The project supported by this award is part of a large scale effort to understand the homotopy groups of spheres, one of the fundamental problems in topology. The spheres, which are among the simplest geometric objects, are building blocks for more complex entities. The homotopy groups of spheres are collections of continuous functions between spheres considered up to deformations. They are used to study the connections between these basic components. These groups and the tools used to study them have deep connections with other fields. For example, there are connections with number theory as interesting functions and sequences appear as patterns and structure in the homotopy groups of spheres. There are connections with differential geometry, as these groups are used in the classification of geometric objects. Recently, the tools used to study the homotopy groups groups of spheres have also been used in physics in the classification of phases of matter. Although the study of the homotopy groups of spheres is notoriously difficult, a bridge has been built between algebraic geometry and algebraic topology that allows us to use sophisticated algebraic theory to enable calculations of the homotopy groups of spheres. This bridge is known as chromatic homotopy theory and is the central theme of this project. Duality and invertibility are ubiquitous concepts in mathematics which are central to understanding relationships between mathematical objects. The project supported by this award studies these phenomena within chromatic homotopy theory. The project also proposes theory and computations that will provide data to help study these fundamental phenomena.
At the heart of chromatic homotopy theory is the study of a higher analogue of the homotopy category of spectra local with respect to mod p K-theory: The K(n)-local stable homotopy category. It is equivalent to the homotopy category of K(n)-local E-modules in G-equivariant spectra, where Morava E-theory E is a higher analogue of p-complete K theory and the stabilizer group G is a generalization of the group of Adams operations. The strategy of the proposal is to restrict problems to finite subgroups F of G and use the theory of finite resolutions to pass from information obtained from finite subgroups to information about the K(n)-local category. An important circle of ideas in the proposal concerns duality and invertibility. The PI and collaborators study K(n)-local Spanier-Whitehead duality using techniques from equivariant homotopy theory. They use an analogue of the J-homomorphism to compute the duals of homotopy fixed point spectra for the action on E of some finite subgroups F of G. This homomorphism is also used to study Picard groups of categories of K(n)-local E-module F-spectra. The PI is an expert in computations at chromatic height 2 for p = 2 and proposes projects to further our understanding in this difficult case. These include the computation of the Picard group of the K(2)-local category, the study of K(2)-local Brown-Comenetz duality and computations of K(2)-local homotopy groups for finite spectra. Advancements in chromatic homotopy theory at height 2 have informed our understanding of K(n)-local phenomena. The PI proposes various projects to generalize this insight at higher heights. Finally, the project also examines one of the most important problems in chromatic homotopy theory, the chromatic splitting conjecture, which plays a central role in the relationship between chromatic homotopy theory and the algebraic geometry of formal groups.
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.
