Structured Ring Spectra and Chromatic Stable Homotopy Theory

Northwestern University, Evanston IL

Investigators

Abstract

DMS-0204129 Paul G. Goerss Homotopy theory is the study of topological phenomena that remain invariant under fairly flexible sorts of continuous deformations. Most basically, we seek to compute and to understand homotopy classes of maps between finite simplicial or CW complexes, perhaps after some sort of stabilization. Computation and understanding are not the same: we have forty or more years of computations, but only lately have we begun to have the language to in which to make a coherent theory out of the data we have. This project is, therefore, as much about exploring and refining the language as it is about calculations -- although there will be a mixture of both. The entry point is the observation that any cohomology theory with a good theory of Chern classes gives rise to a formal group law and, hence, a formal group. Conversely, given a formal group we can ask whether there is an associated cohomology theory. These cohomology theories then delineate the finer structure of stable homotopy theory. This is the chromatic picture of stable homotopy: we seek to use the algebraic geometry of formal groups to organize and direct investigations into the deeper structure of computations and theory. This project seeks to develop this point of view in two directions, one local and one global. The first, or local, direction is an investigation into K(n)-local homotopy theory in general and into the K(n)-local sphere in particular. The second, or more global, direction, would be to make systematic our knowledge of structured ring spectra using stacks and the moduli stack of formal groups as the basic parametrizing device.