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I have been in York since 2008. Before coming here, I held a Junior Research Fellowship at Christ Church in Oxford. My PhD is from Birmingham.
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 work focuses on the structure and representation theory of these groups and how they act on other varieties. A common theme 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: Many classical mixing problems, like how many times you should shuffle a deck of cards to randomise it, can be formulated as random walks on groups. My work in this area, in collaboration with my colleague Stephen Connor, has explored certain types of shuffle as well as random walks on other algebraic structures like finite rings. A particular focus is the existence (or not) of a so-called cutoff phenomenon for such walks.
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; random walks on symmetric and hyperoctahedral groups.