In the past few years I have worked with: Alejandro Adem, Wenfeng Gao, and Jan Minac on the relations between cohomology of groups and field theory.
With Alejandro Adem, Jon Carlson, Jim Milgram, and Kris Umland I have been involved in a project to compute the cohomology of finite simple groups. The latest paper in this series is about the Higman-Sims group; we believe that the Mathieu group M24 can also be studied using our techniques.
Marty Isaacs and I have studied the character theory of groups arising from nilpotent algebras. In an ideal world, the goal of these investigations would be to develop something analogous to the Kirillov theory of coadjoint orbits and representations of nilpotent Lie groups, in circumstances where there is no exponential map. This problem seems very difficult; perhaps one day we will write a survey of the problems in this area.
Peter Symonds and I have studied the module structure of a group action on a polynomial ring. One would think that there would be nothing new to say about group actions on polynomial rings, but in fact we are trying to prove the following conjecture: Let G be a group, acting by linear transformations on a vector space V over a finite field. Extend this action to the polynomial ring k[V] and write this ring as a direct sum of indecomposable G-modules. Then only finitely many isomorphism classes of indecomposable modules appear as summands.