Reductive algebraic groups
I am interested in the subgroup structure of reductive algebraic groups; in particular, the properties of completely reducible subgroups. A subgroup H of a reductive algebraic group G is said to be completely reducible if whenever H is contained in a parabolic subgroup P of G, H is contained in some Levi subgroup of P. This agrees with the usual notion of complete reducibility if G=GLn. I am collaborating with Gerhard Röhrle and Michael Bate. Some of our work is or was supported by Marsden grants UOC0501 and UOA1021, EPSRC grant EP/C542150/1 and DFG grant SPP1388.
Lattices in rank one groups over local fields
Let G be a rank one algebraic group over a local field K (for example, G=PSL2(K)). Lisa Carbone and I are studying the geometry of spaces of lattices in G. Such a lattice is also a lattice in the automorphism group of the Bruhat-Tits tree of G, so the theory of automorphism groups of trees plays an important part. These spaces of lattices can also be interpreted in terms of representation varieties.
Let F be a finitely generated group. In the theory of subgroup growth, one counts the number a(n) of index n subgroups of F and investigates the asymptotic behaviour of the function a(n). I am interested in representation growth, where one counts instead the number of irreducible complex characters of F. This involves ideas from number theory and model theory.
A representation variety is the space of homomorphisms from a finitely generated group to a Lie group or algebraic group, and a character variety is the space of conjugacy classes of homomorphisms. I study the geometry of representation varieties and character varieties. My PhD thesis was on representation varieties of surface groups.
My other research interests include geometric invariant theory, quantum field theory and cryptography.