Equivariance and Higher Algebra in Motivic Homotopy Theory

代数几何涉及对多项式方程的解的理解。人们早就知道，一般来说，人们不能指望找到多项式方程的精确解，即使是一个变量的单一方程。然而，通常可以回答关于解决方案集的更多定性问题。特别是，代数几何中最基本和最困难的问题之一是解是否存在。动机同伦理论是解决这类问题的一种相对较新的方法，在过去的二十年里，它已经帮助解决了许多悬而未决的问题。它从不同的数学领域——拓扑学——借鉴了许多思想和技术，并以意想不到的方式成功地将它们应用于代数几何。遵循这一趋势，本项目旨在将一些富有成效的思想从拓扑学带到代数几何的世界。本项目由两部分组成。在第一部分中，PI将从参数化同伦理论的角度发展等变动力同伦理论。在第二部分，PI将研究动机同伦理论中的高度结构化交换代数，并希望解决识别动机环空间的问题。更具体地说，PI将通过构建代数堆栈上光纤同伦理论的Grothendieck六函子形式，为等变动力同伦理论奠定坚实的基础。这种形式的存在对堆栈的经典不变量有有趣的结果，比如代数k理论；它也可以用来构造代数堆栈的新的性能良好的上同调不变量，并更好地理解它们之间的关系。在本项目的第二部分，PI将引入e_∞环概念的动机改进，并证明许多代数变异的经典上同不变量具有这种改进的乘法结构。本文还将介绍动机无穷空间的相关概念，并研究它们在无限动机循环空间的潜在识别原理中的作用。为了实现这些目标，PI将使用现有的以及来自等变代数几何、普通动机同伦理论、经典代数拓扑和高等范畴理论的新方法。

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.