Karin Baur: Combinatorial approaches to Grassmannian cluster structures
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.
Arend Bayer TBA
Ian Grojnowski Recognising Flag Varieties
Lucien Hennecart: Global BPS Lie algebra and tautological generation
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.
Dominic Joyce Hilbert schemes of points on surfaces and Geometric Representation Theory
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.
Tudor Padurariu TBA
Olivier Schiffmann Chi-independence of the BPS sheaf for symplectic surfaces
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.