Peter Angelinos (University of Toronto) : p-adic integration for abstract Hitchin systems
I will describe the application of p-adic integration to the moduli spaces of SLn- and PGLn- Higgs bundles, as well as my current work in expanding these methods to more general abstract Hitchin systems.
Dhyan Aranha (Universität Duisburg-Essen) : Some ideas about (Virtual) Localization
I would like to speak on joint work with Adeel Khan, Alyosha Latyntsev, Hyeonjun Park and Charanya Ravi about using homotopical methods to study torus localization. One of the consequences of this is a general form of Graber and Pandharipande's virtual localization formula with no hypothesis about global resolutions or global embeddings. Along the way I will describe a general concentration theorem for torus actions on stacks and a homotopy invariance result for quasi-smooth cone stacks.
Francesca Carocci (EPFL) : BPS invariant from p-adic integrals
We consider moduli spaces of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes. Working over a non-archimedean local field F, we define a natural measure on the F-points of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on the moduli spaces with respect to this measure is independent of the Euler characteristic. Analogous statements hold for (meromorphic or not) Higgs bundles. Recent results of Maulik-Shen and Kinjo-Koseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles. This is a joint work with Giulio Orecchia and Dimitri Wyss.
Ben Davison (University of Edinburgh) : Stacks of semistable sheaves on K3 surfaces
I'll explain some new results on the Borel-Moore homology of stacks of semistable coherent sheaves on K3 surfaces, as well as intersection cohomology of coarse moduli spaces. For nonprimitive Chern classes, these spaces can be highly singular. Nonetheless, considering all multiples of a given class simultaneously, we (in joint work with Lucien Hennecart and Sebastian Schlegel Mejia) have a cohomological upgrade of the integrality theorem from DT theory, connecting the intersection cohomology of the coarse spaces with the Borel-Moore homology of the stacks. Aside from the integrality theorem itself, applications include a new description of Maulik-Toda-style GV invariants of local K3 surfaces, a proof of the Halpern-Leistner purity conjecture for the BM homology of the stack, and wall-crossing invariance results.
Frédéric Déglise (ENS de Lyon) : Decompositions of motivic complexes
As one of the standard conjectures, decompositions of motives is at the heart of motivic theory, from Grothendieck to Beilinson’s point of view. One of the nice properties of Voevodsky motives is the existence of a weight structure, as discovered by Bondarko. This structure has been exploited by Wildeshaus in his study of motives of Shimura varieties to settle a theory of (motivic) intermediate extensions, a mirror (or rather « pre-realization ») of the perverse theory.
In this talk I will explain how one can exploit both Bondarko’s and Wildeshaus’ theory to deepen the (various) decomposition conjectures for motives, and to obtain new ones for certain families of varieties and schemes. This is a work in collaboration with Mattia Cavicchi and Johannes Nagel.
Jens Eberhardt (Bergische Universität Wuppertal) : A K-theoretic Approach to Geometric Representation Theory
Perverse sheaves and intersection cohomology are central objects in geometric representation theory. This talk is about their long-lost K-theoretic cousins, called K-motives. We will discuss definitions and basic properties of K-motives and explore potential applications to geometric representation theory. For example, K-motives shed a new light on Beilinson-Ginzburg-Soergel's Koszul duality — a remarkable symmetry in the representation theory and geometry of two Langlands dual reductive groups. We will see that this leads to a new “universal” Koszul duality that does not involve any gradings or mixed geometry which are as essential as mysterious in the classical approaches.
Andres Fernandez Herrero (Colombia University) : Moduli spaces of gauged maps into projective GL_n-schemes
Given a projective variety X equipped with an action of the general linear group GL_n, a gauged map into X is a morphism from a smooth projective curve into the quotient stack X/GL_n. In this talk, I will present the construction of a stack compactification of the moduli of gauged maps of fixed genus and "degree" in terms of maps from nodal curves into X/GL_n satisfying an appropriate stability condition. This stack admits a proper good moduli space, and can be thought of as a version of stable maps into the algebraic stack X/GL_n. This talk is based on joint work in progress with Daniel Halpern-Leistner.
Adeel Khan (Academia Sinica) : Derived specialization on derived moduli spaces
I’ll talk about a construction called derived specialization, whose most general incarnation lives in the world of stable motivic homotopy theory on stacks. For quasi-smooth morphisms, it gives rise to virtual pullbacks on motives and thus motivic lifts of 2-dimensional CoHAs. Applied to moduli spaces with (-1)-shifted symplectic structures, it can be used to construct some examples of 3-dimensional CoHAs. Applied to moduli spaces with (-2)-shifted symplectic structures, it can be used to construct the Oh-Thomas class of a Calabi-Yau fourfold in oriented BM homology theories. Partially based on a joint work in progress with T. Kinjo and H. Park, and another joint work in progress with H. Park and C. Ravi.
