Algebraic K-theory and Equivariant Homotopy Theory

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The theme of this project is to use the tools of equivariant stable homotopy theory to study algebraic K-theory, particularly the K-theory of singular and filtered rings. Although the definition of algebraic K-theory is not inherently equivariant, the tools of equivariant stable homotopy theory have proven useful for K-theory computations. In particular, one fruitful approach exploits the equivariant structure of topological Hochschild homology (THH) to compute algebraic K-theory. In the case of certain singular rings, this approach reduces the computation of K-theory to the computation of equivariant stable homotopy groups of THH, graded by the real representation ring of the circle. To compute K-theory one needs to determine which equivariant homotopy groups arise, compute these groups, and then assemble them to recover K-theory. Each of these steps is difficult and understood only in a small number of cases. This project seeks to address these issues for various specific K-theory computations, as well as defining an abstract algebraic object embodying structures that arise in these computations. Other specific goals of the project include developing an approach for the K-theory of filtered rings, and answering several questions about the structure of THH. Algebraic K-theory is an invariant which can be applied to study basic objects from several fields of mathematics. In particular, algebraic K-theory can be used to study properties of fundamental objects in algebra, called rings. Although higher algebraic K-theory was defined more than 30 years ago, computational progress has been slow. Indeed, even for some very basic rings, the algebraic K-theory is still not known. K-theory computations, however, have important applications to many areas of mathematics: algebraic number theory, classification of manifolds, motivic homotopy theory, special values of L-functions, etc. An approach to these important computations lies in the field of algebraic topology, and more specifically, in the study of equivariant homotopy theory. The goal of this project is to use these tools to not only produce new algebraic K-theory computations, but also to develop the framework and theory to facilitate future computations.