## Research

*“Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root and it would be hard to find a better one on which to demonstrate the working of mathematical intellect.”*

– Hermann Weyl

My main research focus is in group theory, the mathematical study of symmetry. I am interested in properties of abstract groups, such as generating sets and subgroup structure, and I am also interested in properties of group actions, such as questions concerning derangements and bases of permutation groups. I am particularly happy when these two topics intersect. While group theoretic questions are usually the motivation for my work, some of my projects would be better described as representation theory, geometric group theory, Lie theory or combinatorics. Lately, I have also been interested in formalising proofs in Lean.

A major strand of my work has concerned generating pairs for groups. This began with finite groups, and a recent highlight here is my paper with Tim Burness and Robert Guralnick [*Annals of Mathematics* (2021)] which answered the question: “In which finite groups does every nontrivial element belong to a generating pair?” More recently, I have been considering the ways that these sorts of results do and do not extend to infinite groups, and this is the subject of my current EPSRC Postdoctoral Fellowship.

I wrote a survey article based on my plenary lecture at Groups St Andrews in 2022 (a preprint is available at arxiv:2210.09635). This gives an overview of recent work concerning generating pairs for finite and infinite groups.

A full list of my publications can be found on my Publications page.

For a friendly nontechnical introduction to my work, take a look at my poster I presented at STEM for Britain. For further information about my work, but still requiring no specialist information at all, have a read of my leaflet The Art of Measuring Symmetry.