- Adeel Khan (AS): Shifted microlocalization
- The classical theory of constructible sheaves "microlocalizes" onto the cotangent bundle, and gives rise to a new invariant of exact symplectic manifolds called microsheaves. I will speculate on what n-shifted symplectic versions of this story should look like. My joint work with Kinjo, Park, and Safronov, which Hyeonjun will discuss, can be regarded as implementing (-1)-shifted microlocal sheaf theory.
- Young-Hoon Kiem (KIAS): Generalized sheaf counting
- Counting bundles or sheaves with desired properties is a
fundamental problem in algebraic geometry. There are three major
issues concerning sheaf counting, namely compactification,
singularity and automorphism. Firstly, the issue of compactifying
moduli spaces has many solutions such as Hilbert schemes and moduli
spaces of stable sheaves, and we have to deal with the comparison
problem of the moduli spaces and invariants. Geometric invariant
theory and various wall crossing techniques are of key importance in
this direction. Secondly, it is well known that moduli spaces may
acquire arbitrarily bad singularities and ordinary intersection
theory does not make sense. However, we can often utilize a deeper
structure (sometimes called the hidden smoothness) to construct a
nice homology class, called the virtual fundamental class, which
enables us to define sheaf counting invariants like Donaldson,
Seiberg-Witten, Donaldson-Thomas and DT4 invariants in algebraic geometry. In this talk, our main focus lies in the third issue.
Difficulties arise in sheaf counting due to automorphism groups even when the moduli stacks are smooth as in the case of semistable vector bundles over smooth projective curves because the moduli stacks are not Deligne-Mumford and we don’t know how to integrate cohomology classes on Artin stacks in general. To remove infinite automorphism groups, we should modify the moduli stack and there are several ways such as Kirwan’s partial desingularization (with intrinsic blowups for quasi-smooth stacks or canonical stabilizer reduction in derived algebraic geometry) and constructing smooth morphisms from Deligne-Mumford stacks as in the works of T. Mochizuki and Joyce-Song. After these modifications, we obtain virtual fundamental classes which provide us with generalized sheaf counting invariants, and it is an open problem to compare these generalized invariants. For instance, we want to compare the generalized Donaldson-Thomas invariants of Kiem-Li-Savvas (by partial desingularization) and Joyce-Song (by stable pairs) for Calabi-Yau 3-folds.
In this talk, I will focus on the curve case (singular moduli spaces of vector bundles over curves) with a prospect towards higher dimensional cases. I will show how we can compare all the generalized intersection pairings by partial desingularization, parabolic bundles, pairs and Joyce's wall crossing mechanism. Based on a joint work with Chenjing Bu.
- Emile Bouaziz (AS) : Elliptic objects and non-commutative geometry:
- I'll discuss some recent joint work with Adeel Khan which constructs a very general theory of elliptic loop spaces and uses this to give a clean construction of equivariant elliptic cohomology. There will be a lot of motivation coming from more classical objects, and I will include a variety of classical computations. I will also sketch a cochain model coming from the theory of meromorphic loop spaces. This latter is part of an ongoing project with Scherotzke, Sibilla and Tomasini constructing non-commutative versions of (equivariant) elliptic cohomology. I will attempt to pepper the talk with some vague speculations about the rôle of vertex algebras and conformal blocks to this theory, which I do not yet understand to my satisfaction.
- Hyeonjun Park (KIAS) : Lagrangian classes, Donaldson-Thomas theory, and gauged linear sigma models
- In this talk, I will explain the construction of Lagrangian
classes for perverse sheaves in cohomological Donaldson-Thomas
theory, whose existence was conjectured by Joyce. The two key
ingredients are a relative version of the DT perverse sheaves and a
hyperbolic version of the dimensional reduction theorem. As a
special case, we recover Borisov-Joyce/Oh-Thomas virtual classes in
DT4 theory.
As applications, I will explain how to construct the following
structures from the Lagrangian classes: (1) cohomological field
theories for gauged linear sigma models, (2) cohomological Hall
algebras for 3-Calabi-Yau categories, (3) relative
Donaldson-Thomas invariants for Fano 4-folds with anti-canonical
divisors, (4) refined surface counting invariants for Calabi-Yau
4-folds.
This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.
- Tsung-Ju Lee (NCKU): GKZ systems and their solutions
- A GKZ system, introduced by Gelfand, Graev, Kapranov, and Zelevinsky, is a system of linear partial differential equations generalizing the hypergeometric structure which can be traced back to Euler and Gauss. As observed by Batyrev, periods of Calabi--Yau hypersurfaces or complete intersections in Fano toric varieties are solutions to a certain type of GKZ systems. To calculate periods, a well-known strategy is to compute the full set of solutions to the relevant GKZ system first, and then try to restore periods among the solutions. In this talk, I will explain how to solve the GKZ systems, especially those arising from B-side in mirror symmetry, by describing their Riemann--Hilbert partners.
