Local and Global Chromatic Stable Homotopy Theory

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. In essence, the basic program is to gather local information – the data that can been seen from formal groups of a single height – 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 derived algebraic geometry; these allow us to interpolate among heights. This proposal focuses on four 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; that is, what we can see at height 2. A great deal of information has been collected by Shimomura and his collaborators, but there is a need for a different organizing principle to bring order out of this wealth of facts. This long-standing project, with Hans-Werner Henn and others, is now nearing completion. We are seeing beautiful and interesting v2-periodic phenomena. A second, closely related project, is to investigate the fixed point spectra of Morava E-theory for certain closed subgroups of the Morava stabilizer group. These are much simpler than the sphere itself, but capture a great deal of the important homotopy theory. The other two projects are more global in nature. One 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. The point here is to do systematic investigation of the Gl2-equivariant structure. The other project is to view the Chromatic Splitting Conjecture through the lens of p-divisible groups. The key observation is that localizations of the spectra of complex oriented cohomology theories transform formal groups into p-divisible groups. Intellectual merit: While the focus of this proposal is in stable homotopy theory, there is an interplay of ideas across algebraic topology and algebraic geometry – the cohomology of profinite groups, the geometry of formal groups, the theory of elliptic curves, and the theory of p-divisible groups all come into play. Broader impact: The PI has a strong record of graduating doctoral students, including women, for both postdoctoral and teaching positions at the university level. He works successfully with undergraduate students who go on to graduate school. He currently has 5 active doctoral students, and 2 who graduated in the past 3 years. He is Director of Graduate Studies for Mathematics at Northwestern University, a program with about fifty students. He runs a weekly seminar at Northwestern and regularly co-organizes conferences. Both the seminar and conferences have a tradition of support for graduate students and other junior research mathematicians.