Computations in Classical Chromatic Homotopy Theory, Algebraic K-Theory, and Motivic Homotopy

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project seeks to computationally approach the Hopkins-Miller higher real K-theory spectra, the algebraic K-theory of structured ring spectra, and motivic homotopy. In all three cases, the computations themselves are strongly related: there is a subtle interplay between the algebraic action of a finite group (be they automorphisms of a fixed formal group, the circle action on topological Hochschild homology, or the action of a Galois group) and the geometry of Thom spectra over the classifying spaces. In particular, our goals are threefold: (1) find a procedure to systematically determine the homotopy ring of the Hopkins-Miller higher real K-theory spectra EO_n(G) (this is joint with Hopkins and Ravenel), (2) better standing the Bokstedt-Hsiang-Madsen TR and TC machinery and the algebraic K-theory of basic chromatic spectra, and (3) use the standard techniques of algebraic topology to provide foundational computations in motivic homotopy.The goal of algebraic topology is to systematically build a connection between algebraic objects like numbers and topological objects like spaces. These connections are self-reinforcing: problems in algebra become problems in topology which are further refined into algebra, and much of modern algebraic topology relies heavily on the ways spaces themselves can be described more algebraically. This project exploits this connection in multiple ways. First, in trying to understand how to build spaces out of spheres, one encounters the problem of computing a large family of invariants: the homotopy groups of spheres. This has been a very active part of algebraic topology since the 1930s, and the first part of the project is to compute other, related rings which act as increasingly good approximations. Second, the recent developments allow a two-way interchange between classical questions about rings and structured spectra. In particular, this project seeks to better understand the algebraic objects using more geometrical constructions.