I was an undergraduate at Trinity College in Oxford and did my PhD in Birmingham. I returned to Oxford for a Junior Research Fellowship at Christ Church in 2005 before moving to York in 2008.
Linear Algebraic Groups: These are, loosely speaking, groups of matrices which also carry the structure of an algebraic variety. The study of algebraic groups therefore mixes typical group theoretic tools with ideas from algebraic geometry. My main focus is the structure of these groups and their subgroups, but this naturally involves thinking about their representations and more generally their actions on other varieties. I have also used many ideas from geometric invariant theory and the theory of buildings in my work. A common theme through my work is to consider what happens to these groups as one varies the field one works over -- for example, one might move from characteristic zero to positive characteristic, or from an algebraically closed field to one which is not algebraically closed.
Random Walks on Algebraic Structures: Together with my colleague Stephen Connor, I have recently begun to think about random walks on algebraic structures. Specifically, we have been considering random walks on finite rings -- like the ring of integers mod n -- and how these walks converge to their equilibrium. A particular focus is the existence (or not) of a so-called cutoff phenomenon for such walks.
Algebra Research Group
Geometry and Analysis
Available PhD research projects
Although my main research focus is on the areas described above, I am happy to supervise PhD projects across a wide range of algebraic topics. Typically students do not arrive with a fully formed project idea (although you are welcome if you do have one!), rather we start off with some reading in topics of mutual interest and converge on a project over the course of time. Current and former research students have studied a multitude of topics, including: representations of reductive groups and symmetric groups; Lie algebras and Chevalley groups; geometric invariant theory; counting representations of finite groups; cohomology of small categories; spherical buildings.