Abstracts

Noah Arbesfeld - Computing vertical Vafa-Witten invariants 

I'll present a computation in the algebraic approach to Vafa-Witten invariants of projective surfaces, as introduced by Tanaka-Thomas. The invariants are defined by integration over moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components. The physical notion of S-duality translates to conjectural symmetries between these contributions. One component, the "vertical" component, is a nested Hilbert scheme on a surface. I'll explain work in preparation with M. Kool and T. Laarakker in which we express invariants of this component in terms of a certain quiver variety, the instanton moduli space of torsion-free framed sheaves on P^2. Applying an identity of Kuhn-Leigh-Tanaka, we deduce constraints on Vafa-Witten invariants conjectured by Göttsche-Kool-Laarakker. One consequence is a formula for the contribution of the vertical component to refined Vafa-Witten invariants in rank 2. 

Dori Bejleri - A moduli-theoretic approach to heights on stacks

A theory of heights of rational points on stacks was recently introduced by Ellenberg, Satriano and Zureick-Brown as a tool to unify and generalize various results and conjectures about counting problems over global fields. In this talk I will present a moduli theoretic approach to heights on stacks over function fields inspired by twisted stable maps of Abramovich and Vistoli. For some well-behaved class of stacks, we obtain moduli spaces of points of fixed height whose geometry controls the number of rational points on the stack. I will outline an approach for more general stacks which is closely related to the geometry of the moduli space of vector bundles on a curve. This is based on joint work with Park and Satriano.

Ben Davison - BPS cohomology for moduli of sheaves on K3 surfaces

I'll explain a recent proof of the cohomological integrality theorem for stacks of semistable sheaves on K3 surfaces, obtained in joint work with Hennecart and Schlegel Mejia.  The theorem states that the Borel-Moore homology of this stack, which is very large, is generated via a kind of half-Yangian procedure from a specific generalised Kac-Moody Lie (GKM) algebra, with Chevalley generators identified with intersection cohomology of coarse moduli schemes.  The GKM Lie algebra with the same Cartan data is (by definition) the BPS cohomology of the category.  If time permits, I'll discuss applications to cohomological wall-crossing and Gopakumar-Vafa invariants.

Pierre Descombes - Toric localization and perverse sheaves

We will here present the hyperbolic localization functor and its use in categorified enumerative geometry. This functor was introduced by Braden for any scheme with an action of C*, as the composition of the restriction to the attracting variety with the projecion on the fixed variety. One of its main interest is that it behaves well with tperverse sheaves and mixed Hodge modules: it preserves Deligne's weight, and the well known Bialinicky-Birula decomposition can be expressed as a kind of decomposition theorem for intersection cohomology. Moreover, it can be used to obtain a toric localization formula for cohomological DT invariants, defined by gluing the perverse sheaves of vanishing cycles of critical charts on a -1 shifted symplectic scheme with orientation. We will present a proof of this localization formula, and some application to compute cohomological DT invariants of toric quivers.

Camilla Felisetti - On the intersection cohomology of vector bundles

Intersection cohomology is a topological notion adapted to the description of singular topological spaces, and the Decomposition Theorem for algebraic maps is a key tool in the subject. The study of the intersection cohomology of the moduli spaces of semistable bundles on Riemann surfaces began in the 80’s with the works of Frances Kirwan. Motivated by the work of Mozgovoy and Reineke, in joint work with Andras Szenes and Olga Trapeznikova, we give a complete description of these structures via a detailed analysis of the Decomposition Theorem applied to a certain map from parabolic bundles. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning. In the talk, I will give an introduction to the subject, and describe our results.

Naoki Koseki -Categorical blow-up formula for Hilbert schemes of points on surfaces

I will explain the construction of semi-orthogonal decompositions on the Hilbert schemes of points on a blown-up surface, which categorify the blow-up formula of the Euler characteristics. I will present two independent proofs. The first one is based on Jiang-Toda’s Quot formula. The second one is based on the derived McKay correspondence and the notion of symmetric products of a dg-enhanced triangulated category.

Henry Liu - The K-theoretic DT/PT vertex correspondence

On smooth quasi-projective toric 3- and 4-folds, vertices are the contributions from an affine toric chart to the enumerative invariants of Donaldson-Thomas (DT) or Pandharipande-Thomas (PT) moduli spaces. Unlike partition functions, vertices are fundamentally torus-equivariant objects, and they carry a great deal of combinatorial complexity, particularly in equivariant K-theory. In joint work with Nick Kuhn and Felix Thimm, we give two different proofs of the K-theoretic 3-fold DT/PT vertex correspondence. Both proofs use equivariant wall-crossing in a setup originally due to Toda; one uses a Mochizuki-style master space, while the other uses ideas from Joyce's recent universal wall-crossing machine. A crucial new ingredient is the construction of *symmetrized* pullbacks of symmetric obstruction theories on moduli stacks, using Kiem-Savvas' étale-local notion of almost-perfect obstruction theory. I believe our techniques, particularly the Joyce-style approach, can also be applied to related questions such as DT/PT descendent transformations, the DT crepant resolution conjecture, and the 4-fold DT/PT vertex correspondence.

