Equivariance and Higher Algebra in Motivic Homotopy Theory

代数几何是关于理解多项式方程的解。人们早就知道，一般来说，人们不可能希望找到多项式方程的精确解，即使是一个单一的方程在一个变量。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。动机同伦理论是一个相对较新的方法来解决这些问题，它已经帮助解决了许多开放的问题在过去的二十年。它借用了许多思想和技术，从不同的数学领域，拓扑，并成功地将它们应用到代数几何以意想不到的方式。本计画的目的是将拓扑学中的一些有成果的想法带到代数几何的世界。在第一部分中，PI将从参数化同伦理论的角度发展等变动机同伦理论。在第二部分中，PI将在motivic同伦理论中研究高度结构化的交换代数，并希望解决识别motivic循环空间的问题。更具体地说，PI将奠定坚实的基础等变动机同伦理论通过构建Grothendieck的六函子形式主义的纤维同伦理论代数堆栈。这种形式主义的存在对于经典的栈不变量（如代数K理论）有着有趣的影响;它也可以用来构造新的行为良好的代数栈上同调不变量，并更好地理解它们之间的关系。在这个项目的第二部分中，PI将引入E_infinity环概念的motivic精化，并证明代数簇的许多经典上同调不变量都具有这种精化的乘法结构。PI还将引入动机E_infinity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将使用现有的以及新的方法，从等变代数几何，普通motivic同伦理论，经典代数拓扑，和更高的范畴理论。

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.