Chromatic Stable Homotopy Theory and Derived Algebraic Geometry

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The chromatic picture of stable homotopy uses the algebraic geometry of formal groups to organize and direct investigations into the deeper structures of the field. The basic program is to gather local information and then try to assemble that data into a more global picture. It is in the second step where we can use constructions and information from the emerging field of derived algebraic geometry. This proposal focuses on three projects, all growing out of this local-to-global mixture. The most computational is an investigation of the homotopy groups of the $K(2)$-local sphere; this is local by nature and we seek a complete calculation. The other two projects are more global. The first is to investigate the existence and non-existence of derived schemes (or stacks) with level structure; that is, structured versions of the Hopkins-Miller topological modular forms. Of interest here are the bad primes where interesting homotopy theory arises from supersingular curves. The other project here is a look at duality. A form of Serre-Grothendieck duality should hold in the derived setting, but it will be homotopy theoretic in nature, not simply algebraic geometry.This project is in homotopy theory, which is a branch of topology, a 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. 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.