Anton Mellit -

Alexei Oblomkov - HOMFLYPT homology and categorical Heisenberg action

The talk is based on joint work with L. Rozansky. In our previous work we explained how one  can associate to a braid  on $n$ strands complex of coherent $C^* \times C^*$ -equariant sheaves on $Hilb_n(C^2)$. In my talk, I explain how the categorical Nakajima operators fit into the picture. Roughly, the Nakajima operators correspond to adding extra strands to the braid. I will also show how this observation allows us to compute the homology of torus link (and more).

Tudor Pădurariu - Quasi-BPS categories for K3 surfaces

BPS invariants and cohomology are central objects in the study of (Kontsevich-Soibelman) Hall algebras or in enumerative geometry of Calabi-Yau 3-folds. 

In joint work with Yukinobu Toda, we introduce and study a categorical version of BPS cohomology for local K3 surfaces, called quasi-BPS categories. For a generic stability condition, we construct semiorthogonal decompositions of (Porta-Sala) Hall algebra of a K3 surface in products of quasi-BPS categories. When the weight and the Mukai vector are coprime, the quasi-BPS category is smooth, proper, and with trivial Serre functor etale locally on the good moduli space. Thus quasi-BPS categories provide (twisted) categorical (etale locally) crepant resolutions of the moduli space of semistable sheaves on a K3 surface for a generic stability condition and a general Mukai vector. We also discuss a categorical version of the \chi-independence phenomenon for BPS invariants/ cohomology. 

Andrea Ricolfi - Structures on the Quot scheme of points of a Calabi-Yau 3-fold

We discuss recent (and less recent) progress on the geometry of Quot schemes of points on a Calabi-Yau 3-fold. We will outline what is (not) known. Then we will discuss in detail one special aspect, namely the d-critical structure(s) on the Quot scheme. Such structures play a central role in Donaldson-Thomas theory, and they typically occur as truncations of (-1)-shifted symplectic derived  structures; the problem of constructing the d-critical structure on a "DT moduli space" without passing through derived geometry (which is hard) is wide open. We discuss this problem, and new results in this direction, when the moduli space is the Quot scheme of points on a Calabi-Yau 3-fold. Joint work with Michail Savvas.

Francesco Sala - Cohomological Hall algebras and affine Yangians

After a brief introduction to the theory of (2-dimensional) cohomological Hall algebras of quivers, curves, and surfaces, I will discuss the cohomological Hall algebra COHA(S, Z) of coherent sheaves on a smooth quasi-projective complex surface S set-theoretically supported on a closed subscheme Z. When S is the minimal resolution of an ADE singularity and Z is the exceptional divisor, I will describe how to characterize COHA(S, Z) via the Yangian of the corresponding affine ADE quiver. (joint project with Emanuel Diaconescu, Mauro Porta, Oliver Schiffmann, and Eric Vasserot.)

Andras Szenes -

Junliang Shen - Perverse filtrations and Fourier transforms

The perverse filtration captures interesting homological information of algebraic maps. Recent studies of Hitchin systems, compactified Jacobians, and enumerative geometry suggest two mysterious features of the perverse filtration of an abelian fibration. They are (1) the multiplicativity of the perverse filtration with respect to the cup product, and (2) the perversity of a tautological class is given by its Chern grading. In this talk, I will discuss these features and a uniform method proving them. For the approach, we establish a theory of Fourier transform extending the Beauville decomposition from abelian varieties/schemes to certain abelian fibrations with singular fibers. Based on joint work with Davesh Maulik and Qizheng Yin.

Martin Ulirsch - Archimedean and non-Archimedean P=W phenomena on abelian varieties and beyond

 Let $X$ be a smooth projective complex variety. Simpson's non-abelian Hodge correspondence provides us with a real analytic isomorphism between the Betti moduli space of characters of $\pi_1(X)$ and the moduli space of topologically trivial semistable Higgs bundles on $X$. The P=W conjecture, recently proved by Maulik--Shen and Hausel--Mellit--Minets--Schiffmann, predicts that the perverse filtration on the cohomology of the Dolbeault moduli space agrees (up to index shift) with the weight filtration on the cohomology of the Betti moduli space, when $X$ is a compact Riemann surface. In this talk I will report on a project, in which we extend this $P=W$ phenomenon to $X$ being a complex abelian variety, where it takes a particularly simple form. The insights gained in this situation lead us to a non-Archimedean incarnation of the $P=W$ phenomenon on the $\ell$-adic cohomology of the Betti/Dolbeault moduli space of an abelian variety $X$ over an algebraically closed non-Archimedean field $K$ of characteristic zero with maximally degenerate reduction. A central new insight is that the reduced cohomology of the tropicalization of the moduli space of topologically trivial vector bundles on $X$ plays a role as a correction term. I will conclude with some speculation on how to extend this story to the setting of Mumford curves over non-Archimedean fields. This talk is based on joint work with B. Bolognese and A. Küronya and joint work in progress with A. Gross, I. Kaur, and A. Werner. 

Dimitri Wyss -BPS-functions from non-archimedean integrals

In previous work with F. Carocci and G. Orecchia we discovered, that BPS-invariants appearing in Donaldson-Thomas theory for moduli of sheaves on del Pezzo surfaces admit a natural interpretation as non-archimedean integrals. Motivated by this, we develop an integration theory for smooth Artin stacks and obtain as application a new expression of these BPS invariants. In particular this gives a new proof of Maulik-Shen’s $\chi$-independence result for del Pezzo surfaces. This is joint work in progress with Michael Groechenig and Paul Ziegler.