Past Meetings: 2021–22

Alan Logan — On the conjugacy problem for HNN-extensions of free groups

The conjugacy problem is one of the most important decision problems in Geometric Group Theory. It is known to be undecidable in general, but decidable for many important classes of groups. However, it remains open for many classes of groups, such as HNN-extensions of free groups. I will define and motivate the terms in the previous 3 sentences, before explaining a link between the conjugacy problem for HNN-extensions of free groups and an undecidable problem from computer science, called Post's Correspondence Problem. Joint work with Laura Ciobanu.

Ross Paterson — Average Ranks of Elliptic Curves

Fix a field K, which is a finite extension of the rational numbers.  If we pluck a random elliptic curve defined over the rational numbers E out of a bag, how "large" should the set E(K) of K-points on E be?  Moreover, how much does our choice of K matter?  Is E(K) typically larger for some K than for others?  This talk will make these questions formal, and then discuss what I can prove on them so far!

Nicolas Vaskou — Parabolic subgroups of large type Artin groups.

We will see what is an Artin group and what is a parabolic subgroup of an Artin group. We will then introduce a geometric complex whose structure is given by the parabolic subgroups of a given Artin group. If the Artin group is large, this complex will come with an appropriate kind of negatively curved structure (namely, it is systolic). We will then give ideas of how one can recover from the geometric study of this complex, nice algebraic properties on the parabolic subgroups of the Artin group.

Patrick Kinnear — Invertibility in Morita categories

In this talk, we will introduce a family of higher categories called Morita categories. We will explain their relevance to the study of topological field theories, and will discuss the importance of understanding invertible 1-morphisms in this regard. We will report on work in progress to characterise invertibility in such categories, and if time allows will indicate some future geometric applications of this work to character stacks.

Hannah Dell — Bridgeland stability and the Beauville surface 

Stability conditions on triangulated categories were introduced by Tom Bridgeland in 2007 as a mathematical way to understand Douglas' work on Π-stability for D-branes in string theory. In the same paper Bridgeland also showed that the space of stability conditions on a given triangulated category can be viewed as a complex manifold (called the stability manifold), giving us a way to extract geometry from homological algebra. However an explicit description of the stability manifold is only known in a few cases, and there are few general theorems about their geometry. If we restrict our attention to stability conditions on derived categories of algebraic varieties, then in all known examples the stability manifold contains an open set of so-called geometric stability conditions. This year Lie Fu, Chunyi Li, and Xiaolei Zhao showed that if a smooth projective variety X has finite Albanese morphism, then the stability manifold of D^b(X) is only comprised of geometric stability conditions. The question remains as to whether the converse of this is true.

In this talk I will motivate and introduce Bridgeland stability conditions, and then describe my work with the Beauville surface: a variety with non-finite Albanese morphism that I hope to use as a test case for this open question.

Sam Lewis — Preprojective spherical subalgebras of Kronecker quivers

Draw a graph with two vertices and n >= 2 arrows between them, all going in the same direction. If you now add n more arrows in the opposite direction and fix some d>=0, how many “paths” (possibly modulo some relations) are there in this graph with length d? In this talk I will make this game precise by introducing quivers, preprojective algebras, and their spherical subalgebras, before discussing a recent result on the latter. Finally, if time permits, I will talk about the geometric motivation of this work and current open questions.

Gemma Crowe — The Conjugacy Problem in virtual right-angled Artin groups

The conjugacy problem in RAAGs was first solved by Green in the 90s, and has a quick linear time solution provided by Crisp, Godelle and Wiest. This decision problem remains open however for a closely related group, namely virtual RAAGs. In my talk I will introduce the methods used to solve the conjugacy problem in RAAGs, and some of my ideas so far to tackle the conjugacy problem in virtual RAAGs.

David Murphy — The Grothendieck Group of the Discrete Cluster Category of Dynkin Type A∞

There is a well-known connection between cluster categories of Dynkin type An and the combinatorics of the arcs between vertices of an (n+3)-gon. In this talk we shall look at a well-studied extension of this, namely the discrete cluster category of Dynkin type A∞, or, equivalently, the category with indecomposable objects being the arcs on a 1-sphere with infinitely many discrete marked points on the boundary. We also provide a brief overview of the Grothendieck group of a triangulated category, before giving our main results in a forthcoming paper of the Grothendieck group for the discrete cluster categories of Dynkin type A∞.

Charalampos Verasdanis — Classification of localizing tensor-ideals of tensor-triangulated categories

Given a tensor-triangulated category T, a longstanding open problem is the classification of localizing tensor-ideal subcategories of T in terms of subsets of some topological space. While this problem remains elusive in absolute generality, the subject has recently seen important progress in the classification of smashing tensor-ideals with the introduction of a space called the smashing spectrum. We will discuss what the situation looks like when the smashing spectrum classifies the localizing tensor-ideals of T. The concepts under consideration will be illustrated upon the derived category of a commutative noetherian ring.

Antoine Goldsborough — Random walks and Quasi-Isometries

The study of random walks on various groups is a very rich area of mathematics and a considerable amount of research has been done on these. In geometric group theory, a crucial notion is that of a “quasi-isometry” between two spaces. Sadly, random walks and quasi-isometries do not behave well together. In this talk, we will propose the study of a more general process, namely of a Markov chain in order to resolve this issue. This leads to interesting results about random walks on groups quasi-isometric to specific groups, including a Central Limit Theorem. This is joint work with Alessandro Sisto.

