Transchromatic homotopy theory

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Topological spaces are some of the most fundamental objects in mathematics. Algebraic topology associates algebraic objects to topological spaces in order to tell them apart. Some of the most powerful algebraic tools for doing this are "cohomology theories". A certain sequence of cohomology theories called the Morava E-theories have a deep and somewhat understood connection to algebraic geometry and an important conjectural relationship with geometry. The first two Morava E-theories are classical; however, the rest are quite mysterious. The goal of this project is to use the connection to algebraic geometry to play different Morava E-theories off each other in order to expose properties of the conjectural geometry. The PI plans to study the relationship between the chromatic layers in order to expose the geometry that lurks behind the scenes in chromatic homotopy theory. He will do this by pursuing three interrelated programs. The PI will use the algebraic geometry of p-divisible groups, generalized character theory, and field theories (in the sense of Stolz-Teichner) to attack the problem. In joint work with Schommer-Pries, the PI hopes to give a field theoretic construction of higher chromatic cohomology theories such as Morava E-theory. Also of particular interest are "transchromatic" proofs of classical theorems that lead to more general results. For instance, in work with Tomer Schlank, the PI has given a new proof of Strickland's theorem on the Morava E-theory of symmetric groups that naturally leads to a generalization of the theorem to wreath products of finite abelian groups with symmetric groups. One of the main tools in the proof is a character map from E-theory to p-adic K-theory that allows one to reduce certain problems in E-theory to representation theory. This map deserves further study.