#### 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*=GL* _{n}*. 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*=PSL_{2}(*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.

#### Representation growth

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.

#### Representation varieties

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.

#### Other interests

My other research interests include geometric invariant theory, quantum field theory and cryptography.