I am a first year PhD student under the supervision of Marcel Ortgiese and Tim Rogers. My core mathematical interests are interacting particle systems and evolutionary games. I am interested in (too) many areas of applied maths so have also gained some working knowledge of Bayesian inference, inverse problems, statistical physics during my time at SAMBa.

Before SAMBa I studied maths at Warwick and Bristol, undertaking research projects on percolation theory and graph embeddings, respectively. Any probabilistic or statistical problem with a discrete flavour is extremely likely to appeal to me.

Outside of maths I enjoy playing piano, Texas Hold ‘em and Pot Limit Omaha, sports and games which require good balance; reading fiction; watching films. The Eight Mountains (2023) is a film I saw recently and really liked.

Research project title:
Interacting particle systems with an infinite type space.

Supervisor(s):
Marcel Ortgiese, Tim Rogers

Project description:

My PhD project will examine interacting particle systems (IPSs) which have an infinite type space. Evolutionary games in which each particle can take any mixed strategy can often be considered as IPSs with an infinite type space. Here, the strategy a particle takes defines the particle type. Equations for the dynamics of such systems (replicator dynamics for instance) obtained by taking the number of particles to infinity can obfuscate whether the particle population is (primarily) composed of a single mixed strategy (is “monomorphic”) or a mixture of pure strategies (is “polymorphic”). However, for finite stochastic systems either monomorphic or polymorphic populations can be favoured depending on the micro-dynamics! For starters, I want to find out when polymorphism is favoured and why.

For readers who didn’t follow much of the above I aim to provide essential context below. Namely, I explain what is meant by type space and give an example of an IPS.

Imagine the two-dimensional integer lattice for which a coloured dot (i.e. particle) has been placed at every lattice point. The type space is set of all possible colours (i.e. types) each particle can take. Consider a finite type space {red, blue} to begin with, that is, imagine each particle is coloured either red or blue. Our particles will not move, but they can change type (i.e. colour) over time. Only particles which are neighbours on the lattice can directly affect each other’s type, and only according to some collection of interaction rules we specify beforehand. One classic interaction rule involves every particle having their own Poisson clock and copying the type of a neighbour chosen at random whenever their clock rings. When these Poisson clocks are independent and have the same rate an IPS called the voter model is obtained. Imagine starting the voter model with particles in a red-blue chessboard arrangement. Large regions of a single colour will form and merge as time progresses.

Visit Bill’s website:
https://billnunn.github.io/website/