Research

Analysis of random geometric models derived from point processes (e.g., tessellations and polytopes) and dynamical geometric objects (e.g., random graphs and evolutions of point/particle processes), and further development of optimal transport techniques for point processes.

Identification of effective equations for vortex driven mixing in fluid dynamics, exploitation of new variational approaches to study non-linear stochastic homogenisation problems such as Dirichlet problems in randomly perforated domains in the free discontinuity setting, and investigation of HJB equations on continuum percolation clusters.

Analysis of the concept of self- organised criticality for spin systems, construction of Gibbs measures for systems of self-interacting Brownian motions, convergence of interface growth models using regularity structures, and further development of large deviation results for random walks in random environment through their relation to HJB equations.

Analysis of SPDEs using the Polchinski flow, leverage of tools from the theory of Gaussian multiplicative chaos to understand continuous directed polymers, and analysis of numerical approximations for irregular SPDEs by combining regularity structures and machine learning algorithms.

Analysis of and limit theorems for sophisticated training algorithms for machine learning (e.g., Adam algorithm), and design of new algorithms.