Homotopy Theory and Ring Spectra

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The goal of this project is to study phenomena relating strictly commutative ring spectra to the theory of formal groups. One part of the project is to elucidation of the structure of the algebra of power operations for Morava E-theory; and in particular a proof of the conjecture that this power-operation algebra is Koszul. Another part of the project is the study of E-infinity orientations of commutative ring spectra, extending the work of Ando, Hopkins, and the PI on the string orientation of topological modular forms. 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". It is a surprising fact that cohomology theories are illuminated by the theory of formal groups, which in turn are closely related to problems in algebraic number theory. The goal of this project is to understand this relationship.