Equivariance and Higher Algebra in Motivic Homotopy Theory

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Algebraic geometry is concerned with understanding solutions of polynomial equations. It has long been known that in general one cannot hope to find exact solutions to polynomial equations, even for a single equation in one variable. Nevertheless, it is often possible to answer more qualitative questions about the set of solutions. In particular, one of the most fundamental and notoriously difficult questions in algebraic geometry is whether solutions exist at all. Motivic homotopy theory is a relatively new approach to such questions, and it has already helped solve many open problems in the past twenty years. It borrows many ideas and techniques from a different field of mathematics, topology, and successfully applies them to algebraic geometry in unexpected ways. Following this trend, this project aims to bring several fruitful ideas from topology to the world of algebraic geometry.This project consists of two parts. In the first part, the PI will develop equivariant motivic homotopy theory from the point of view of parametrized homotopy theory. In the second part, the PI will study highly structured commutative algebras in motivic homotopy theory and hopes to address the problem of recognizing motivic loop spaces. More specifically, the PI will lay solid foundations for equivariant motivic homotopy theory by constructing Grothendieck's six-functor formalism for fiberwise homotopy theory over algebraic stacks. The mere existence of this formalism has interesting consequences for classical invariants of stacks, like algebraic K-theory; it can also be used to construct new well-behaved cohomological invariants of algebraic stacks, and to better understand the relations between them. In the second part of this project, the PI will introduce a motivic refinement of the notion of E_infinity ring and show that many classical cohomological invariants of algebraic varieties possess this refined multiplicative structure. The PI will also introduce the related notion of motivic E_infinity space and investigate their role in a potential recognition principle for infinite motivic loop spaces. To achieve these goals, the PI will use existing as well as new methods from equivariant algebraic geometry, ordinary motivic homotopy theory, classical algebraic topology, and higher category theory.