Algebraic invariants of structured ring spectra, arithmetic, and geometry

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This proposal describes several major research thrusts related to algebraic invariants of structured ring spectra. In algebraic K-theory, the PI describes an approach to resolving the central conjectures of Waldhausen and Rognes that underpin a program to describe the K-theory of the sphere spectrum via a chromatic filtration, relating arithmetic and manifold geometry. In topological Hochschild homology, the proposal describes an approach to realize a vision of Hesselholt to build an ``additive'' motivic spectral sequence in which the deRham-Witt complex and TR-theory play the respective roles of Milnor K-theory and algebraic K-theory. In string topology, the proposal describes an project which utilizes some of the technology developed to study algebraic K-theory to describe algebraic invariants of ``brane'' categories; this is related to the burgeoning connections between string topology and symplectic topology. In orientation theory, the proposal describes a series of projects associated to the relationship between units of ring spectra, orientations, and transfers, particularly in the equivariant setting.These latter projects have potential applications to mathematical physics.Algebraic topology began as the study of algebraic invariants of geometric objects which are preserved under certain smooth deformations. Gradually, it was realized that these 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 cohomologytheories) 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. The project funded by this grant carries out this program in the setting of a rich invariant called algebraic K-theory, studying foundational properties of this theory from the perspective of homotopy theory and applying the results to a broad range of questions in manifold geometry, algebraic geometry, and string topology.