Generalized Steenrod operations and equivariant geometry

Topology is a subject that studies shapes. This subject was revolutionized in 1947 by Norman Steenrod when he introduced the Steenrod operations. These operations led to some of the most enigmatic results in geometry which popularized homotopy theory, a subbranch of topology, in the fifties and the sixties. This project will lead to advances in equivariant homotopy theory, an enhancement of homotopy theory which is sensitive to the symmetries of shapes. The resulting equivariant analogs of Steenrod operations will lead to new applications in equivariant geometry. The award will also be used to stimulate the research culture and support graduate students at New Mexico State University. The PI will establish a new method that generalizes the construction of classical Steenrod operations to equivariant homotopy theory and constructs equivariant Steenrod operations for all finite groups. Equivariant Steenrod operations will immediately lead to equivariant analogs of Stiefel-Whitney classes which can find many applications in the study of equivariant vector bundles and equivariant smooth manifolds. Atiyah Real vector bundles, which are important examples of equivariant vector bundles, will be studied. In particular, the James periodicity number of tautological Atiyah Real vector bundles will be determined using an equivariant analog of the Adams conjecture formulated using Atiyah Real K-theory. The PI will also continue to study periodic self-maps of chromatic homotopy theory and extend them to equivariant as well as motivic settings. This project is jointly funded by the Topology program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.