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.
The primary goal of this project was to use the tools from a field of mathematics called equivariant stable homotopy theory to study algebraic K-theory. Algebraic K-theory groups are mathematical objects that are often very difficult to compute. Despite the fact that higher algebraic K-theory was defined over 40 years ago, even for some basic cases the K-theory groups are not yet computed. Such computations, however, are important to a variety of areas of mathematics including algebraic topology, algebraic geometry, number theory, geometric topology, etc. In this project the PI and her collaborators used new tools and techniques in equivariant stable homotopy theory to compute some algebraic K-theory groups that were previously not accessible. They also made advances studying other mathematical objects, such as topological Hochschild homology, which are closely related to algebraic K-theory. During the award period the PI also co-organized several workshops and conferences to bring experts together and disseminate knowledge in the field. These workshops included two summer schools for graduate students and a semester-long Algebraic Topology research program at the Mathematical Sciences Research Institute (MSRI). In addition to the purely intellectual merits of these programs, they also had key components (in the form of introductory workshops, panel discussions, programs aimed at young women and other underrepresented minorities, etc.) which were aimed at the education and mentoring of early-career researchers.
