Past Meetings: 2022–23

Simon Crawford — Geometric resolutions of some noncommutative singularities

Kleinian singularities are a family of surface singularities which have been well-studied in ring theory, algebraic geometry, and representation theory. Kronheimer showed that it is possible to construct the minimal resolution of a Kleinian singularity via a process known as quiver GIT. Using this, Bridgeland-King-Reid proved a celebrated result which showed that the category of coherent sheaves on this resolution is equivalent to modules over a certain noncommutative ring. Recently, noncommutative analogues of Kleinian singularities were introduced by Chan-Kirkman-Walton-Zhang, called quantum Kleinian singularities. In this talk, I will describe how quiver GIT can be adapted to extend the results of Kronheimer and Bridgeland-King-Reid to some families of quantum Kleinian singularities. This is joint work with Susan Sierra.

Juan Carlos Morales Parra — The quantum dilogarithm and Elliptic quantum groups

Faddeev introduced the modular double of a quantum group, generalising Drinfeld’s construction, with the aim of deriving a simpler expression of the R-matrix. The construction strongly relies on a special function: the quantum dilogarithm. In this talk we will present its key properties, some applications, and the construction of the modular double for U_{p,q}(sl₂).

Lucas Buzaglo — Submodule-subalgebras of the Witt algebra

I will describe some Lie algebraic properties of certain subalgebras of the Witt algebra known as submodule-subalgebras. Interestingly, properties of these Lie algebras are, in some sense, controlled by the Witt algebra, even if we forget that they are subalgebras of the Witt algebra. This can be explained by the fact that the Witt algebra can be abstractly reconstructed from any of its submodule-subalgebras, which can be described as a universal property satisfied by the Witt algebra.

Charlotte Llewellyn — Flops and local systems

A flop X --> X' is a special type of birational map between algebraic varieties. They are well-studied because of their connection to the minimal model programme, but also because they induce non-trivial autoequivalences of the bounded derived category D(X). Mirror Symmetry predicts that one can consider D(X) as a stalk in a local system of categories over the Stringy Kahler Moduli Space (SKMS), and thus obtain an action of the fundamental group of the SKMS on D(X). In this talk I aim to: show how this works in an explicit example; explain the prediction from mirror symmetry for why this should work more generally; give an overview of some existing results in this area. 

Theodoros Lagiotis — Noncompact 3D TQFTs

The study of topological quantum field theory originates in a definition by Atiyah and Segal, inspired from mathematical physics. I will review some well known results regarding constructions and classifications of 3-dimensional TQFTs. I will then proceed to explain how my work serves as a natural continuation of those results. 

Parth Shimpi — What do Coxeter groups know about homological algebra?

The Tits cone of a Coxeter group is a topological space carrying naturally a faithful action of the group by reflections. Bridgeland (2007) showed that if the Coxeter group is of spherical or affine type, then this representation arises naturally in the study of homological properties of surface singularities. More precisely, the moduli space of Bridgeland stability conditions is a topological covering space of the Tits cone. I will describe the construction, and ongoing work on how this allows for a complete classification of t-structures up to well-understood automorphisms. 

Isambard Goodbody — Finite dimensional DGAs

Differential Graded Algebras are tools that appear in geometry, topology and representation theory. One reason for this is that many derived categories (for example: of reasonable schemes) are equivalent to derived categories of DGAs. Sometimes these DGAs which appear from geometry are finite dimensional algebras. In this case we can apply the powerful theory of representation theory of finite dimensional algebras. However we can't always be so lucky. I'll explain some results which generalise the theory of finite dimensional algebras to a wider class of DGAs: those whose underlying chain complexes are finite dimensional. Up to quasi-isomorphism this class includes the more commonly studied class of proper and connective DGAs. There are applications to some invariants of DGAs. 

Nafeesa Khalil — Twisting biquadratic extensions

We explore how a Drinfeld twist can be used to twist a biquadratic field extension into a quaternion algebra.

Luke Naylor — Narrowing down possible Bridgeland stability walls

The notion of stable vector bundles have been studied for over a half century, being motivated by physics, but also fitting into the story of classifying vector bundles. In more recent history, these notions have been translated, and researched in a much more homological algebraic setting by defining stability conditions on triangulated categories (such as the derived category of coherent sheaves on a space). These are known as Bridgeland stability conditions, which in certain settings, are known to be parametrized by certain spaces. Some of these stability conditions give the same moduli space as Gieseker stability, however other moduli spaces appear when we cross over "walls". I aim to introduce this transition of stability conditions to triangulated categories, talk about walls, and methods to narrow them down, and their connections to some classical number theory.

Scott Harper — Generating sets for finite groups

There is a long history of studying generating sets for groups. Beginning with a conjecture of Netto around 150 years ago, this talk will give some highlights of this subject, before discussing some recent work with Tim Burness and Robert Guralnick. I will showcase some applications of this work to other areas of group theory, explain some of the key techniques and pose some open questions.

Thomas De Fraja — The relation between T-duality and Courant algebroids

T-duality originated as symmetry of theories in string theory, but has since found its way into many areas of mathematics. The problem is, no one really knows how to define it. In this talk, we will attempt to resolve this. After reviewing some consequences of T-duality, we will introduce Courant Algebroids (CA) and CA relations. While CA relations generally behave wildly, they can be tamed, and upon doing so, give us our desired definition. A few consequences we will see are an attempt to go beyond a Fourier-Mukai transform (in the simple case of fibre bundles), and even an extra set of arrows for the category of (exact) Courant algebroids. In this talk we will be working with vector bundles, but mostly on the level of linear algebra, so we will assume little knowledge of differential geometry.

