**Date & Time:** Monday, January 11, 2010, 17:00-18:00

**Venue:** Ramanujan Hall

**Title:** Equivariant Infinite Loop Spaces

**Speaker:** Rekha Santhanam, Johns Hopkins University

**Abstract:** Generalized equivariant cohomology theories such as equivariant K-theory and equivariant singular cohomology are represented by equivariant spectra. \textit{Connective} equivariant spectra have several space level models in the case when the group acting is a finite.

We describe two such models, equivariant $\Gamma$-spaces defined by Shimakawa and equivariant $E_\infty$-spaces defined by Constenoble, May and Waner. We show that they are Quillen equivalent and therefore, both the categories have the same homotopy theory.

In the non-equivariant case, the twistings of a cohomology theory, for example Twisted K-theory (which finds applications in String theory) are classified by the units of the ring spectrum representing the cohomology theory. Motivated by the notion of twisted equivariant K-theory, we define the group of \textit{units } of equivariant ring spectra in terms of equivariant $\Gamma$-spaces.

We then give a generalization of Segal's construction which gives a method to write down examples of equivariant $\Gamma$-spaces starting with symmetric monoidal categories enriched with the group action.