Rational and equivariant phenomena in chromatic homotopy theory

Many of the most productive themes in modern mathematics sit at the intersection of several different fields. This project involves problems in between algebraic topology, number theory, and group theory with some applications to differential geometry and input from algebraic geometry. One of the primary goals of this project is to solve a problem suggested by the work of Morava in the 1970s. A solution to this problem will be an important step toward understanding how spheres of different dimensions can wrap around each other -- a problem that is central to modern algebraic topology. The research will be integrated with the PI's educational efforts at the undergraduate and graduate level. The goal of this project is to make use of recently developed tools to attack several open problems in chromatic homotopy theory. The PI will work with collaborators to show that the rationalization of the monochromatic layers of the sphere spectrum are exterior algebras over the p-adic rationals on certain generators. This problem dates back to the 1970s and is one of the central open problems in chromatic homotopy theory. The PI will also address a question concerning the kernel of the canonical map from the Burnside ring of a finite group to its monochromatic cohomotopy. This question is bound up in the theory of power operations and the theory of fusion systems. Together with graduate students, the PI will develop tools to help produce a universal exponential relationship between multiplicative and additive power operations. Finally, the PI will work with collaborators to build on previous work and advance understanding of the multiplicative properties of global equivariant complexified elliptic genera. This project is jointly funded by the Topology program, and the Established Program to Stimulate Competitive Research (EPSCoR). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.