Sarunas Kaubrys — Counting fundamental group representations of 3-manifolds

Donaldson-Thomas theory starts with the study of virtual counts of sheaves on a Calabi-Yau 3-fold. Through the work of Joyce, these invariants can be upgraded to the form of a sheaf on the moduli space. One can obtain the original invariants by taking Euler characteristic. It is also possible to make sense of a 3 Calabi-Yau category, generalising sheaves on a Calabi-Yau 3 -fold and try to calculate DT invariants for the moduli space of objects in this category. For representations of a quiver with potential this sheaf is well understood via the cohomological integrality theorem of Davison-Meinhardt. I will explain how one can think about DT invariants for fundamental group representations of a 3-manifold, which will be 3 Calabi-Yau. In particular, I will explain my ongoing work to connect DT invariants of the tripled Jordan quiver with potential and fundamental group representations of the 3 torus via an exponential type map.

Franco Rota — Del Pezzo surfaces with quotient singularities

If one wants (as we do!) to understand the derived category of a singular surface X, one needs control over the singularities of X, and over its birational geometry. In an explicit example, we’ll go through some of the combinatorics of the singularities and relate the resolution of X to a classical problem in planar geometry. My talk will be based on my collaboration with Giulia Gugiatti.

Lewis Dean — Double Affine Weyl Semigroups & The Demazure Product

Hecke algebras are generated by elements whose multiplication is controlled by the corresponding Weyl group W. In the q = 0 specialisation, it is determined fully by the Demazure product in W, which is well understood in the finite and affine cases. Recently, a more explicit expression for this product has been developed, and in this talk we discuss conjectures and results so far on extending this to a double-affine setting.

Karim Réga — The Moduli Zoo

Moduli problems are ubiquitous in mathematics, leading to the meta-theory of defining and examining moduli spaces. There are various notions of moduli space out there, all with different degrees of satisfaction. Focusing on three prevalent ones, fine, coarse and good moduli spaces, we will try to explain what properties one desires for a moduli space and how whether they can be satisfied leads to different notions of a moduli space. On the way there, we'll also pick up some facts about the deep link between taking quotients and moduli problems, and algebraic stacks.

Augustinas Jacovskis — Geometry from derived categories

I will discuss how the derived category of coherent sheaves on a variety X (in particular a special subcategory of it) encodes lots of the geometric information of X. I’ll mention a few applications to Hodge theory. 

Marina Purri Brant Godinho — Cluster algebras and categories

Cluster algebras were introduced by Formin and Zelevinsky in 2002 in order to study canoncial bases of quantum groups. For the moment, the link between these algebras and canoncial bases of quantum groups is still a matter of open research. However, connections between cluster algebras and several areas of mathematics have been discovered. Notably, these algebras find applications in Poisson geometry, integrable systems, algebraic geometry, and the representation theory of quivers. The link between cluster algebras and the representation theory of quivers passes through a categorification of these algebras.

In this talk, I will present and introduction to cluster algebras, and discuss their categorification via cluster categories.

Charlotte Bartram — Spherical objects and spherical functors

Spherical objects were first introduced in 2000 by Seidel and Thomas as mirror symmetric analogues of Lagrangian spheres on a symplectic manifold. Each such object produces an autoequivalence called a spherical twist, the analogue of a Dehn twist around a Lagrangian sphere. In this talk I will introduce the notion of a spherical object and explain how this notion can be generalised to that of a spherical functor.

Sebastian Schlegel Mejia — BPS Lie algebras for 2-Calabi–Yau categories

BPS Lie algebras are and object of the (categorified) enumerative geometry of 2-(and 3-)Calabi–Yau categories. Examples of 2-Calabi–Yau categories are: categories of representations of preprojective algebras, representations of fundamental group algebras of Riemann surfaces, Higgs bundles on a curve, or coherent sheaves on a K3 surface. The idea for BPS Lie algebras comes from BPS state counts in physics and their original, very technical, mathematical definition comes from cohomological Donaldson–Thomas theory.

In this talk I will present the partially conjectural "algebraists" definition of BPS Lie algebras of 2-Calabi–Yau categories akin to the definition of semisimple Lie algebras via Dynkin diagrams, that is via "generators and relations" given by auxiliary data. We will see that the generators are governed by the topology of moduli spaces and the relations are governed by the Euler form of the given category. I hope to give some intuition about how BPS Lie algebras "count" simple objects in the 2-Calabi–Yau category. Parts of the talk are based on joint work in progress with Ben Davison and Lucien Hennecart.

Stefanie Zbinden — The Morse boundary: capturing hyperbolic features of non-hyperbolic groups

Isambard Goodbody — Differential graded algebras and derived equivalences

Many invariants of a ring depend only on its category of modules ModR. Hence it is useful to study when ModR = ModS, that is when two rings R and S are Morita equivalent. Morita Theory gives a complete picture of when this happens in terms of the existence of certain modules. The next observation is that in fact many of these invariants depend only on the derived category of the ring D(R), so we study when R and S are derived equivalent, i.e. D(R) = D(S). Rickard provided the exact conditions for this to occur in terms of the existence of certain modules. The conditions on these modules are more restrictive than in the classical case, and removing this extra restriction moves us away from rings and into DGAs. In this talk I'll define DGAs and talk about their relation to derived equivalences using an example. 

Luke Naylor — From Fourier to Fourier–Mukai

When you look at the definition for the Fourier–Mukai Transform, it's hard to see what this has to do with integral transforms, let alone the one in its name. Yet, this analogy was obvious to Shigeru Mukai, who first introduced it in the 80s as "a Fourier functor". During this talk I will present the Fourier Transform, and the Fourier-Mukai transform, concentrating on the similarities and how they are used for the same purpose. Then, finally, a peek at what role this plays in finding walls for Bridgeland stability conditions.