In my joint work with Anssi Lahtinen, we have constructed a new system of (modular) characteristic classes in mod-p cohomology for representations (or, equivalently, vector bundles) over the finite field of order p^r. These vanish on decomposable representations. The construction of the classes is explicit and simple. We are able to evaluate them on some representations of elementary abelian p-groups, and thereby show that many of them are nonzero. This means that there are new families of nonzero classes in the modular cohomology of the general linear groups over finite fields, about which very little is known to date. In particular, the only previously known nonzero elements (except for small n) occur in degrees at least exponential in n; we produce nonzero classes in degrees linear in n.

For the talk, I will mostly restrict to the field of order 2, as the results there are simpler.