Equivariance and Higher Algebra in Motivic Homotopy Theory

代数几何与理解多项式方程的解有关。人们很久以前就知道，一般人不能指望找到多项式方程的精确解，即使对于一个变量的单一方程也是如此。然而，通常可以回答有关解决方案集的更多定性问题。特别是，代数几何中最基本也是出了名的困难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。它从数学、拓扑学的不同领域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。顺应这一趋势，该项目旨在将拓扑学中的几个富有成效的想法带到代数几何的世界中。在第一部分中，PI将从参数同伦理论的角度发展等变动机同伦理论。在第二部分中，PI将研究高结构交换代数的基元同伦理论，并希望解决识别基元环空间的问题。更具体地说，PI将通过为代数堆栈上的纤维同伦理论构造Grothendieck的六函子形式，为等变基元同伦理论奠定坚实的基础。这种形式的存在对于经典的堆栈不变量，如代数K-理论，具有有趣的结果；它还可以用来构造新的代数堆栈的行为良好的上同调不变量，并更好地理解它们之间的关系。在这个项目的第二部分，PI将引入对E_无穷环概念的动机精化，并证明了许多经典的代数簇的上同调不变量都具有这种精化的乘法结构。PI还将引入Motivic E_无穷空间的相关概念，并研究它们在无限Motivic环空间的潜在识别原理中的作用。为了实现这些目标，PI将使用现有的和新的方法，从等变代数几何，普通动机同伦理论，经典代数拓扑，和更高范畴理论。

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.