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 number theory. The aim of this project is to understand aspects of this relationship, with the prospect of creating new computational tools in homotopy theory.
This project concerns the theory of power operations in equivariant cohomology theories. Several interrelated projects are proposed, which aim to draw connections between the theory of ultracommutative ring spectra, which generalize commutative ring spectra to the equivariant setting, and the algebraic geometry of isogenies of elliptic curves and formal groups. These connections will advance understanding of topological invariants such as elliptic cohomology.