- You-Cheng Chou (KIAS): Permutation-equivariant quantum K-theory of a point
- In this talk, I will start with two motivations for this
project. The first is to establish a K-theoretic version of the
Witten-Kontsevich theorem. The second comes from localization
computation: localization computation for a general target usually
reduced to the point case.
Then I will present ongoing work on two key aspects of this project. First is a pushforward formula applied to the universal cotangent line bundle and the pluri-Hodge bundle. This technique is combinatorially easier for not using Kawasaki-Hirzebruch-Riemann-Roch theorem. Second, I will discuss the permutation-equivariant string equation, which gives a recursive formula when forgetting multiple marked points with unit insertions. This talk is based on joint works (some in progress) with Y.-P Lee, Leo Herr, Irit Huq-Kuruvilla, and Kamyar Amini.
- Borislav Mladenov (AS) : Algebraically coisotropic subvarieties in holomorphic symplectic varieties
- If (Χ,σ) is a holomorphic symplectic variety, a subvariety Z
is coisotropic if the restriction of the holomorphic sympelctic form
σ to Z has constant minimal rank. In particular hypersurfaces are
coisotropic. It turns out that being coisotropic is a cohomological
property, hence the notion of coisotropic classes. This leads
naturally to the definition of coisotropic Hodge classes. We say Z
is algebraically coisotropic if the foliation defined by the kernel
of σ is algebraically integrable.
A conjecture of Voisin states that the space of coisotropic Hodge classes over \Q is generated by classes of (special) algebraically coisotropic subvarieties. In this talk, I'm going to introduce the main characters, discuss constructions of examples of these algebraically coisotropic subvarieties, and present old and new cases of this conjecture, focusing on moduli of sheaves on K3 surfaces.
- Jeongseok Oh (SNU) : Regarding the blowup conjecture
- This talk is a small update of Y.P.'s talk last year about the blowup conjecture. There are no proofs in this talk; rather I will introduce an unverified idea for the conjecture: genus 0 primary GW invariants of the blowup and blowdown are the same if their virtual dimensions are the same. This is a joint work with Y.P. Lee and Seungjae Yun.
- Yuki
Koto (AS):
Quantum cohomology of nonabelian GIT quotients and Fourier transforms
- Quantum cohomology of GIT quotients has been extensively studied by several approaches. Among them, I focus on two guiding conjectures: Abelian/Nonabelian Correspondence and Reduction Conjecture. In this talk, I introduce a conjectural Fourier transform that relates quantum cohomologies of prequotients and of quotients, and propose a new conjecture that connects the two conjectures. I will also discuss my proposal on partial flag bundles and its application. Part of this talk is based on joint work with Ionut, Ciocan-Fontanine.
- Irit Huq-Kuruvilla (AS): Gromov-Witten Invariants and Euler Characteristics
- We discuss some work-in-progress on a new class of Gromov-Witten invariants modeled on the topological Euler characteristic, and formulas to compute them. As a special case of these formulas, we derive the Bini-Harer formula for the ordinary Euler characteristics of $M_{g,n}$ in terms of their orbifold Euler characteristics.
- Maxime Cazaux (AS) : Quantum K theory of Fermat Singularities
- The Landau-Ginzburg/Calabi-Yau correspondence relates the Gromov-Witten invariants of a hypersurface in projective space and the so-called FJRW invariants of the associated singularity.
The FJRW invariants are defined as integrals of characteristic classes over the (smooth) stack of r-spin curves, classifying (twisted) curves with an r-th root of the canonical bundle.
Both theories admit a natural extension to K-theory, where cohomological classes are replaced by classes of vector bundles. A natural question is whether the LG/CY correspondence still holds in the K-theoretic setting.
In this talk, we will present an explicit computation of the genus-0 FJRW invariants of the Fermat singularities, and provide a link to the quantum K-theory of the associated hypersurface.
- Woonam Lim (Yonsei U): On the \chi-independence phenomenon
- The moduli space of one-dimensional sheaves on surfaces and Calabi–Yau 3-folds has been extensively studied due to its connections with the Hitchin integrable system and curve counting theories. These moduli spaces depend in a subtle way on the holomorphic Euler characteristic, often denoted by \chi. Although the geometry and topology of the moduli spaces vary significantly with \chi, certain cohomological phenomena are known or conjectured to be \chi-independent. In this talk, I will review the \chi-independence phenomenon and explain some recent results. If time permits, I will also discuss some speculations about how a (-1)-shifted Lagrangian correspondence via Hecke modifications may provide a useful framework for studying \chi-independence in the Calabi–Yau 3-fold case. This talk is based on joint works (some in progress) with Y. Kononov, M. Moreira and W. Pi.
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