This page contains a (hopefully!) non-technical description of these activities. Some of them are recent, others less so. The more recent activities are described first. While the emphasis here is on completed work, there is inevitably an overlap between topics described here, and possible future projects. References in [square brackets] are to my list of publications. References such as [R18] are to my list of refereed papers, while those such as to [O3] are to other work.

My work in papaers such as [R18] and [R19] has concerned the structure of the Euclidean minimal spanning tree. This is the graph connecting a given set of points in d-dimensional space with minimal total length, and is of interest in computer science, the physical sciences and in statistics. The structure of the minimal spanning tree on random points is of interest, for example because of certain statistical tests (see [R22], [R25]), and is only partially understood. The random points could be uniformly distributed on the unit cube, or have some general probability density function.

In [R21], I worked on the minimal spanning tree in the n-cube, a discrete structure with some analogies to high-dimensional Euclidean space. I hope to continue to work on such discrete structures in the near future. There may be applications in mathematical genetics.

For an account of aspects of the related topic of interacting particle systems, see my set of lecture notes [O3]. Typically, these systems involve a collection of particles living on a lattice, evolving in a random way, with neighbouring sites on the lattice interacting. For example, certain models of spatial epidemics can be formulated naturally in this way. Often, particle system models, and also percolation, exhibit a phase transition at a critical value of some parameter, such as the rate of infection.

Another lattice-based mathematical model is the self-avoiding walk. Imagine a particle moving around in a random way among neighbouring sites in a 3-dimensional lattice, constrained never to visit the same site twice. This has been studied as a model for long polymer chains. An important quantity in self-avoiding walk is the so-called `connective constant'.

I have also worked on a variety of lattice-based models of the type described above. Usually the precise critical value or connective constant is not analytically known. In [R8], [R11], [R15], and [R17], I developed exact asymptotics for these critical values in certain limiting cases for the lattices.

The main topic of my PhD thesis [O1], also the subject of in [R2,R3], is the behaviour of mathematical Brownian motion. Of special case in this work was the study of self-avoiding Brownian paths in three-dimensional space, a continuous-space analogue of the self-avoiding walk described above. In [R2] is a result about the non-existence, in some sense, of these self-avoiding paths.

One major application of the theory of Brownian motion and stochastic analysis is in the study of certain partial differential equations. In [R9], the theory of brownian motion is used to analyse some equations arising in statistical physics, and to obtain new information on the deviation from the ideal gas law for a particular model of molecular interaction. Some probabilistic questions concerned, with the three-dimensional Brownian bridge (Brownian motion constrained to be at a specific position at a specified time) were studied in [R7].

Last update 1 June 1999