I am interested in algebra, topology and combinatorics. You can find most of my research articles on the arXiv.

Homological methods in algebra

An ongoing project, joint with Paul Turner and more recently Michael Bate, on combinatorial sheaves on categories and their (co)homology. These objects play a similar role in homological algebra to that played by modules in other parts of algebra: they provide a unifying framework into which a variety of different problems can be shoehorned. Much of this stuff grew out of work that we did on the Khovanov homology of knots and links. Here are some recent lecture notes on Khovanov homology for beginners. More recently we have been looking at hyperplane arrangements (here and here) and matroids.

Partial symmetry

Another ongoing project, joint with John Fountain and more recently James East. If by “mirror symmetry” one means the theory of reflection groups, then by “partial mirror symmetry” one means the theory of reflection monoids. Reflection groups lie at the intersection of many parts of algebra and mathematical physics. Reflection monoids were invented by John Fountain and myself in 2010.

Representation theory

Quite a few years ago now I proved that every Fuchsian group has among its homomorphic images all but finitely many of the alternating groups. This answered an old conjecture of Graham Higman that had its origins in the study of groups of conformal automorphisms of Riemann surfaces. Here is a very old talk on this and its connection with the congruence subgroup problem. So I have always been interested in the representation theory of infinite discrete groups. More recently, and as an off-shoot of the partial symmetry programme, I have become interested in the representation theory of inverse monoids.  Here are some recent lecture notes on the representations of inverse monoids, written mainly for group theorists.

Geometric manifolds

I work less on this these days, although there are a few open problems that still bug me (particularly this old chestnut: what is the minimum volume of a compact hyperbolic 4-manifold?). Over the years I worked with John Ratcliffe, Steve Tschantz and Colin Maclachlan (Aberdeen). Most of this work has been on the volume spectra of orientable hyperbolic manifolds in a given dimension, usually an even dimension, where the presence of an Euler characteristic makes life a lot easier. A highlight was a solution to the Siegel problem in 6-dimensions, with the construction of a minimal volume orientable (but non-compact) hyperbolic 6-manifold. The construction is essentially algebro-combinatorial and uses Coxeter groups.

Former PhD students:

• Majed Albaity
• Nouf Alqahtani
• Sanaa Bajri
• Guinevere Dyker
• Sam Ford
• Mihail Hurmuzov
• Sajida Koussar
• Jim Woodward