The main theme of my research career has been the study of various kinds of stochastic processes, principally Markov processes, and the analysis of how the distributions of such processes are affected by various operations such as path transformations and conditionings. In recent years I have become interested in interfaces between this core of ideas and ideas in other areas of mathematics, especially combinatorics, special functions, and analytic number theory.
I have worked on various random combinatorial objects, such as permutations, partitions, and trees, and how the asymptotic behaviour of such structures over a large set of elements can be described in probabilistic terms, often involving Brownian motion and related processes. This has led me to the study of various measure-valued and partition-valued Markov processes whose behaviour may be understood in terms of combinatorial constructions involving random trees.
I view this line of research largely as pure mathematics, but mathematics of a concrete kind which is often motivated and influenced by applications, for instance the use of Bayesian nonparametric statistic in some models for machine learning. Stochastic models with a natural probabilistic structure typically turn up in different disguises in diverse fields. The study of their mathematical structure allows ideas and results developed in one context to be transferred to another. Numerous examples of this phenomenon can be found in the course Combinatorial Stochastic Processes on my publication page.