**Title:** Equivariant algebraic K-theory

**Speaker:** Mona Merling

**Speaker Info:** University of Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

In the early 1980's, Dress and Kuku, and Fiedorowicz, Hauschild and May introduced space level equivariant versions of the plus and Q constructions in algebraic K-theory. However, their methods did not allow for nontrivial group action on the input ring or category. We generalize these definitions to the case in which a finite group G acts nontrivially on a ring (or an exact or Waldhausen category) and we construct a genuine equivariant K-theory spectrum with good properties from a G-ring. For instance, we recover equivariant topological K-theory, Atiyahâs Real K-theory and the statement of the Quillen-Lichtenbaum conjecture relating the K-theory spectrum of a field to the homotopy fixed points of the K-theory spectrum of any Galois extension.The equivariant constructions rely on finding categorical models for classifying spaces of equivariant bundles (a joint project with Guillou and May) and the use of equivariant infinite loop space machines such as the one developed by Guillou and May, or the equivariant version of Segal's machine, as generalized by Shimakawa. The comparison of these machines, which allows their interchangeable use in algebraic K-theory constructions, is a joint project with May and Osorno. New ideas are needed since, among other things, the comparison theorem of May and Thomason fails equivariantly.

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