Motivic Symmetries

Over the past century, algebraic topology has developed many sophisticated methods and tools to study geometric "shapes". Motivic homotopy theory, provides a framework to apply tools from algebraic topology, to the study of algebraic varieties and to important algebro-geometric invariants such as vector bundles, quadratic forms, algebraic cycles, and rational points. This project will study structural and computational aspects of homotopy theory for algebraic varieties with an emphasis on geometric applications. Broader impacts of this project include work with incarcerated individuals, as well as work with undergraduate and graduate students. In this project, the PI will combine equivariant and higher categorical techniques to study ramifications of recent developments within motivic homotopy theory. In the first part of the project, the PI proposes to further develop tools to study normed motivic spectra, with a focus on orientations. In the second part, the PI proposes to focus on computations of equivariant motivic invariants and slice spectral sequences. In the third the PI proposes to develop a theory of equivariant framed correspondences and apply this to the study of equivariant motivic loop spaces. 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.