University of Oxford

A big mapping class group is the mapping class group (MCG) of a surface of infinite type. Although several aspects of big MCGs remain mysterious, their geometric definition allows some simple, interesting arguments. In this talk, we will use big MCGs as an excuse to survey some (more or less) classical results in geometric group theory: we will present a quick introduction to infinite type surfaces, highlight differences between standard and large MCGs, and use Higman’s embedding theorem to deduce that there exists a big MCG that contains every finitely presented group as a subgroup.

Following on from Christoph's talk last week, I will present a version of the supercooled Stefan problem with noise. I will start by discussing the physical intuition and then give a probabilistic representation of solutions. From there, I will identify a simple relationship between the initial heat profile and a single parameter for how the liquid solidifies, which, if violated, forces the temperature to develop a discontinuity in finite time with positive probability. On the other hand, when the relationship is satisfied, the temperature remains globally continuous with probability one. The work is part of a new preprint that should soon be available on arXiv.

TBC

We develop a new online algorithm for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm optimizes over the parameters in the multi-dimensional SDE model in order to minimize the distance between the model's stationary distribution and the target statistics. We rigorously prove convergence for linear SDE models and present numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the unbiased descent direction under the stationary distribution. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds on a new class of Poisson partial differential equations, which are then used to analyze the parameter fluctuations in the algorithm. This presentation is based upon research with Ziheng Wang.
 

We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash to a subset of the entities in order to limit defaults to a given proportion of entities. We prove that the value of the agent's control problem converges as the number of defaultable agents goes to infinity, and it satisfies  a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a forward-backward coupled system of PDEs. Our simulations show that the agent's optimal strategy is to subsidise banks whose asset values lie in a non-trivial time-dependent region. Finally, we study a linear-quadratic version of the model where instead of the losses, the agent optimises a terminal loss function of the asset values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically. Joint work with Christa Cuchiero and Stefan Rigger.

We introduce a method for estimating the roughness of a function based on a discrete sample, using the concept of normalized p-th variation along a sequence of partitions. We discuss the consistency of this estimator in a pathwise setting under high-frequency asymptotics. We investigate its finite sample performance for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of Fractional Brownian motion and other fractional processes.
We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations.
Through a detailed numerical experiment based on a stochastic volatility model, we show that even when instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rougher behaviour corresponding to a Hurst exponent significantly smaller than 0.5. Similar behaviour is observed in financial data, which suggests that the origin of the roughness observed in realized volatility time-series lies in the `microstructure noise' rather than the volatility process itself.

Hermitian matrix models with non-trivial covariance will be introduced. The Kontsevich Model is the prime example, which was used to prove Witten's conjecture about the generating function of intersection numbers of the moduli space $\overline{\mathcal{M}}_{g,n}$. However, we will discuss these models in a different direction, namely as a quantum field theory. As a formal matrix model,  the correlation functions of these models have a unique combinatorial/perturbative interpretation in the sense of Feynman diagrams. In particular, the additional structure (in comparison to ordinary quantum field theories) gives the possibility to compute exact expressions, which are resummations of infinitely many Feynman diagrams. For the easiest topologies, these exact expressions (given by implicitly defined functions) will be presented and discussed. If time remains, higher topologies are discussed by a connection to Topological Recursion.

In this talk, we use tools from representation theory and symmetric function theory to compute correlations of eigenvalues of unitary invariant ensembles. This approach provides a route to write exact formulae for the correlations, which further allows us to extract large matrix asymptotics and study universal properties.

TBA