The general focus of my research is the algebraic structures that underlie physical theories. My favourite physical theories are two-dimensional conformally invariant quantum field theories (a.k.a. conformal field theories), mostly because the underlying algebras are so captivating. These are of course those things called chiral algebras in physics and vertex operator algebras (or conformal vertex algebras) in maths. (If you don't know what these are, Michael Penn has a bunch of introductory youtube videos here.) Much of what I do centres around exploring the representation theory of these algebras, but anything involving cool math is fine with me!
Most of my conformal field theory research is devoted to logarithmic examples. This means that the representation theory is non-semisimple, i.e. that the physically relevant category of modules over the given vertex operator algebra includes modules that are reducible but indecomposable. A consequence is that there are correlation functions in the theory that exhibit logarithmic singularities, whence the name. Such things arise quite generically, for example when studying non-local observables in statistical lattice models (e.g. percolation or even the Ising model) as well as in the description of string theories with fermionic target spaces. The bosonic ghost system is a particularly important example.
There are not many well-understood logarithmic conformal field theories. My work is helping to rectify that. At the moment, I'm particularly interested in reducible but indecomposable modules for affine vertex operator algebras (which are obtained from affine Kac–Moody algebras) and the W-algebras obtained from them by quantum hamiltonian reduction. These are not only interesting as fundamental examples, but also because they finding increasing relevance in the exciting new dualities relating two-dimensional conformal field theories to higher-dimensional supersymmetric gauge theories.
If you're interested in working on any of these things, please drop
me a line! I'm always happy to consider new MSc/MPhil/PhD students and
every now and again I even have positions for postdocs open (usually
when the grant gods have been kind — see below).
However: I currently get so many requests for PhD supervision that I can't respond to them all. If you're serious, then please take the time to explain why you want to work on the stuff I do and why you have at least some of the background to do so. Also, please don't send me your CV/résumé — this is not helpful. Instead, please send me your transcripts.
One thing that I need to stress is that I do maths. There's lots of exciting physics words here, but if you want to work with me, then you have to be willing to learn maths. Not maths as taught in theoretical physics courses, but maths as taught by pure mathematicians. The thing about math physics research is that you can only get so far with the tricks you learned at uni. Either you do the easy bits and then move onto a different topic, or you learn some advanced maths (or develop it yourself) that allows you to do the bits that aren't so easy. I'm firmly in the latter camp and, for me, this is what keeps research interesting.
Anyway, you can find definitive versions of my publications here and a selection of slides/notes from my talks over the years here. If you're really keen, you can find a (probably not particularly up-to-date) CV here. For your entertainment, I also offer some of my (successful) grant applications — the unsuccessful ones I shall keep for when I'm feeling maudlin.