Abstract: Skew braces are an algebraic structure with close historical and practical ties to the set-theoretic Yang-Baxter equation. We will discuss existing results about skew braces, covering gamma functions, the connection between skew braces and regular subgroups of the holomorph of a group, results about bi-skew braces, and ideals. We will use this to completely classify the finite bicyclic skew braces, and, among them, which are bi-skew.
Abstract: A D-set is a relational structure on the leaves of a tree, consisting of a quaternary relation defined such that we have D(x,y,z,w) if and only if the path from x to y is disjoint from the path from z to w. In this talk I will introduce several model-theoretic notions, namely ultrahomogeneity, indiscernible sequences, dp-minimality and distality, and I will discuss how they arise during the study of D-sets and what they tell us about structures in general.
Abstract: Elliptic PDEs are a widely studied class of partial differential equations. I will discuss a collection of interesting results in regularity and existence, some of them rather remarkable! In doing this I will discuss the H\"older space approach and the Sobolev space approach and hopefully give a couple of the key ideas of the proofs.
Cluster algebras were introduced in 2002 by Fomin and Zelevinsky in the context of Lie theory as commutative algebras defined combinatorially through an iterated mutation process. Since then, they have appeared in many areas of mathematics, including mathematical physics, algebraic combinatorics, and algebraic geometry. Buan, Marsh, Reineke, Reiten, and Todorov later introduced the cluster category, providing a natural categorical model for the combinatorics of cluster algebras. This talk will be a gentle, pictorial introduction to cluster algebras and their categorification via cluster categories, illustrated with plenty of examples.
That the angles of a plane triangle sum to \(\pi\) is a consequence of the parallel postulate. Hiding, therefore, in this formula from primary school is a curvature assumption. It was only a couple of millennia later that mathematicians decided to start considering curved geometries, without the parallel postulate, yielding for example formulae for the sum of the angles of spherical triangles. The conclusion of all this is the shocking Gauss—Bonnet theorem, which states that the curvature (a local property) of a surface determines its topology (a global property).
This talk will survey the longstanding programme to analyse the complex exponential by means of model theory. This ambition has coalesced around Zilber's quasi-minimality conjecture, which we will motivate and define. We will then cover subsequent progress and analogous results by Zilber and Gallinaro-Kirby. Expect a little logic, algebra and complex analysis.
The (Artin) braid group is an important and well-studied group with applications in algebra, geometry, and mathematical physics. I will introduce some of the ways the braid group presents itself and discuss two well-known representations coming from topology: the Artin representation and the Burau representation. These representations will help to demonstrate the connection between braids and knots. I will then go on to introduce a natural generalisation of the braid group, called the loop braid group, and discuss how the Artin and Burau representations lift for this group.
Sub-Riemannian geometry is a generalization of Riemannian geometry and it has been extensively studied in the past 50 years. In this context, Carnot groups play a central role, as they are the sub-Riemannian analogue of Euclidean spaces. They are connected, simply connected Lie groups whose Lie algebra admits a stratification. Thanks to this stratified structure, a particular subcomplex of the de Rham complex, known as the Rumin complex, can be defined on Carnot groups. Introduced by Rumin in the 1990s, it provides a more natural cohomological framework for these groups by selecting a distinguished class of differential forms, often referred to as "intrinsic" forms. In this talk, I will give an overview of Carnot groups and construct the Rumin complex for a special class of Carnot groups: the Heisenberg groups.
Harmonic Maps between Riemannian Manifolds are some of the most widely studied maps in Geometric Analysis. They appear as critical points of the Dirichlet Energy Functional, perhaps the simplest functional between Manifolds one can write down. They can also be viewed as generalisations of Harmonic Functions which I will introduce and discuss some of their remarkable properties. Then I will go on to generalise to Harmonic Maps and discuss some of the problems that arise from them.
Coxeter-Conway friezes and their generalisations are of interest to many mathematicians, largely thanks to their well-known connections to cluster theory and representation theory. Many such connections are motivated by the beautiful correspondence, discovered by Coxeter and Conway, between these frieze patterns and triangulations of polygons. By way of this this same correspondence, we may also make connections between these friezes and other areas of mathematics - in this talk, we discuss a connection between (particularly nice) plane curve singularities and the theory of Coxeter-Conway friezes via an invarant known as the lotus of a plane curve singularity. We will also see how the continued fraction representations of rational numbers appear in this construction.
