Year 3 Travel Scope: New Tools in Chromatic Homotopy Theory

The goal of this project is to tackle two open problems in chromatic homotopy theory. The rst is with regard to the asymptotic behavior of chromatic homotopy theory and the second is a conjecture of Ravenel's. Intuition towards both of these problems comes from the study of character theory and power operations and, as part of this project, we will continue to develop these tools. With Barthel, we have shown that chromatic homotopy theory is asymptotically alge- braic. The rst goal of this project is to show that chromatic homotopy theory can be asymptotically approximated by algebra in characteristic p closely connected to the theory of formal Drinfeld modules. Due to the existence of Drinfeld elliptic modules at all heights, success in this endeavor should produce interesting higher height variants of topological modular forms. Ravenel conjectured that the kernel of the canonical map from the Burnside ring of a nite group to the K(n)-local cohomotopy of the group contains certain virtual G-sets. With Reeh, we will develop a theory of K(n)-local fusion systems. By understanding the relationship between these combinatorial objects and K(n)-local cohomotopy, we hope to settle Ravenel's conjecture. Both of these problems interact with a variety of other problems in chromatic homotopy theory, thus we expect that the tools developed to tackle these problems will be widely applicable. 1