Dora Puljic — Hopf–Galois extensions and Braces

Hopf-Galois theory was introduced by S. Chase and M. Sweedler in 1969 as a generalisation of classical Galois theory. Later, it was shown by C. Greither and B. Pareigis that the classification of Hopf-Galois extensions amounts to studying certain regular subgroups. On the other hand, in 2007, W. Rump introduced an algebraic object called a brace in order to help study non-degenerate, involutive, set-theoretic solutions to the Yang-Baxter equation. Later, skew braces, a generalisation of braces, were introduced to aid the study of non-involutive solutions. Initially discovered by D. Bachiller, and perhaps surprisingly, there is a connection between skew braces, an object studied within quantum group theory, and Hopf-Galois extensions, a number theoretic object. In this talk I will motivate Hopf-Galois theory and brace theory and briefly explain how they connect to each other.

Simone Castellan — Automorphisms of Kleinian singularities of type A and D and of their quantisations

Given a Poisson algebra A and an associative quantisation Q, a natural question is wether the group of Poisson automorphisms of A is isomorphic to the group of automorphisms of Q. In the greatest generality, this is a difficult open problem. In the case where Q is the Weyl algebra and P is the symmetric algebra (both on n generators) this the well known Kontsevich conjecture. In general, this isomorphism means that every Poisson automorphism can be quantised. In this talk, I will give an introduction to the topic of quantisation of Poisson algebras. I will also present some preliminary results I have on the isomorphism problem, in the case of Kleinian singularities of type A and D.

Brian Makonzi — The Artin component of cyclic surface singularities

I will discuss how to use noncommutative resolutions of non-Gorenstein singularities to construct classical deformation spaces, by computing the Artin component of the deformation space of a cyclic surface singularity using only the quiver of the corresponding reconstruction algebra. This extends work of Brieskorn, Kronheimer and Cassens–Slodowy into the setting of singularities ℂ^2/G with G≤GL(2,ℂ),and also gives a prediction for what is true more generally.

Lucas Buzaglo — Universal enveloping algebras of Krichever-Novikov algebras

Universal enveloping algebras of finite-dimensional Lie algebras are fundamental examples of well-behaved noncommutative rings. On the other hand, enveloping algebras of infinite-dimensional Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. In this talk, I will introduce a class of infinite-dimensional Lie algebras known as Krichever-Novikov algebras, and talk about my recent paper where I proved that their enveloping algebras are not noetherian, providing a new family of non-noetherian universal enveloping algebras.

Peter J Cameron (St Andrews) — Graphs defined on groups

The phrase “graphs defined on groups” might suggest Cayley graphs to you.  These graphs are of great importance both in finite and infi-nite group theory,  as well as in construction of graphs and networks with good properties (such as Ramanujan graphs).  However, I mean something a bit different: these are graphs constructed directly from the group G, and invariant under the automorphism group of G. The first such graph to be considered was the commuting graph, in which x and y are joined  if xy = yx,  introduced  in  the  seminal  paper  of Brauer and Fowler, arguably the first step towards the classification of finite simple groups.  Next was the generating graph, in which x and y are joined if〈x, y〉=  G, useful in studying probabilistic generation of almost simple groups.

In the first lecture, I will give a general account of these and related graphs, which form a hierarchy (the edge set of each graph contained in that of the next).  It turns out to be very productive to look at the groups for which two of these graphs coincide.  Among such classes we find Dedekind groups, 2-Engel groups, EPPO groups (all elements of prime power order), groups with cyclic or generalised quaternion Sylow subgroups, minimal non-abelian (or non-nilpotent, or non-soluble) groups,  etc.  I will also tell the story of how,  during the Covid lockdown, I found myself running a research discussion in South India, at which many of these results were discovered.

In the second lecture, I will look in more detail of some of the plums that grow on this tree.  I will explain how we strengthened a theorem of Landau from 1903; how we can strip away the rubbish from some of these graphs to find hidden jewels; how the results turn up old and new number-theoretic results and problems, including the strange constant c = 2.6481017597. . .; and how Andrea Lucchini modified the notion of generating graph to handle groups which are not 2-generated.

Willow Bevington (Edinburgh) — Cohomology and the Axiom of Choice

With the advent of homotopy type theory we're seeing more and more used of algebraic topology in logic, where "types" in axiomatic type theory are modelled by homotopy types. It is, however, often hard to see how topology and logic interact. In this talk I will give a largely elementary account of how cohomology can be used to study a famous mathematical axiom; the axiom of choice. This justifies and motivates the use of homotopy theory in logic, and is in any case a beautiful stand-alone result.

Rhys Davies (Glasgow) — The composition factors of some rational Cherednik algebra modules

This short talk will outline the structure of some modules which appear in the representation theory of rational Cherednik algebras (with t=1). Though they arise from some rather specialised theory, the modules in question can be boiled down to some quite simple algebra. It is the discussion of these modules -- and the way in which one may compute their composition factors -- which will form the basis of this talk.

Arman Sarikyan (Edinburgh) — On the Rationality of Fano-Enriques Threefolds with terminal cyclic quotient singularities 

A three-dimensional non-Gorenstein Fano variety with at most canonical singularities is called a Fano-Enriques threefold if it contains an ample Cartier divisor that is an Enriques surface with at most canonical singularities. There is no complete classification of Fano-Enriques threefolds yet. However, L. Bayle has classified Fano-Enriques threefolds with terminal cyclic quotient singularities in terms of their canonical coverings, which are smooth Fano threefolds in this case.

The rationality of Fano-Enriques threefolds is an open classical problem that goes back to the works of G. Fano and F. Enriques. In this talk we will discuss the rationality of Fano-Enriques threefolds with terminal cyclic quotient singularities.

Lewis Dean (Glasgow) — Affine Demazure Products and the Quantum Bruhat Graph

The Demazure product for Weyl groups is an important concept with regards to certain Hecke algebras - it is well understood in the finite case, but only recently have explicit expressions been found for the affine case. In this talk, we look at affine Weyl groups and how the Demazure product can be calculated via the Quantum Bruhat Graph.