## 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.

Status | Active |
---|---|

Effective start/end date | 8/1/20 → 7/31/23 |

### Funding

- National Science Foundation: $221,203.00

## Fingerprint

Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.