NSF Award
2301520
Algebraic topology is the study of topological objects, such as topological spaces, through algebraic invariants that do not change when spaces are continuously deformed. Stable homotopy theory, a central part of algebraic topology, expands from the study of spaces to that of spectra, which are "stabilized spaces" that also represent generalized cohomology theories, or ways to assign algebraic invariants to spaces or other spectra. Equivariant stable homotopy theory further adds to spectra the actions of groups, which can be thought of as ways to map a spectrum to itself in composable and invertible ways. Equivariant stable homotopy theory has grown to be an important tool that offers insights into many deep questions in algebraic topology. The techniques of equivariant stable homotopy theory have also found applications in other areas of mathematics, including algebraic geometry and number theory. The broader impact aspect of the project includes mentoring of graduate and undergraduate students in mathematical research. The principal investigator (PI) will also continue outreach efforts by working to make her research area accessible to the public.<br/><br/>This project includes a circle of ideas in equivariant stable homotopy theory. The PI will continue her ongoing work on equivariant complex cobordism spectra, in particular the extension of her previous calculation of the coefficients of such spectra for primary p-groups to more general groups. This has important implications to the study of equivariant formal group laws, another part of the project that the PI will pursue. The PI will also investigate applications to her recent calculation, together with her collaborators, of the equivariant Mackey Steenrod algebra for odd primes. Specifically, one such application the PI is pursing is the construction of odd-primary versions of the Real Brow-Peterson spectrum. A closely related question is the construction and understanding of equivariant elliptic and Barsotti-Tate cohomologies, as well as the formal group laws associated with these spectra. The PI will also continue her ongoing project, along with her collaborators, in the calculation of self-conjugate and double-real cobordism spectra.<br/><br/>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.
