Equivariant Stable Stems

Grants and Contracts Details

Description

The PI proposes several projects in R-motivic and C2-equivariant stable homotopy theory. The R-motivic projects are a useful stepping stone towards C2-equivariant com- putations. Intellectual Merit The C2-equivariant stable homotopy groups were determined out to the 13-stem in the early 1980's. Dan Dugger, Dan Isaksen, and the PI have pioneered a new approach to C2-equivariant homotopy theory, passing rst through C-motivic homotopy theory and next through R-motivic homotopy theory on the way towards C2-equivariant calculations. This has already paid o, providing new insight into previous equivariant calculations. The PI proposes to continue this program; the new, motivic techniques should allow the PI and Isaksen to advance the calculation of C2-equivariant stable homotopy groups much further than the current knowledge base. This is expected to also have important consequences in classical, nonequivariant computations. For instance, this should lead to better understanding of the Mahowald root invariant in stable homotopy groups. In another, closely related project with Bhattacharya and Li, the PI proposes to con- struct explicit nite complexes in the R-motivic and C2-equivariant stable homotopy categories that support periodicity operators. In general, producing such nite complexes with periodicity operators leads to periodic families of elements in the stable homotopy groups of spheres. A third project, with Bonventre and Stapleton, investigates power operations in equi- variant multiplicative cohomology theories. In particular, the PI and his collaborators propose to determine a minimal quotient for which the power operation becomes additive in the equivariant sense. Their project will include examples of interest from the point of view of transchromatic stable homotopy theory. Broader Impact The PI is currently advising two Ph.D. students who are working on research closely connected with this proposed work. The PI plans to take on more Ph.D. students in the near future. The PI has mentored two postdocs over the last ve years and plans to recruit another during the period covered by this proposal. The PI plans to work with undergraduate students through the UK Geometry Lab on a project centered around the Steenrod Algebra. The PI has made a habit of posting detailed course notes on his website. The PI regularly hears from faculty and graduate students at other institutions that these course notes have helped them personally. The PI plans to continue to post course notes as he teaches new advanced topics courses. For example, the PI plans to teach a topics course on computations in equivariant stable homotopy theory in the 2020- 2021 academic year. The PI is developing a website with Niles Johnson that will display the actions of the R-motivic and C2-equivariant Steenrod algebras on certain standard nite modules.
StatusActive
Effective start/end date8/1/207/31/24

Funding

  • National Science Foundation: $221,203.00

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