |
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Speaker: Qingyuan JangTitle: Derived projectivizations of two-term complexesVenue: VirtualAbstract: For a given two-term complex of vector bundles on a derived scheme (or stack), there are three natural ways to define its “derived projectivizations”: (i) as the derived base-change of the classical projectivization of Grothendieck; (ii) as the derived moduli parametrizing one-dimensional locally free quotients; (iii) as the GIT quotient of the total space by $\mathbb{G}_m$-action. In this talk, we first show that these three definitions are equivalent. Second, we prove a structural theorem about the derived categories of derived projectivizations and study the corresponding mutation theory. Third, we apply these results to various moduli situations, including the moduli of certain stable pairs on curves and the Hecke correspondences of one-point modification of moduli of stable sheaves on… |
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Speaker: Mark ShoemakerTitle: A mirror theorem for GLSMsVenue: VirtualAbstract: A gauged linear sigma model (GLSM) consists roughly of a complex vector space V, a group G acting on V, a character \theta of G, and a G-invariant function w on V. This data defines a GIT quotient Y = [V //_\theta G] and a function on that quotient. GLSMs arise naturally in a number of contexts, for instance as the mirrors to Fano manifolds and as examples of noncommutative crepant resolutions. GLSMs provide a broad setting in which it is possible to define an enumerative curve counting theory, simultaneously generalizing FJRW theory and the Gromov-Witten theory of hypersurfaces. Despite a significant effort to rigorously define the enumerative invariants of a GLSM, very few computations of these invariants… |
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Speaker: Alan ThompsonTitle: The Mirror Clemens-Schmid SequenceVenue: VirtualAbstract: I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a “mirror P=W” conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting. This is joint work… |