Motivic Homotopy Theory, Group Actions, and K-theory

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This project studies motivic homotopy theory, which is a relatively new tool which allows us to utilize methods from algebraic topology to understand the objects of interest in algebraic geometry, namely algebraic varieties - sets of solutions of polynomial equations. The theory has had several spectacular successes in resolving open problems. In this project, the PI plans to develop analogous tools which are well suited for studying not just algebraic varieties, but also their symmetries. The PI, together with collaborators, will study structural aspects of motivic homotopy theory. This includes a study of the equivariant motivic sphere spectrum and the algebraic structure of its homotopy sheaves. They will develop an equivariant motivic slice spectral sequence, focusing on examples of interest such as Hermitian K-theory. They will continue their study of tensor triangular geometry of motivic homotopy and its implications for chromatic motivic homotopy theory. Lastly they will study aspects of homotopy algebraic K-theory of ring spectra.