Research in Group Theory includes combinatorial and geometric group theory, semigroup theory, presentations, decidability, string rewriting systems and related homological finiteness conditions, and groups acting on graphs and other relational structures. (Dr R Gray)
Research in Algebraic Combinatorics and finite permutation groups includes the invariant theory and homology of partially ordered sets and reconstruction problems. (Dr J Siemons)
Research in Representation Theory includes: modular representation theory of the symmetric groups and related algebras, including the Hecke algebras of type A, the q-Schur algebras and the Ariki-Koike algebras (Dr S Lyle); representation theory of finite dimensional algebras, algebraic groups and related algebras, connections with Lie theory, homological methods, categorification (Dr V Miemietz).
For details of potential PhD research topics in this area, please see Mathematics PhD Projects in Algebra and Combinatorics.