Homotopy Theory and Ring Spectra

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The goal of this project is to study phenomena relating strictlycommutative ring spectra to the theory of formal groups. Onepart of the project is to elucidation of the structure of thealgebra of power operations for Morava E-theory; and in particular aproof of the conjecture that this power-operation algebra is Koszul.Another part of the project is the study of E-infinityorientations of commutative ring spectra, extending the work of Ando,Hopkins, and the PI on the string orientation of topological modularforms. Homotopy theory is a branch of topology; it arose as the study ofcertain invariant properties of spaces, namely those left unchanged bycontinuous deformations. The most powerful tools for studying suchproperties are what are called "cohomology theories". It is asurprising fact that cohomology theories are illuminated by the theoryof formal groups, which in turn are closely related to problems inalgebraic number theory. The goal of this project is to understandthis relationship.