Algebraic K-Theory, Topological Hochschild Homology, and Equivariant Homotopy Theory

Michigan State University, East Lansing MI

Investigators

Abstract

The mathematical fields of algebra and topology are deeply intertwined. Indeed, tools from algebra can be used to study objects in topology, and vice versa. One illustration of this deep interaction is through algebraic K-theory. Algebraic K-theory is an invariant of rings, fundamental objects in algebra. There is great interest in algebraic K-theory due to its significant applications in the fields of algebraic geometry, number theory, and topology. While algebraic K-theory is difficult to compute, and many open questions remain, there is a powerful approach using tools from topology. In recent years, exciting advances in algebraic topology have made it possible to study questions in algebraic K-theory which were previously thought to be inaccessible. A goal of this project is to produce new algebraic K-theory computations. A key step in computing algebraic K-theory is studying a related invariant called topological Hochschild homology. Another goal of this project is to further develop the framework and theory around variants of topological Hochschild homology, and study applications to several other areas of mathematics. In addition to the mathematics research goals, the project also includes work in undergraduate and graduate education, undergraduate research, conference organization, and efforts to support the participation of women and other underrepresented groups in mathematics. This project uses the tools of equivariant stable homotopy to study algebraic K-theory and topological Hochschild homology. Algebraic K-theory is an invariant of a ring which is generally very difficult to compute. A fruitful approach to the study of algebraic K-theory is the trace method approach, which approximates algebraic K-theory by theories that are more computable, such as topological Hochschild homology and topological cyclic homology. The trace method approach relies on tools from equivariant stable homotopy theory. This project explores the intricate relationship between equivariant homotopy theory, algebraic K-theory, and topological Hochschild homology. Specific research goals of the project are organized into three broad objectives: One, use recent developments in trace methods and equivariant stable homotopy theory to compute algebraic K-theory groups which were previously inaccessible. Two, use equivariant homotopy theory to study algebraic and topological Hochschild homologies such as twisted topological Hochschild homology and Real topological Hochschild homology. Three, study applications of topological Hochschild homology theories to questions in geometry and low-dimensional topology. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.