Tasuki Kinjo (IPMU) : Global critical chart for the moduli stack of G-Higgs bundles
In this talk, I will describe the (-1)-shifted cotangent stack of the moduli stack of G-Higgs bundles on a Riemann surface as a critical locus inside the moduli stack of twisted G-Higgs bundles. As a consequence, we describe the Borel—Moore homology of the moduli stack of G-Higgs bundles as the vanishing cycle cohomology. This talk is based on an ongoing work with Pavel Safronov.
Robert Laterveer (Université de Strasbourg) : The standard conjectures for some Lagrangian fibrations
This is a report on joint work with Giuseppe Ancona, Mattia Cavicchi and Giulia Sacca. We establish a general criterion stating that certain Lagrangian fibrations verify the standard conjectures. The main application is that the Laza-Sacca-Voisin hyperkaehler 10folds verify the standard conjectures. In my talk, I will attempt to combine a superficial introduction to the Laza-Sacca-Voisin 10folds with a superficial sketch of our arguments.
Simon Pepin Lehalleur (University of Amsterdam) : Introductory talk - motives and moduli
This opening talk will consists of an overview of some fundamental aspects of the theory of motives related to the themes of the week. Moduli spaces and representation theory give plentiful examples of highly structured spaces and correspondences where sophisticated cohomology and sheaf theories can be computed explicitly, and these computations in turn can reveal deeper properties of the geometry and arithmetic of moduli spaces. I will illustrate this with some simple examples, as a warm-up for the more advanced examples we will see later this week.
Francesco Sala (University of Pisa) : Categorified Hall algebras and their representations
Cohomological Hall algebras play a preeminent role in the theory of moduli spaces, representation theory, and gauge theory. After a gentle introduction to the theory of two-dimensional cohomological Hall algebras and their relation to algebraic geometry and representation theory, I will describe their "categorification", called categorified Hall algebras. The second part of the talk is devoted to the introduction of representations of categorified Hall algebras via torsion pairs of abelian categories. (The talk is based on joint works with Emanuel Diaconescu and Mauro Porta.)
Olivier Schiffmann (Université de Paris-Sud Orsay) : Cohomological Hecke operators on surfaces
We will provide a description of the cohomological Hall algebra of zero-dimensional sheaves on a smooth surface $S$, and relate it to a W_{1+\infty}-algebra modeled on the cohomology ring of $S$. We will also describe applications of this algebra on various moduli spaces / stacks of sheaves on $S$. We will give a (partly conjectural) picture of the cohomological Hall algebra of Higgs sheaves on a smooth projective curve. This is based on joint work with Minets, Mellit and Vasserot, and with Davison and Hennecart respectively.
Jakob Scholbach (University of Padova) : Central Motives
In this talk, I will report on an integral motivic Satake equivalence and a motivic refinement of Gaitsgory's central sheaves functor. This is joint work (partly in progress) with Thibaud van den Hove and Robert Cass.
Junliang Shen (Yale University) : Cohomology of the moduli of vector bundles and the moduli of Higgs bundles on a Riemann surface
The moduli of Higgs bundles on a curve can either be viewed as a variant of the moduli of vector bundle on a curve --- a very classical moduli space that has been studied for decades, or the non-abelian Dolbeault cohomology of the curve in view of the non-abelian Hodge theory. In this talk, I will discuss some interesting symmetries of the cohomology of the moduli of Higgs bundles that do not show up for the moduli of vector bundles. I will then explain how different viewpoints lead to completely different proofs of this statement. If time permits, I may discuss some open questions.
Lisanne Taams (Radboud University Nijmegen) : Torsion sheaves on stacky curves
The goal of this talk is to show that the motive of the stack of sheaves on a stacky curve is generated by the motive of the coarse space of the curve. The most challenging case is that of torsion sheaves on stacky curves, which are closely related to representations of cyclic quivers. We highlight some of the geometric constructions that lie at the hart of the proof. If there is time, we will relate this to a graded version of the Grothendieck-Springer resolution.
Olga Trapeznikova (Université de Genève) : Intersection cohomology of moduli spaces of semistable bundles on curves
The study of the intersection cohomology of moduli spaces of semistable bundles on Riemann surfaces began in the 80’s with the works of Frances Kirwan. In this talk, I will describe joint work with Camilla Felisetti and Andras Szenes, in which, motivated by the work of Mozgovoy and Reineke, we give a complete description of these structures via a detailed analysis of the Decomposition Theorem applied to the parabolic forgetful map. As a result, we obtain a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning.
Matthias Wendt (Bergische Universität Wuppertal) : Aspects of Witt-sheaf cohomology of classifying spaces
In the talk I want to outline work in progress to understand Witt-sheaf cohomology of classifying spaces: how the real cycle class map and homotopy fixed points provide intuition, how symmetric representations provide cohomology classes and what this looks like in the example of orthogonal groups.
