University of Oxford

Finitely presented groups are a natural algebraic generalisation of the collection of finite groups. Unlike the finite case there is almost no hope of any kind of classification.

The goal of random groups is therefore to understand the properties of the "typical" finitely presented group. I will present a couple of models for random groups and survey some of the main theorems and open questions in the area, demonstrating surprising correlations between these probabilistic models, geometry and analysis.

We consider the coordinate-iterated-integral as an algebraic product on the shuffle algebra, called the (right) half-shuffle product. Its anti-symmetrization defines the biproduct  area(.,.), interpretable as the signed-area between two real-valued coordinate paths. We consider specific sets of binary, rooted trees known as Hall sets. These set have a complex combinatorial structure, which can be almost entirely circumvented by introducing the equivalent notion of Lazard sets. Using analytic results from dynamical systems and algebraic results from the theory of Lie algebras, we show that shuffle-polynomials in areas-of-areas on Hall trees generate the shuffle algebra.

I will elaborate on some recent developments on the theory of special functions which are relevant to the calculation of Feynman integrals in perturbative quantum field theory, highlighting the connections with some recent ideas in pure mathematics.

Arctic sea ice is one of the most sensitive components of the Earth’s climate system. The underlying ocean plays an important role in the evolution of the ice cover through its heat flux at the ice-ocean interface. Despite its importance, the spatio-temporal variations of this heat flux are not well understood. In this talk, I will take the following approach to study the variations in the heat flux. First, I will consider the problem of classical Rayleigh-Bénard convection and systematically explore the effects of fractal boundaries on heat transport using direct numerical simulations. And second, I will analyze time-series data from the Surface Heat Budget of the Arctic Ocean (SHEBA) program using Multifractal Detrended Fluctuation Analysis (MFDFA) to understand the nature of fluctuations in the heat flux. I will also discuss developing simple stochastic ODEs using results from these studies.

The sequence of so-called Signature moments describes the laws of many stochastic processes in analogy with how the sequence of moments describes the laws of vector-valued random variables. However, even for vector-valued random variables, the sequence of cumulants is much better suited for many tasks than the sequence of moments. This motivates the study of so-called Signature cumulants. To do so, an elementary combinatorial approach is developed and used to show that in the same way that cumulants relate to the lattice of partitions, Signature cumulants relate to the lattice of so-called "ordered partitions". This is used to give a new characterisation of independence of multivariate stochastic processes.