Projects and Grants per year
Grants and Contracts Details
Description
Project Summary
Motivic to C2equivariant homotopy and beyond
Overview
The PI proposes several projects in Rmotivic stable homotopy theory and equivariant
stable homotopy theory, for various pgroups. Several of the Rmotivic projects have
direct consequences in C2equivariant homotopy theory.
Intellectual Merit
The computation of the stable homotopy groups of spheres is a central problem in
homotopy theory. This problem is now often viewed through the lens of chromatic homo
topy theory, where the homotopy groups are filtered in terms of levels of periodicity. This
chromatic filtration on the stable homotopy groups has become a powerful organizational
principle. Meanwhile, in the last two decades, the field of motivic stable homotopy the
ory emerged and saw rapid development, while equivariant stable homotopy theory saw
an explosion of interest after the HillHopkinsRavenel solution of the Kervaire invariant
problem.
The PI proposes work linking these areas of stable homotopy theory. One of the central
themes is that motivic homotopy theory over the real numbers R gives an efficient stepping
stone towards C2equivariant homotopy theory. The PI proposes to use recent advances in
Rmotivic homotopy theory to compute C2equivariant stable homotopy groups in a range
larger than what had been previously computed. These computations then lead to greater
understanding of nonequivariant stable homotopy groups, for example via Mahowald’s
root invariant.
Extending beyond C2equivariant calculations, the PI has previous work in equivariant
stable homotopy theory for the groups K4 and Q8. The PI proposes to use equivariant mo
tivic homotopy theory as well as the equivariant slice filtration to approach computations
of those equivariant stable homotopy groups.
In previous work with Bhattacharya and Li, the PI began to develop the theory of v1
selfmaps in Rmotivic and C2equivariant homotopy theory. This is the beginning of the
chromatic filtration in these homotopy theories. The PI proposes to work out an analogue
of the Telescope Conjecture in these areas, extending work of Culver and Quigley in the
Cmotivic case.
Broader Impacts
The PI will participate in various mentoring activities in the electronic Computational
Homotopy Theory community, including being an official mentor for two eCHT NSF
funded postdocs and serving as an administrator for reading courses, graduate student
seminars, online courses, and more. The PI is developing a website with Niles Johnson
that will display the actions of the Rmotivic and C2equivariant Steenrod algebras on
certain standard finite modules.
Status  Active 

Effective start/end date  8/15/24 → 7/31/27 
Funding
 National Science Foundation
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.
Projects
 1 Active

Motivic to C2Equivariant Homotopy and Beyond (ALN 47.049)
8/15/24 → 7/31/27
Project: Research project