Abstract. Modular representation theory studies actions of finite groups (Lie algebras, algebraic groups, finite group schemes) on vector spaces over a field of positive characteristic. The simplest example is an action of the cyclic group Z/p on a vector space. Such an action is described by a single matrix which, in turn, is classified by its Jordan canonical form.

I shall describe an approach to the study of modular representations via their restrictions to certain elementary subalgebras which are analogs of one-parameter subgroups. As an application, we can recover the algebraic variety associated to the cohomology ring of a finite group scheme $G$ by purely representation-theoretic means, generalizing Quillen's stratification theorem" for group cohomology. As another application, we construct new numerical invariants of representations. These invariants are expressed in terms of Jordan forms.

Most of our results apply to any finite group scheme, but they are non-trivial even in the case of the finite group Z/p x Z/p, which is a baby example that will be used for illustrative purposes throughout the talk.