Classical Methods in Motivic Homotopy Theory

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The subject of algebraic geometry concerns itself with the study of systems of polynomial equations. As it turns out, solving such systems exactly is, apart from a few very special cases, essentially impossible. We thus attempt to describe the solution sets more qualitatively. One classically fruitful approach to this is to consider the set of solutions as a topological space, an object sitting in some higher dimensional space. For instance, the equation x^2+y^2=1 describes a circle of radius 1. Topologists have invented invariants for qualitatively describing topological spaces, and we can just attempt to work out these invariants for the special spaces we are interested in. Taking this idea to its extreme, one arrives at a field called motivic homotopy theory. It provides novel ways for qualitatively describing solutions of systems of polynomial equations, and has in the past twenty years been successfully applied to several longstanding open problems in algebraic geometry. This project aims to deepen our understanding of motivic homotopy theory by exploring further not only its parallels with classical topology, but also what happens where these parallels break down. This project consists of several parts. One set of parts explores properties of motivic highly structured ring spectra, called normed spectra (introduced in collaboration with M. Hoyois). The Principal Investigator (P.I.) will (1) study power operations in normed spectra and deduce splitting results for normed spectra of positive characteristic and stability results for the motivic homology of symmetric groups and (2) construct further normed spectra, such as a normed spectrum structure on the motivic spectrum KO representing hermitian K-theory. Another set of parts explores the passage from unstable to stable motivic homotopy theory. Specifically the P.I. will (3) study the motivic Barratt-Priddy-Quillen map, and (4) study an unstable motivic homology Whitehead theorem. These goals will be achieved by using techniques from higher category theory, classical stable homotopy theory, algebraic topology and algebraic geometry. The results in parts (1) and (2) can be used to more effectively study algebraic varieties using cohomology theories; for example by exploiting the cohomology operations present in theories represented by normed spectra. Parts (3) and (4) are more useful for studying the motivic (stable) homotopy category itself, and hence the totality of all cohomology theories for algebraic varieties at once. 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.