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Younghan Bae (ETH Zurich) - TBA
Arkadij Bojko (ETH Zurich)
Title: Wall-crossing for Calabi-Yau fourfolds and applications
Abstract: Joyce's vertex algebras are a powerful new ingredient added to the existing theory of wall-crossing for sheaves on surfaces. My work focuses on proving wall-crossing in two dimensions higher - for Calabi-Yau fourfolds. It is desirable that the end result can have many concrete applications to existing conjectures. For this purpose, I introduce yet another new structure into the picture - formal families of vertex algebras. Apart from being a natural extension of the theory, they allow to wall-cross with insertions instead of the plain virtual fundamental classes. To make the whole machinery work with (polynomial) Bridgeland stability conditions and sheaf-counting classes for fourfolds, I require a different approach compared to the surface case. In the talk, I will discuss the main difficulties that I encountered, and I will present examples using the complete package.
Christopher Brav (Moscow Institute of Physics and Technology) - TBA
Yalong Cao (RIKEN iTHEMS)
Title: From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials
Abstract: I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau 4-folds. For non-compact CY4 like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.
Jeroen Hekking (University of Regensburg) - TBA
Benjamin Hennion (University of Paris-Saclay) - TBA
Yunfeng Jiang (University of Kansas)
Title: 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. If time permits, we also talk about the possible methods to construct a virtual fundamental class on the moduli space of stable maps from semi-log-canonical surfaces to projective varieties, and especially Calabi-Yau 4-folds.
Dominic Joyce (University of Oxford)- TBA
Adeel Khan (Academia Sinica)- TBA
Tasuki Kinjo (Kyoto University)
(2)Title: Microlocal methods in enumerative geometry
Abstract: In this talk, I will present a new method for constructing virtual fundamental classes of quasi-smooth derived schemes, using the perverse sheaves of vanishing cycles. This approach is motivated by the interplay between derived geometry and microlocal geometry. Additionally, we will demonstrate how this idea leads to the construction of the critical cohomological Hall algebras for the canonical bundle of algebraic surfaces, refining the cohomological Hall algebras of Porta-Sala. This talk is partially based on joint work with Adeel Khan.
Martijn Kool (Utrecht University)- TBA
Woonam Lim (ETH Zurich)
Title: Virasoro constraints, vertex operator algebras, and wall-crossing.
Abstract: In enumerative geometry, Virasoro constraints were first conjectured for the moduli of stable curves (the Witten conjecture) and stable maps. Recently, the analogous constraints were conjectured in several sheaf theoretic contexts; stable pairs on 3-folds and torsion-free sheaves on surfaces. In joint work with A. Bojko and M. Moreira, we generalize and reinterpret Virasoro conjecture in sheaf theory using Joyce’s vertex algebra. A new interpretation makes use of a conformal element and primary states of vertex algebras which are classical subjects in representation theory. As an application, we prove the constraints for any moduli of torsion-free sheaves on curves and surfaces via Joyce's wall-crossing formulas.
Jeongseok Oh (Imperial College)- TBA
Tony Pantev (University of Pennsylvania)- TBA
Hyeonjun Park (Korea Institute for Advanced Study)
Title: A Darboux theorem and virtual Lagrangian cycles for (-2)-shifted symplectic fibrations
Abstract: There are two crucial discoveries for (-2)-shifted symplectic derived schemes: (1) Darboux theorem of Brav-Bussi-Joyce/Bouaziz-Grojnowski (2) virtual cycles of Borisov-Joyce/Oh-Thomas. In this talk, we extend these to families of (-2)-shifted symplectic derived schemes.
Firstly, we show that (-2)-shifted symplectic fibrations are (-1)-shifted Lagrangians on the derived critical loci of functions on the bases when the obstructions of exactness come from the functions. Based on this, we provide a Darboux theorem for (-2)-shifted symplectic fibrations that are not necessarily exact.
Secondly, we construct virtual Lagrangian cycles for (-2)-shifted exact symplectic fibrations. A key technical lemma is a generalized cone reduction for (-p)-shifted exact p-forms. We also present a virtual pullback formula for Lagrangian correspondences.
As an application, we prove that the Donaldson-Thomas invariants of Calabi-Yau 4-folds are invariant along the deformations for which the (0,4)-Hodge pieces of the second Chern characters remain zero. This assures that the reduced virtual cycles for counting surfaces detect the variational Hodge conjecture.
Finally, we propose a refined Donaldson-Thomas theory of Calabi-Yau 4-folds in terms of the vanishing cycle cohomology of the Hodge loci, relying on the Joyce conjecture for (-1)-shifted Lagrangians.
Marco Robalo (Sorbonne Universite)- TBA
Pavel Safronov (University of Edinburgh) - TBA
Michail Savvas (University of Texas at Austin)
Title: Stabilizer reduction and sheaf-theoretic invariants
Abstract: I will discuss a canonical blowup procedure for Artin stacks in derived algebraic geometry, which eliminates positive-dimensional stabilizer groups and generalizes the partial desingularization algorithm developed by Kirwan for smooth varieties in the context of Geometric Invariant Theory. This procedure applies in particular to moduli stacks of sheaves, allowing us to construct intersection-theoretic generalized sheaf-counting invariants on surfaces and Calabi-Yau threefolds. Based on joint work with Jeroen Hekking and David Rydh.