In this talk, I will say a little about a cohomology theory for Z/2-spaces graded on the ring of virtual real representations of Z/2. Examples computations will include spheres and real projective spaces. I will also introduce an RO(Z/2)-graded version of the Leray-Serre spectral sequence, which contains a surprising amount of information about these equivariant spaces. If time permits, I will speak some about my attempts to further a theory of equivariant characteristic classes for equivariant vector bundles. |