The coordinate ring of the Grassmannian Gr(k,n) of k dimensional subspaces of \mathbb{C}^n
has the structure of a cluster algebra. An additive categorification of it has been given by Jensen-King Su, using a quotient of a preprojective algebra of a cyclic quiver. This algebra is infinite dimensional and its module categories is wild in general. We show how this cluster category can be given using dimer models in the disk (certain quivers with faces)
and describe the cluster categories using combinatorial tools.
Given a family of K3 surfaces, what is the right notion of an action of a group on their derived categories? This question is surprisingly subtle, and disambiguating the different possible notions naturally leads to obstructions to the existence of such actions. I will also discuss effective criteria to achieve the vanishing of all obstructions. As one type of example, this leads to families of "non-commutative abelian surfaces", and to the construction of every polarised Kummer-type Hyperkaehler variety through a moduli space.
The moduli stack and space of sheaves on quasi-projective symplectic surfaces (cotangent bundle of a curve, K3 or Abelian surface) are of high interest in algebraic geometry and enumerative geometry. Their cohomology encodes various enumerative invariants. The cohomology of the stack has the advantage of carrying extra structures: the cohomological Hall algebra structure.
In this talk, I will explain how to construct a bialgebra structure on the cohomology of the moduli stack, and how to use it to define a global BPS Lie algebra without requiring the existence of a good moduli space. Then, I will explain how to generalize Markman's tautological generation theorem for smooth moduli spaces of sheaves on symplectic surfaces to non-smooth moduli spaces. These can be obtained from non-primitive Mukai vectors.
This is joint work with Ben Davison, Tasuki Kinjo, Olivier Schiffmann and Eric Vasserot.
Let X be a projective complex surface, and M the moduli stack of objects in the derived category D^bcoh(X). The homology H_*(M,Q) has an explicit superpolynomial description due to Jacob Gross, and work by me gives H_*(M,Q) the structure of a vertex algebra, and in a slightly different way, a "quantum vertex algebra".
The Hilbert schemes of points Hilb^n(X) have natural maps to M. Using these, we regard the fundamental classes [Hilb^n(X)]_\fund as elements of H_*(M,Q), and the homology H_*(Hilb^n(X),Q) as a vector subspace of H_*(M,Q).
I use work of Ellingsrud-Gottsche-Lehn 2001 to show that the formal generating function Hilb(X,q) = \sum_{n\ge 0} q^n [Hilb^n(X)]_\fund is the unique solution of an o.d.e. in H_*(M,Q)[[q]], written using operators in my quantum vertex algebra structure on H_*(M,Q). Using this I give an explicit formula for Hilb(X,q) in terms of universal functions.
In 1995, Grojnowski and Nakajima gave an action of a Heisenberg algebra on \bigop_{n\ge 0} H_*(Hilb^n(X),Q). In 2023 this was upgraded by Mellit-Minets-Schiffmann-Vasserot to an action of a "deformed W_{1+\infty} algebra”. I explain how to write these algebra actions in terms of operators from my quantum vertex algebra on H_*(M,Q), which act on H_*(M,Q) preserving the vector subspace \bigop_{n\ge 0} H_*(Hilb^n(X),Q).
I think this quantum vertex algebra (or perhaps better, the associated quantum vertex Lie algebra) should play an important part in the Geometric Representation Theory of surfaces. I am looking forward to discussing all this with people at the conference.
I will give an overview of categorical Hall algebras of quivers with potential and of their decompositions into certain finitely generated
subcategories, called quasi-BPS categories, whose rational Grothendieck groups have dimension related to BPS invariants. For the tripled quiver with potential, which is a canonical pair associated to a quiver, the Grothendieck group of the categorical Hall algebra is the positive half of a quantum affine algebra, and the above decomposition is compatible with the perverse t-structure recently introduced by Cautis. In particular, it induces a finite length t-structure on each quasi-BPS category.
I will then discuss two families of examples and consequences of this construction. For finite type quivers, we expect quasi-BPS categories to be equivalent to categories of modules over KLR algebras. For the Jordan quiver, they can be used to prove conjectures of Negut and Gorsky–Negut on categorifications of the quantum toroidal algebra. Further, we obtain t-structures on categories of equivariant sheaves on Nakajima quiver varieties. We also obtain filtrations of quasi-BPS categories in graded
pieces indexed by nilpotent orbits. The results discussed are based on work in progress with Sabin Cautis, and with Sabin Cautis and Yukinobu Toda.
