Pure PGR

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.

No! (Okay, it’s slightly more complicated than that). Inspired by a recent expository paper on arXiv of the same name, my talk will give the necessary set theoretic background to consider questions about the independence of measurability of certain functions from ZFC. If time permits, I’ll give further results using large cardinal assumptions and in the context of o-minimality.

Spin of a particle is an integral notion in Quantum theory. Understanding this idea has directly influenced the development and enrichment of many topics in pure mathematics for the last century, including Algebra, Lie theory, Representation theory, Operator theory, Differential geometry, to name a few. In this talk I shall give an overview of the mathematical aspects of Spin, and discuss various pure mathematical ideas associated to it.