Abstracts
Jenny August
Title: Stability Conditions for Contraction Algebras
Abstract: For a finite dimensional algebra, Bridgeland stability conditions can be viewed as a continuous generalisation of tilting theory, providing a geometric way to study the derived category. Describing this stability manifold is often very challenging but in this talk, I’ll look at a special class of symmetric algebras whose tilting theory is determined by a related hyperplane arrangement. This simple picture will then allow us to describe the stability manifold of such an algebra, and along the way provide a new proof of a classically topological result, known as the K(pi,1) conjecture.
Gwyn Bellamy
Title: Projective generators for equivariant D-modules.
Abstract: In this talk we will consider the question of existence of projective generators for the category of (coherent) equivariant D-modules on a smooth affine variety. This may seem like a rather technical question, but I will explain how it is motivated by (a) quantum Hamiltonian reduction (b) a "character theory" for admissible D-modules (c) existence of regular invariant eigen-distributions on polar representations. I will show that one can use Luna's slice theorem to show that projective generators do exist in this category when the G-variety is "visible". This is based on joint work with Sam Gunningham.
Tobias Dyckerhoff
Title: The symplectic geometry of algebraic K-theory
Abstract: In this talk, I will explain a connection between the symplectic geometry of symmetric products, the algebraic K-theory of stable infinity-categories, and higher Auslander-Reiten theory. Exploiting this interplay, we will discuss how to implement the local-to-global behaviour for Fukaya categories of symmetric products via factorization homology of certain decorated E_2-algebras.
Based on joint work in progress with G. Jasso and Y. Lekili.
Lena Gal
Title: The bialgebra structure on Hall categories
Abstract: The theorem of Green states that a Hall algebra associated to a nice enough abelian category C is a bialgebra up to a twist. We will formulate a categorified version of this statement when C is the category of representations of a simply laced quiver. This approach allows us to construct a categorification of the positive half of the quantum group as a bialgebra object in a braided 2-category. We will discuss how the above can conjecturally be used to construct a 4d TQFT with defect.
Travis Mandel
Title: Strong positivity for quantum cluster algebras
Abstract: I will describe joint work with Ben Davison in which we construct ``quantum theta bases'' for skew-symmetric quantum cluster algebras. This settles some of the central conjectures from quantum cluster theory. E.g., the structure constants for our bases are Laurent polynomials in the quantum parameter with non-negative integer coefficients (strong positivity), as are the coefficients of the theta functions expressed in any cluster (universal positivity). Our approach combines deep results from DT-theory with the scattering diagram techniques used by Gross-Hacking-Keel-Kontsevich.
Sven Meinhardt
Title: Integrality of Donaldson-Thomas invariants
Abstract: Donaldson-Thomas invariants, also called BPS invariants, as defined by Joyce/Song and Kontsevich/Soibelman are conjectured to be integers. After providing a short introduction to these invariants, I will show how results obtained in collaboration with Ben Davison can be used to prove the famous "Integrality Conjecture". If time permits, I will also sketch a path towards categorification of BPS invariants.
Aleksander Minets
Title: Cohomological Hall algebras and sheaves on surfaces
Abstract: I will start with a brief overview of cohomological Hall algebras, with the emphasis on the ones associated to ruled surfaces. In the second half of the talk, I will discuss the action of these CoHAs on some moduli spaces, closely related to framed sheaves on surfaces. If time permits, I will also explain how one can see the product explicitly via shuffle presentation.
Clélia Pech
Title: Geometry of rational curves on some varieties with a Lie group action
Abstract: In this talk I will describe a family of algebraic varieties with actions of Lie groups which are closely related to homogeneous spaces (which for instance include projective spaces, quadrics, Grassmannians). After describing the geometry and the orbit structure of these varieties, I will explain how to understand rational curves on these varieties, as well as an algebraic structure encoding the intersection theory of these curves, called the quantum cohomology ring. This is joint work with R. Gonzales, N. Perrin, and A. Samokhin.
Markus Reineke
Title: The Cohomological Hall Algebra of the Kronecker quiver
Abstract: We describe the semistable Cohomological Hall Algebra of the Kronecker quiver (for the central slope) by generators and relations, and speculate on generalizations and relations to R-matrices, differential operators and Yangian symmetries. This is joint work with H. Franzen.
Francesco Sala
Title: continuum Kac--Moody Lie algebras and continuum quantum groups
Abstract: In the present talk, I will define a family of infinite-dimensional Lie algebras associated with a "continuum" analog of Kac--Moody Lie algebras. They depend on a "continuum" version of the notion of the quiver. These Lie algebras have some peculiar properties: for example, they do not have simple roots and in the description of them in terms of generators and relations, only quadratic (!) Serre type relations appear.
I will discuss also their quantizations, called "continuum quantum groups". In particular, in the second part of the talk, I will focus on the case when the "continuum quiver" is a circle: in this case, the continuum quantum group can be realized by means of the theory of classical Hall algebras. If time permits, I will discuss the representation theory of the continuum quantum group of the circle (in particular, the construction of the Fock space). This is based on joint works with Andrea Appel and Olivier Schiffmann.
Olivier Schiffmann
Title: Cohas and cohomology rings
Abstract: We describe the cohomological Hall algebra of the stack of coherent sheaves on a smooth projective curve, in several ways (by generators and relations, via some vertex operator,...). As an application, we give a description of the cohomology ring of the stack of semistable vector bundles of arbitrary rank and degree. Joint work with E. Vasserot.
Balázs Szendrői
Title: Hilbert schemes of points of ADE surface singularities: birational geometry, topology and representation theory
Abstract: I will discuss some old and some more recent results around Hilbert schemes of points on singular surfaces, obtained in joint work with Craw, Gammelgaard and Gyenge.
Matthew Young
Title: Orientifold Donaldson--Thomas theory and Green-type theorems for Hall modules
Abstract: This talk will be an overview of the current status of orientifold Donaldson-Thomas theory. This theory, whose goal is to count quiver-theoretic analogues of orthogonal or symplectic vector bundles, is best formulated in terms of certain modules over cohomological Hall algebra of quivers. One of the key open problems is to understand the compatibility of the comodule structure on this module, that is, to prove a module-theoretic version of Green's theorem. I will explain the recent solution of this problem in the simpler setting of Hall modules of categories which are linear over the field with one element, and discuss its potential implications to DT theory.