Algebraic K-theory, trace methods, and equivariant homotopy theory

There are deep connections between the mathematical fields of algebra and topology. One illustration of these connections is through algebraic K-theory, which uses topological methods to study invariants of fundamental objects in algebra. This approach has garnered wide interest, since it has important applications to a variety of mathematical fields including algebraic geometry, number theory, and geometric topology.

In recent years, exciting advances have made it possible to study questions in algebraic K-theory which were previously thought to be inaccessible. This project applies these developments to produce new computations of algebraic K-theory, leading to a better understanding of the tools themselves and giving further insights on how algebraic and topological invariants are related.

This project will integrate this research with activities around undergraduate and graduate education, undergraduate research, conference organization, and efforts to support the participation of women and other underrepresented groups in mathematics.

This project focuses on the study of algebraic K-theory via the trace method approach, using tools from equivariant homotopy theory to advance our understanding of algebraic K-theory and related invariants such as topological Hochschild homology. The trace method approach, which approximates algebraic K-theory by more computable invariants such as topological Hochschild homology and topological cyclic homology, has led to major advances in our understanding of algebraic K-theory.

This approach relies on tools from equivariant homotopy theory. In recent years, advances in both equivariant homotopy theory and the foundations of trace methods have sparked tremendous progress in this area. This project explores the intricate relationship between equivariant homotopy theory, algebraic K-theory, and trace methods.

Specific research goals of the project include: 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 variants of topological Hochschild homology and related equivariant algebraic structures. Three, develop new applications of trace methods to geometry and topology through interactions with scissors congruence and invariants in geometric 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.