Equivariant cohomology operations, calculus, and geometry

A major objective of algebraic topology is the classification of geometric shapes through computable algebraic invariants. Historically, the most significant breakthroughs in the field were driven by structured techniques called cohomology and homology operations. However, refining these tools so that they can detect inherent symmetries of a shape—which are ubiquitous throughout mathematics and modern physics—has been a decades-old computational bottleneck. Building directly upon the investigator's prior NSF support (DMS-2305016), which introduced the general framework of Eulerian sequences to resolve this barrier, this project will significantly expand this mathematical theory to develop new equivariant homotopy theoretic tools to study symmetries. By developing new algebraic tools in the field of equivariant homotopy theory to study the symmetries of geometric shapes, this project promotes the progress of science. Furthermore, the project advances the nation's STEM workforce by revitalizing the mathematical pipeline in the Southwest region through extensive high school outreach and regional math competitions designed to nurture the next generation scientists. The technical scope of this project advances the framework of Eulerian sequences to construct equivariant Araki-Kudo-Dyer-Lashof operations, acting on the homology of genuine equivariant infinite loop spaces. The investigator will rigorously study the algebraic structure of Eulerian sequences—such as their multiplication, comultiplication, and abstract Adem relations. The proposal will also develop computational methods to identify Eulerian sequences explicitly in both equivariant and non-equivariant settings, leading to new extraordinary homology and cohomology operations. Furthermore, to bridge the gap between stable approximations and unstable geometric realities, the project will establish equivariant analog of orthogonal calculus (Weiss calculus) and investigate unstable problems in equivariant geometry. 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.