Award: DMS 1308916, 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 structures of the field. The program supported by this grant is to gather local information - the data that can been seen from formal groups of a single height - and then 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. This long-standing project, with Hans-Werner Henn and others; we are seeing beautiful and unexpected phenomena at low primes. 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) elliptic curves with level structure; that is, structured versions of the Hopkins-Miller theory of topological modular forms. The point here is to make a systematic investigation of the equivariant structure. The other project is to look at the Chromatic Splitting Conjecture through the lens of p-divisible groups.
All of these projects lie in homotopy theory, a branch of topology. The main aim of this field is to study mathematical phenomena which remain invariant under continuous transformations. Many familiar geometric phenomena - such as angles - are not invariant in this fashion; yet continuous transformations are natural and abundant. Long study has indicated that the among the most fruitful invariant phenomena are classes of maps from circles or, more generally, higher dimensional spheres, into the space to be studied. These are the homotopy groups. Historically these groups were described as a "milling crowd"; however, the recent introduction of techniques and constructions from number theory and algebraic geometry have permitted us to do detailed calculations and to uncover large scale patterns of remarkable regularity and beauty.