We will sketch a proof of Toda's conjecture about $\chi$-independence of the BPS sheaf in the context of moduli stacks of one-dimension sheaves over a symplectic surface. More precisely, we will construct an explicit Hecke operator acting on the pushforward to the Chow variety of the BPS sheaf, which realizes this $\chi$-independence isomorphism.
Our main tools are the action of the cohomological Hall algebra and a microlocal characterisation of the BPS sheaf for symplectic surfaces.
This is joint work with B. Davison, L. Hennecart, T. Kinjo and E. Vasserot.
The Strong Monodromy Conjecture promises a miraculous connection between the theory of algebraic differential operators and the theory of motivic integration, namely it claims the poles of the motivic zeta function are contained in the zeroes of the Bernstein--Sato polynomials. Alas, the miracle has few witnesses; even for homogeneous polynomials in three variables this conjecture is open. We prove it for homogeneous polynomials in three variables whose associated reduced projective divisor has, at worst, quasi-homogeneous singularities. In the analogous higher dimensional situation, we will "solve" the D-module side of the story, giving clues on how to generalize the aforementioned results.
This is joint work with Wim Veys; arXiv:2602.20922.
The component lattice is a generalization of root systems from algebraic groups to algebraic stacks. It governs the structure of parabolic induction and restriction for stacks, and is the universal combinatorial structure that underlies many variants of Hall (bi)algebras, allowing these structures to be defined for general stacks. I will introduce what it is and discuss a few applications. Time permitting, I will briefly discuss categorifying Hall bialgebras.
I will explain a correspondence theorem between the genuinely enumerative count of algebraic elliptic curves in toric varieties and the corresponding count of well-spaced tropical curves, weighted by explicit combinatorial multiplicities. This result provides a complete genus-1 generalization of the celebrated Nishinou–Siebert correspondence theorem. This is joint work with S. Koyama.
I will provide a slightly different perspective on the recent construction of stability conditions on projective schemes over a field, and compare it with the classical double tilting construction on threefolds. I will also explain why this already yields the support property with respect to the full numerical Grothendieck group.
Let 𝕜 be a field with char(𝕜) = p > 0. As a first step towards constructing mirror pairs of Calabi-Yau varieties over 𝕜 via the Gross-Siebert approach, we show that a positive and simple toric log Calabi-Yau space X₀(B,𝒫) → S₀ = Spec(ℕ → 𝕜) has the same type of toric local model as is the case in characteristic zero, provided that the residue characteristic p = char(𝕜) does not divide the total outer monodromy index of (B,𝒫).
Given an arbitrary symmetric quiver equipped with a potential, Davison and Meinhardt constructed a natural filtration on the Kontsevich–Soibelman cohomological Hall algebra (CoHA) by exploiting the hidden properness of the semisimplification morphism. The aim of this talk is to describe, in the case of trivial potential, an explicit realization of this filtration in terms of certain order conditions on the shuffle algebra. We will then explain how this perspective can be used to characterize the BPS Lie algebra of tripled quiver with canonical cubic potential which is known to coincide with the positive half of a generalized Kac–Moody Lie algebra, in terms of certain wheel and limit conditions. If time permits, I will explain how this generalizes to filtrations on cohomology of arbitrary smooth stacks admitting good moduli space satisfying certain symmetricity assumptions. This is based on joint work with Andrei Negut.
In this talk, we show openness for the locus of fibers which an integral transform is fully faithful or an equivalence. Notably, this occurs for singular varieties, as well as schemes in mixed characteristic settings. If time permits, we will discuss the potential ramifications to deforming Fourier-Mukai partnership in moduli theoretic situations where singularities exist. The talk is joint with Elías Guisado Villalgordo.
Compactified Jacobians of singular curves are closely related to enumerative geometry, particularly to the Yau–Zaslow formula of rational curves on K3 surfaces. In this talk, I will discuss geometric and numerical relations between compactified Jacobians and certain moduli spaces of stable maps. I will focus on the Jacobian actions on compactified Jacobians, and the consequences for their Euler characteristics. This includes joint work in progress with Francesca Carocci and Richard Thomas.