The goal of this proposal is to study the relationship between homotopy theory and formal groups, and applications of homotopy theory to category theory. The PI plans to study power operations for Morava E-theory, relating the algebra of such operations to the Bousfield-Kuhn functor, a construction which picks out invariants of a single chromatic layer of unstable homotopy theory. The PI proposes to prove a a series of conjectures, which would lead to some new computational techniques, and would also draw interesting connections between unstable homotopy theory and the algebra of cohomology operations in Morava E-theory. In addition, the PI will study certain models for higher category theory based on the ideas of homotopy theory. Higher categories have recently been applied (through work of Hopkins, Lurie and others) to the classification of topological field theories. The models the PI will study (called Theta-n spaces), are conjectured to be presentations for the theory of (infinity,n)-categories; a goal of this project is to prove this conjecture, and to extend these results to the setting of enriched higher category theory.
Homotopy theory is a branch of topology; it arose as the study of certain invariant properties of spaces, namely those left unchanged by continuous deformations. The most powerful tools for studying such properties are what are called "cohomology theories". Cohomology theories are illuminated by the theory of formal groups, which in turn are closely related to problems in algebraic number theory. The first goal of this project is to understand this relationship, with the prospect of creating new computational tools. A second goal is to understand how to use homotopy theory to shed light on higher category theory.
