Collaborative Research: Algebraic K-theory and Equivariant Stable Homotopy Theory: Applications to Geometry and Arithmetic

Algebraic topology began as the study of those algebraic invariants of geometric objects which 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 (representing objects for multiplicative cohomology theories) which are suitable for performing constructions directly analogous to those of classical algebra. This move has turned out to be incredibly fruitful, both by providing invariants which shed new light on old questions as well as by raising new questions which have unexpected connections to other areas of mathematics and physics. This research proposal studies such constructions in algebraic K-theory, the newly emerging field of Floer homotopy theory, and the foundations of equivariant stable homotopy theory. Broader impacts include workforce development in the form of graduate student advising, undergraduate and postdoc mentorship, high school mathematical science project mentorship, conference organization, and development of new education and training programs.This proposal describes a broad research program to study a wide variety of problems in homotopy theory, geometry, and arithmetic. The PIs' prior work gives a complete description of the homotopy groups of algebraic K-theory of the sphere spectral at odd primes and a canonical identification of the fiber of the cyclotomic trace via a spectral lift of Tate-Poitou duality. In this proposal, the PIs describe a vast expansion of that argument to study the fiber of the cyclotomic trace on more general rings and schemes over algebraic p-integers in number fields and a related K-theory question more generally for other kinds of Artin duality. The prior work also leads to a new approach to the Kummer-Vandiver conjecture based on Bökstedt-Hsiang-Madsen's geometric Soule embedding that the PIs propose to study. The PIs' recent work with Yuan established a theory of topological cyclic homology (TC) relative to MU-algebras, where there are many interesting computations to explore. Prior work of PI Blumberg with Abouzaid building a Morava K-theory Floer homotopy type has opened new lines of research in Floer homotopy theory; this already has been used to resolve old conjectures in symplectic geometry. The PIs propose to extend this construction to its natural generality, building the homotopy type over MUP and KU and rigidifying the multiplication. They propose to obtain spectral models of deformed operations on quantum cohomology (i.e., quantum Steenrod operations). This work will have myriad applications in symplectic geometry and potentially have transformative impact on the Floer homotopy theory program. The PIs propose to resolve the longstanding confusion about the role of multiplicative norm maps in equivariant stable homotopy theory for positive dimensional compact Lie groups. This requires a new foundation for factorization homology and will lead to genuine equivariant factorization homology for positive dimensional compact Lie groups. In prior work, the PIs identified previously unknown multiplicative transfers on geometric fixed points of G-commutative ring spectra. The PIs propose to study how these fit into the foundations of G-symmetric monoidal categories; a first step is to establish new multiplicative splittings that have the feel of multiplicative versions of the tom Dieck splitting.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.