Woonam Lim (Utrecht University)

Title: Nekrasov’s gauge origami via DT4 theory

Abstract: The study of the classical instantons on the spacetime has led to many interesting mathematics through the associated partition functions. In a series of papers, Nekrasov introduced and studied the generalised ADHM equations, whose solutions are instantons on the “origami spacetime”. In this talk, I will explain how to interpret Nekrasov’s gauge origami via DT4 theory. Our main result determines the orientation in the Oh-Thomas localisation formula of the gauge origami partition function. After giving some applications, I will finish by presenting a conjectural sheaf-theoretic description of the gauge origami. This is joint work in progress with N. Arbesfeld and M. Kool.

Tudor Padurariu (CNRS- Jussieu)

Title:  The commuting stack via quasi-BPS categories

Abstract:  I will report on joint work with Sabin Cautis and Yukinobu Toda (in progress).

We study the derived category of the commuting stack of two matrices, alternatively of the moduli stack of dimension zero sheaves on the two dimensional affine space. 

In previous work, we constructed semiorthogonal decompositions of this category in smaller categories, called quasi-BPS categories, which we believed to be indecomposable, and we computed their (localized equivariant or topological) K-theory and explained their relation to BPS invariants of points on the three dimensional affine space.

In the current work, we compute the quasi-BPS categories. As a corollary, we prove a conjecture of Negut about relations between Hecke correspondences, and a conjecture of Gorsky-Negut about the generation of the derived category of the commuting stack. Based on previous joint work with Yukinobu Toda, we obtain a dimension three version of the Bridgeland-King-Reid and Haiman derived equivalence.

Hyeonjun Park (Korea Institute for Advanced Study)

Title: Shifted symplectic pushforwards

Abstract: We introduce how to pushforward shifted symplectic fibrations along base changes. This is achieved by considering symplectic forms that are closed in a stronger sense. Examples include: symplectic zero loci and symplectic quotients. Observing that twisted cotangent bundles are symplectic pushforwards, we obtain an equivalence between symplectic fibrations and Lagrangians to critical loci.

We provide two local structure theorems for symplectic fibrations: a smooth local structure theorem for higher stacks via symplectic zero loci and twisted cotangents, and an etale local structure theorem for 1-stacks with reductive stabilizers via symplectic quotients of the smooth local models.

We resolve deformation invariance issue in Donaldson-Thomas theory of Calabi- Yau 4-folds. Abstractly, we associate virtual Lagrangian cycles for (-2)-symplectic fibrations as unique functorial bivariant classes over the exact loci. For moduli of perfect complexes, we show that the exact loci consist of deformations for which the (0,4)-Hodge pieces of second Chern characters remain zero.

Yalong Cao (Morningside Center of Mathematics)

Title: A degeneration formula of Donaldson-Thomas theory on Calabi-Yau 4-folds

Abstract: We prove a degeneration formula for Donaldson-Thomas theory on Calabi-Yau 4-folds, and apply it to compute zero dimensional invariants on C4 and on any local curve. Joint work with Gufang Zhao and Zijun Zhou.

Adeel Khan (Academia Sinica)

Title: Perverse microsheaves and cohomological DT theory
Abstract: The Donaldson-Thomas invariants of a Calabi-Yau threefold X can be categorified to a perverse sheaf on the moduli stack of coherent sheaves on X.  I will report on work in progress towards a microlocal description of this perverse sheaf through a new theory of microsheaves on derived stacks.  The microlocal perspective seems to clarify certain conjectural aspects of cohomological Donaldson-Thomas theory.  This builds on a previous work, joint with Tasuki Kinjo, where we gave a microlocal description in the case of local surfaces.

Charanya Ravi (Tata Institute of Fundamental Research)

Title: Orbifold Grothendieck-Riemann-Roch for Deligne-Mumford stacks

Abstract:  The Grothendieck-Riemann-Roch theorem for schemes gives an isomorphism between K-theory of coherent sheaves and Chow groups rationally. For algebraic stacks such an isomorphism fails as rational K-theory does not satisfy étale descent in the case of algebraic stacks. This failure can be fixed by either étale sheafifying K-theory or in the case of Deligne-Mumford stacks by enriching the Chow groups to the orbifold Chow groups, i.e, Chow group of the inertia stack. We give a new construction of this map and discuss a virtual version of the orbifold Grothendieck-Riemann-Roch theorem. We then use it to define a virtual topological Euler characteristic for Deligne-Mumford stacks. This is a report on an ongoing joint work with Adeel Khan.

Sergej Monavari (Ecole Polytechnique Fédérale de Lausanne)

Title: Tetrahedron instantons in Donaldson-Thomas theory

Abstract: Tetrahedron instantons were recently introduced by Pomoni-Yan-Zhang in string theory, as a way to describe systems of D0-D6 branes with defects. We propose a rigorous geometric interpretation of their work by the point of view of Donaldson-Thomas theory. We will explain how to naturally construct the moduli space of tetrahedron instantons as a Quot scheme, parametrizing quotients of a torsion sheaf over a certain singular threefold, and how to construct a virtual fundamental class in this setting using quiver representations and the recent machinery of Oh-Thomas (which is in principle designed for moduli spaces of sheaves on Calabi-Yau 4-folds). Furthermore, we will show how to formalize mathematically the invariants considered by Pomoni-Yan-Zhang (initially defined via supersymmetric localization in Physics) and how to rigorously compute them, solving some open conjectures. Joint work with Nadir Fasola.

