Thursday, September 12
10-11: Pavel Safronov
11-11.30: Break
11.30-12.30: Marco Robalo
12.30-2: Lunch break
2-3: Hannah Dell
3-3.30: Break
3.30-4.30: Sarunas Kaubrys
Friday, September 13
9.30-10.30: Pieter Belmans
10.30-11: Break
11-12: Nebojsa Pavic
12-1.30: Lunch break
1.30-2.30: Sebastian Schlegel-Mejia
2.30-3: Break
3-4: Benjamin Hennion
Pavel Safronov, University of Edinburgh
Title: Cohomological Hall algebras for 3CY categories
Abstract: In this talk I will explain the construction of a hyperbolic localization isomorphism for the DT sheaf, following Descombes. This allows one to define the cohomological Hall algebra structure for general 3CY categories (not necessarily deformed Calabi-Yau completions) and a parabolic induction map for DT invariants of general 3-manifolds. This is a report on work joint with Tasuki Kinjo and Hyeonjun Park.
Marco Robalo, Sorbonne University
Title: Contractibility of the Darboux stack of a (-1)-shifted symplectic derived scheme.
Abstract: In this talk I will start by introducing the Darboux stack Darb parametrizing local Darboux charts of a (-1)-shifted derived scheme X as a derived critical locus. This stack carries an action of the monoid of quadratic bundles on X, Quad.
The goal of the talk is to explain the contractibility of the quotient Darb/Quad and its applications in Donaldson-Thomas theory.
This is a joint work with B. Hennion and J. Holstein. This talk serves as a prequel to Hennion’s talk
Hannah Dell, University of Bonn
Title: Bridgeland stability conditions and group actions
Abstract: Bridgeland stability conditions have been constructed on curves, surfaces, and in some higher dimensional examples. In several cases, there are only so-called “geometric” stability conditions which are constructed using slope stability for sheaves, whereas in other cases there are more (e.g. coming from an equivalence with representations of a quiver). Lie Fu -- Chunyi Li -- Xiaolei Zhao were the first to provide a general result explaining this phenomena. In particular, they showed that if a variety has a finite map to an abelian variety, then all stability conditions are geometric. In this talk, we test the converse on surfaces that arise as free quotients by finite groups. To do this, we study stability conditions on equivariant categories. This is joint work with Edmund Heng and Anthony Licata, and based on arxiv:2307.00815 and arxiv:2311.06857.
Sarunas Kaubrys, University of Edinburgh
Title: Cohomological DT invariants of local systems on the 3-torus via the exponential map
Abstract: On any (-1)-shifted symplectic stack with appropriate orientation data one can define a perverse sheaf called the DT sheaf. The cohomology of this sheaf is called the cohomological DT invariant and can be seen as a categorification of DT invariants, originally defined by Thomas. In this talk we will focus on the (-1)-shifted symplectic stack of local systems on Sigma_g x S^1 for Sigma_g a Riemann surface. I will explain work in progress on how to compute the DT invariants of this stack by relating them to the DT invariants of a (-1)-shifted cotangent bundle, via an exponential map. For the case g >1, we will also use the Cohomological Hall algebra structure on the cohomology of the DT sheaf, as defined by Kinjo-Park-Safronov. In the case of the 3-Torus, this computation leads to Langlands duality between SLn and PGLn DT invariants for prime n.
Pieter Belmans, University of Luxembourg
Title: Hochschild cohomology of Hilbert schemes of points
Abstract: I will present a formula describing the Hochschild cohomology of symmetric quotient stacks, computing the Hochschild–Kostant–Rosenberg decomposition of this orbifold. Through the Bridgeland–King–Reid–Haiman equivalence this allows the computation of Hochschild cohomology of Hilbert schemes of points on surfaces. These computations explain how this invariant behaves differently from say Betti or Hodge numbers, which have been studied intensively in the past 30 years, and it allows for new deformation-theoretic results. This is joint work with Lie Fu and Andreas Krug.
Nebojsa Pavic, University of Graz
Title: Semiorthogonal decompositions of derived categories of singular varieties
Abstract: We study semiorthogonal decompositions of derived categories of singular projective varieties. We focus our attention to projective varieties with 1/n(1,...,1) singularities. We give sufficient conditions for when derived categories of such varieties admit a semiorthogonal decomposition into two components; a component containing the information of the singularity and a component encoding the smooth information. We give various examples satisfying our condition and we also give criteria which do not allow such decompositions. This is joint work in progress with M. Kalck and Y. Kawamata.
Sebastian Schlegel-Mejia, EPFL
Title: Socles and E-series of moduli stacks
Abstract: I will explain a stratification of moduli stacks of objects in abelian categories arising from the notion of a socle of an object.
As an application, we obtain, in good cases, e.g. if the category is hereditary or 2-Calabi—Yau, a recursive procedure to compute E-series of moduli stacks in terms of the E-polynomials of the moduli spaces of simple objects.
Benjamin Hennion, University of Paris-Saclay
Title: Towards gluing matrix factorizations
Abstract: Thanks to the work of Behrend, Donaldson--Thomas invariants can be constructed by gluing from local models of the moduli space of sheaves on a Calabi--Yau 3-fold.
In this talk, we will describe how to construct a categorification starting from matrix factorizations on said local models.
This will be achieved by specializing the results explained in Robalo's talk, but the focus on matrix factorization should make this talk independently accessible.
This is work in progress with J. Holstein and M. Robalo.