Most of my current research is focused on the problem of identifying Morita equivalence classes of blocks of finite groups. This is part of the study of the representation theory of finite groups with respect to fields of prime characteristic. Briefly, Morita equivalence is an equivalence of module categories, preserving the structure of modules for an algebra.
This problem is fundamental to the area, and ties in with another of my areas of interest, global-local relationships in finite groups.
Problems range from Donovan's conjecture, which is a finiteness conjecture concerning the number of Morita equivalence classes, to classification of Morita equivalence classes in specific cases.
A tool I use frequently is the classification of finite simple groups, but there is scope for a variety of projects suited to different interests. The precise nature of the project would be open for discussion with the prospective student.