Collaborative Research: Algebraic K-Theory, Arithmetic, and Equivariant Stable Homotopy Theory

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Algebraic topology began as the study of those algebraic invariants of geometric objects that are preserved under certain smooth deformations. Gradually, it was realized that the algebraic invariants, called cohomology theories, could themselves be represented by geometric objects known as spectra. A central triumph of modern homotopy theory has been the construction of categories of ring spectra that are suitable for performing constructions analogous to those of classical algebra. This has proved fruitful by providing invariants, which shed new light on old questions. In addition, it has raised new questions that have unexpected connections to other areas of mathematics and physics. This project works in the setting of an invariant called algebraic K-theory and related theories. The project studies applications of these theories to a broad range of questions in number theory, algebraic geometry, and geometric topology, as well as algebraic topology itself. The award provides support for students who will be engaged in parts of this research.This grant funds a broad research program aimed at applying recent work of the PIs on algebraic K-theory, trace methods, and equivariant stable homotopy theory to study a wide variety of problems in homotopy theory. Prior work of the PIs studied the algebraic K-theory of the sphere spectrum and the fiber of the cyclotomic trace for algebraic number rings. The current project expands the investigation to the fiber of the cyclotomic trace on more general rings over algebraic p-integers in terms of Tate-Poitou duality and a related K-theory question more generally for other kinds of Artin duality. The project explores a connection between the geometric Soule embedding and the Kummer-Vandiver conjecture discovered in the PIs' prior work. The PIs' prior work also gives a splitting that is consistent with and gives evidence for the existence of p-adically interpolated Adams operations on the algebraic K-theory at least in the context of regular rings. The project investigates specific conjectures and approaches to this problem. The project advances a new approach to the the Hatcher-Waldhausen map that would have implications in geometric, differential, and symplectic topology. The project proposes a construction of multiplicative norm maps in equivariant stable homotopy theory for positive dimensional compact Lie groups. This requires a new foundation for non-equivariant factorization homology and holds the promise of constructing a genuine equivariant factorization homology theory for positive dimensional compact Lie groups. The project includes a collaboration of the PIs with Basterra, Hill, and Lawson to study equivariant TAQ as part of a broader program to develop the foundations for equivariant derived algebraic geometry. If successful, this program will provide an organizing principle for phenomenological data coming from work on topological modular forms.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.