Bill studied pure maths at Warwick but has been tiptoeing towards the ‘applied side’ ever since.
Bill studied pure maths at Warwick but has been tiptoeing towards the ‘applied side’ ever since. At Warwick Bill’s research project was on percolation theory, and after this he did a project on graph embeddings and modelling network data in Bristol, for which he gained an MSc. Any probabilistic or statistical problem with a discrete flavour is very likely to appeal to Bill. But he’s also highly interested in machine learning (particularly from a more statistical perspective) and hopes to deepen his understanding of numerical analysis while part of SAMBa.
Bill’s interests outside of maths are quite typical among those interested in maths: he plays piano (lots of Bach); and really enjoys bouldering, reading, and pubs. Bill likes surreal humour and his favourite surreal film is ‘Film Film’ (1982).
Research project title: Interacting particle systems with an infinite type space.
Supervisor(s): Marcel Ortgiese, Tim Rogers
Project description:
What is an interacting particle system, what is meant by particle type?
Imagine the two-dimensional integer lattice for which a ‘red’ or ‘blue’ particle has been placed at every lattice point. The colour of the particle is called the ‘type’. Specifying a collection of local interaction rules can lead to a plethora of interesting stochastic systems. For one example, a particle could wait for an exponentially distributed time and then change type to that of a randomly chosen neighbour. When the waiting times are independently sampled at each particle, we obtain the famous interacting particle system (IPS) known as the ‘voter model’. In the two-dimensional voter model large red and blue regions form and slowly grow as time progresses.
The voter model is one tip of a terrific multidisciplinary iceberg, from statistical physics (Ising model, other spin systems) to spatial ecology (Durrett’s bushes-grass-trees model, evolutionary games on graphs).
The project:
Bill’s PhD project will examine IPSs which have an infinity of particle types. An abundant source of infinite type space IPSs are evolutionary games with all mixed strategies permitted. Consider a win lose game like rock-paper-scissors. One would typically study the IPS with three types (‘rock’, ‘paper’, ‘scissors’): particles compete locally, with the losing particle copying the winner’s type. Bill’s ambitions are higher— he would study the system where the particle type is given by any 3-dimensional probability vector. In this infinite type system the particles compete by playing each of rock paper scissors according to their type vector, with competition losers copying the winner’s vector plus a small perturbation. Even this toy system exhibits interesting behaviour. Every particle’s type becomes close to one of the three ‘pure’ probability vectors (1,0,0), (0,1,0), (0,0,1), essentially recovering the finite type system!
Students joining SAMBa in 2023