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).
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.