Henry Liu (IPMU)

Title: The 4-fold Pandharipande-Thomas vertex

Abstract: I will explain a conjectural explicit description of the Oh-Thomas virtual class for the K-theoretic Pandharipande-Thomas vertex on C^4. Interestingly, torus-fixed loci in this setting carry both an Oh-Thomas virtual class as well as a Behrend-Fantechi virtual class, and the conjecture is that they are equal, up to some explicit signs. This generalizes previous conjectures by Nekrasov-Piazzalunga and Cao-Kool-Monavari, and satisfies all low-degree checks of the K-theoretic DT4/PT4 vertex correspondence.

Nick Kuhn (University in Oslo)

Title: Spin Structures on Perfect Complexes

Abstract: We define a notion of spin structure for a perfect complex E on an algebraic stack X outside of characteristic 2. The theory is analogous to the classical case of vector bundles: Given an oriented quadratic structure on E, there exists a natural Z/2Z-gerbe over X carrying a universal spin structure, and providing an obstruction to the existence of a spin structure on E. As an application we show how to construct a "twisted" (in the sense of Caldararu) virtual structure sheaf on moduli spaces of sheaves on Calabi-Yau fourfolds, which refines the K-theory class constructed by Oh-Thomas.

Jeongseok Oh (Seoul National University)

Title: The quantum Lefschetz principle

Abstract: “Quantum Lefschetz” is a pretentious name for understanding how moduli spaces -- and their virtual cycles and associated invariants -- change when we apply certain constraints. (The original application is to genus 0 curves in P^4 when we impose the constraint that they lie in the quintic 3-fold.)

When it doesn’t work there are fixes (like the p-fields of Guffin-Sharpe-Witten/Chang-Li) for special cases associated with curve-counting. We will describe joint work with Richard Thomas developing a general theory.

Qingyuan Jiang (Hong Kong University of Science and Technology)

Title: Derived Projectivizations and Derived Grassmannians

Abstract: Derived Algebraic Geometry (DAG) extends Grothendieck's theory of projectivizations and Grassmannians from sheaves to complexes. This extension is valuable for constructing and studying moduli spaces, particularly in the presence of singularities. In this talk, I will explore the construction and properties of derived projectivizations and Grassmannians. Additionally, I will present structural results for their derived categories, unveiling a formula that simultaneously extends important existing formulas, including those for projective and Grassmannian bundles, blowups, standard flips, and projectivizations of sheaves with homological dimension $le 1$. Furthermore, I will discuss some of the applications of this framework, such as in the study of Abel maps for singular curves and Hecke correspondences.

Hyenho Lho (Chungnam National University)

Title: Tautological relation on Picard stack.

Abstract: Tautological relation on the moduli space of stable curves were studied by several method. I will reveiw the method used by Pandharipande-Pixton to prove the tautological relation on the moduli space of stable curvse using stable quotient. I will explain how to extend the result to get tautological relations on relative Picard stack over the moduli space of stable curves by constructing the stable quotient over Picard stack. After this I will explain how to get the result on the Picard stack which extends the original form of tautological relation given by Faber and Zagier. This talk is based on the joint work in progress with Younghan Bae.

Jemin You (Seoul National University)

Title: Shifted symplectic rigidification

Abstract: We construct shifted symplectic derived enhancements on the rigidified moduli spaces of sheaves on Calabi-Yau varieties as the Hamiltonian reductions of the BGm actions. We give Lagrangian correspondences between the rigidified moduli spaces of sheaves and the fixed determinant moduli spaces of pairs. For Calabi-Yau fourfolds, this is a geometric lift of Park’s virtual pullback formula between the Oh-Thomas classes. Moreover, for hyperkahler fourfolds, we provide a virtual pullback formula between the reduced virtual classes as expected by Cao-Oberdieck-Toda. This is a joint work in progress with Hyeonjun Park.

Arkadij Bojko (Academia Sinica)

Title: Calabi—Yau four dg-quivers and wall-crossing

Abstract: Dg-quivers with potential provide a rich source of examples of Calabi-Yau four categories. Being discrete algebraic objects, dg-quivers offer a combinatorial description of the moduli of their dg-modules and the associated obstruction theories. In the talk, I will explain how one can leverage this perspective to prove wall-crossing for enumerative invariants of Calabi-Yau dg-quivers. This is a toy model for the proof of wall-crossing for sheaves on Calabi-Yau fourfolds.

Denis Nesterov (University of Vienna)

Title: Hilbert schemes of points and Fulton-MacPherson compactifications

Abstract: This talk will be about Hilbert schemes of points, Fulton-MacPherson compactifications, and a relation between the two.

Yang Zhou (Shanghai Center for Mathematical Sciences)

Title: Mixed-Spin-P fields for GIT quotients

Abstract: The theory of Mixed-Spin-P fields was introduced by Chang-Li-Li-Liu for the quintic threefold. Chang-Guo-Li have successfully applied it to prove famous conjectures on its higher-genus Gromov-Witten invariants. In this talk I will explain a generalization of the construction to more spaces. The key is a stability condition which guarantees the separatedness and properness of certain moduli spaces. The stability condition comes directly from the GIT data. It also generalizes the construction of the mathematical Gauged Linear Sigma Model by Fan-Jarvis-Ruan, removing a technique assumption about "good lifitings".

This is a joint work with Huai-Liang Chang, Shuai Guo, Jun Li and Wei-Ping Li.