Computational structures in equivariant chromatic homotopy theory

Spheres are among the simplest geometric objects, serving as building blocks for more complicated topological spaces. The homotopy groups of spheres (collections of continuous functions between spheres, considered up to certain deformations groups) hold fundamental information about maps between topological spaces and have deep connections to number theory, algebraic geometry, differential topology, and geometric topology. Despite their ease of definition, there are few effective methods to compute homotopy groups of spheres. Using equivariant technology the PI will explore the rich connections between equivariant homotopy theory and chromatic homotopy theory and use them to develop powerful new computational techniques. This research will be integrated with conference organization, and graduate students and postdocs mentoring and training. The PI will also be engaged in outreach to local middle school teachers and students. The research involves a range of projects that will leverage recent discoveries in equivariant homotopy theory to advance computations in chromatic homotopy theory. The study of Lubin-Tate theories is one of the most important areas of research in chromatic homotopy theory. In 2009, Hill—Hopkins—Ravenel's resolution of the Kervaire invariant problem elevated equivariant homotopy theory as a potent tool to drive significant progress in chromatic homotopy theory and address classical problems in geometry and topology. The projects involve exploring computational structures in the equivariant slice spectral sequences of Real bordism theories and Lubin-Tate theories at the prime 2. This endeavor extends to achieving analogous results at odd primes. To achieve these goals, the PI plans to employ new equivariant techniques, including transchromatic isomorphisms, stratification results, and the generalized Tate diagram of spectral sequences. These methods will enable extensive equivariant chromatic computations, establish general differential patterns, and reveal a broader range of transchromatic phenomena in the equivariant slice spectral sequences of norms of Real bordism theories and Lubin-Tate theories across various groups and heights. 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.