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.
This is a project in algebraic topology and homotopy theory. Homotopy theory is the branch of mathematics which studies phenomena which remain invariant continuous deformation, while algebraic topology imports tools from other areas of mathematics to detect such phenomena. A very simple example is given by the continuous maps of a circle to itself: up to homotopy these are completely classified by the number of times the circle winds about itself. Defining and calculating this winding number is the business of algebraic topology. A much more profound, but very basic, problem is to calculate the homotopy classes of maps between higher dimensional spheres. Although we know a great deal, the complete answer is far beyond our reach. Current research is a mixture of new calculations a search for large scale patterns. Since the 1970s, one major line of research has used ideas from algebraic geometry to organize and drive this program; this point of view is called chromatic stable homotopy theory, as it dictates that we first calculate monochromatic pieces, which might be called the contributions of a single color. The main outcome for this award was a series of new calculations at chromatic level 2, which is the edge of our current knowledge. For example, in one paper, the PI and his collaborators verified the Chromatic Splitting Conjecture at this level, this demonstrating that expected phenomena did indeed occur, but in another paper, the PI and other collaborators constructed exotic elements in the level 2 Picard group, thereby quantifying some very unexpected behavior. Beyond these specific accomplishments, we have gotten much better at working at level 2 and what was once regarded as impossibly complex has been reduced to accessible order. This is the intellectual merit of the research supported by this grant. The broader impact of this grant was in graduate education. The grant partially supported six graduates, three of which graduated and have assumed positions in academics. The other three have made significant progress towards their doctorates. It should be emphasized that each student has his or her own project, and not all are working in chromatic stable homotopy theory. This diversity of intellectual interests and pursuits helps keep the field fresh and vital.
