Formal Groups, Structured Ring Spectra, and Stable Homotopy Theory

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AbstractAward: DMS-0706705Principal Investigator: Paul G. GoerssThe chromatic picture of stable homotopy uses the algebraicgeometry of formal groups to organize and direct investigationsinto the deeper structure of computations and theory. Thisproject seeks to develop this point of view in two directions,one local and one global. The first, or local, direction is aninvestigation into K(n)-local homotopy theory in general and intothe K(2)-local sphere in particular. The second, or more global,direction, would be to make systematic our knowledge ofstructured ring spectra using stacks and the moduli stack offormal groups as the basic parameterizing device. In particular,a main part of the project is to continue work on the problem ofrealizing families of commutative ring spectra over the modulistack of formal groups. The spectrum of topological modular formsarises from taking the homotopy inverse limit of just such afamily and recent work of Lurie, Behrens, and Lawson had givennew examples. We can ask for systematic results along theselines, and we can ask for a thorough investigation into theexamples we have. An intriguing and novel feature of thesefamilies is that they use the theory of Barsotti-Tate groups tocombine information from formal groups of various heights.This project is in homotopy theory, which is a branch oftopology, a rather modern field that grew naturally out ofgeometry by studying phenomena that remain invariant undercontinuous transformations, rather than rigid (e.g.,angle-preserving) transformations. Of particular importance intopology are the continuous maps between large dimensionalspheres; under a suitable equivalence relation, this is the ringof stable homotopy groups of spheres. This notorious difficult tocalculate, or even to make conjectures about; therefore, in thepast few decades we have focused on trying to understandlarge-scale qualitative phenomena. In summary, this is the mainthrust of this project as well. It has been very fruitful todetect these phenomena using tools from other fields, especiallyalgebraic geometry. The transition from topology to geometry isdone using homology theories, which is a way of linearizingbehavior in topology. Simply sticking to one such theory is aradical process, however, and it loses too much data; therefore,we study families of such theories. Of particular importance isthe family parametrized by the stack of one-parameter formal Liegroups. The theory of stacks is vital here, as this allows us tostudy symmetries across continuous families of geometric objects-- especially when the self-symmetries can vary non-continuouslythroughout the family, as is most certainly the case here.