Formal Groups, Structured Ring Spectra, and Stable Homotopy Theory

Northwestern University, Evanston IL

Investigators

Abstract

Abstract Award: DMS-0706705 Principal Investigator: Paul G. Goerss The chromatic picture of stable homotopy uses 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(2)-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 parameterizing device. In particular, a main part of the project is to continue work on the problem of realizing families of commutative ring spectra over the moduli stack of formal groups. The spectrum of topological modular forms arises from taking the homotopy inverse limit of just such a family and recent work of Lurie, Behrens, and Lawson had given new examples. We can ask for systematic results along these lines, and we can ask for a thorough investigation into the examples we have. An intriguing and novel feature of these families is that they use the theory of Barsotti-Tate groups to combine information from formal groups of various heights. This project is in homotopy theory, which is a branch of topology, a rather modern field that grew naturally out of geometry by studying phenomena that remain invariant under continuous transformations, rather than rigid (e.g., angle-preserving) transformations. Of particular importance in topology are the continuous maps between large dimensional spheres; under a suitable equivalence relation, this is the ring of stable homotopy groups of spheres. This notorious difficult to calculate, or even to make conjectures about; therefore, in the past few decades we have focused on trying to understand large-scale qualitative phenomena. In summary, this is the main thrust of this project as well. It has been very fruitful to detect these phenomena using tools from other fields, especially algebraic geometry. The transition from topology to geometry is done using homology theories, which is a way of linearizing behavior in topology. Simply sticking to one such theory is a radical process, however, and it loses too much data; therefore, we study families of such theories. Of particular importance is the family parametrized by the stack of one-parameter formal Lie groups. The theory of stacks is vital here, as this allows us to study symmetries across continuous families of geometric objects -- especially when the self-symmetries can vary non-continuously throughout the family, as is most certainly the case here.