Equivariant Derived Algebraic Geometry

该项目直接解决了代数拓扑的核心问题：计算不变量，如数字、群和环，以理解空间。代数拓扑的目标是系统地建立代数对象（如数）和几何对象（如空间）之间的联系。这种连接允许双向信息流，代数不变量区分空间和拓扑方法通知代数问题。后者的一个很好的例子是gowers - hopkins - miller的拓扑模形式理论，这是一种用拓扑语言编码椭圆曲线（代数几何中的基本对象）的方法。这建立了新的椭圆曲线种类，并突出了普通代数中不可见的共性。“等变代数拓扑”将空间中固有的对称性集合作为数据的一部分，系统地将具有相同对称性的空间分组，并且产生的数字和不变量必须反映这一点。记住额外的结构会使计算更丰富，但也更复杂，它允许人们梳理出原本相互关联的问题。例如，PI、Hopkins和Ravenel使用等变方法解决了Kervaire不变量1问题，这是代数拓扑中最古老的突出问题，可以追溯到20世纪30年代。这反过来又提供了如何用球体等更简单的空间来构建空间的信息。该项目旨在以解决方案中开发的技术为基础，解决代数和拓扑中的其他计算问题。特别是，该项目试图探索由底层代数数据产生的拓扑模块形式等设置中可见的等方差与拓扑所施加的约束之间的相互作用。现代稳定同伦理论大量利用了稳定同伦范畴类似于模的派生范畴这一事实。这里的地环不是一个普通的环而是一个环谱，球谱。过去二十年的工作描述了如何直接用交换环谱做代数几何：衍生代数几何理论。许多自然发生的例子都是以有限群作用下的交换环谱出现的，所以人们会问，什么时候存在一个可计算的等变交换环谱。这是这个项目的主要焦点，一个新的研究领域被称为“等变衍生代数几何”。从计算的角度来看，目标是了解谱上群作用的不动点的同伦群与谱的底层同伦群之间的相互作用。总的来说，这是一个非常困难的问题。一般切片滤波是解决Kervaire问题最令人兴奋的新工具之一，它是一种直接计算不动点同伦群的方法。对于真正的兰德韦伯精确理论，理论植根于基础的代数几何，这是一个非常有效的工具。对于更大的群体，计算是容易处理的，但更加神秘。该项目的目标之一是确定由等变衍生代数几何产生的各种光谱何时具有与Real Landweber精确理论一样好的切片。等变同伦也是代数k理论的同伦方法的核心。代数K群也非常难以计算，甚至知道它们是否为零也会解决长期存在的数论猜想。同伦的主要方法是通过一个光谱塔，即TR塔，它是由不动点光谱归纳而成的拓扑Hochschild同伦。新的等变机制提供了替代的，更简单的拓扑Hochschild同调结构，使我们能够在Thom光谱上评估它并建立相对版本。

The project addresses directly the heart of algebraic topology: computing invariants like numbers, groups, and rings to understand spaces. The goal of algebraic topology is to systematically build a connection between algebraic objects like numbers and geometric objects like spaces. This connection allows a two-way flow of information, with algebraic invariants distinguishing spaces and topological methods informing algebraic problems. A beautiful example of the latter is the Goerss-Hopkins-Miller theory of topological modular forms, a way to encode elliptic curves (a fundamental object in algebraic geometry) in topological language. This builds new kinds of elliptic curves and highlights commonalities not visible through ordinary algebra. "Equivariant algebraic topology" remembers a collection of symmetries inherent in a space as part of the data, systematically grouping spaces with the same symmetries, and the numbers and invariants produced must reflect this. Remembering the extra structure makes richer, but more complicated, computations, and it allows one to tease apart otherwise interconnected problems. For example, using equivariant methods, the PI, Hopkins, and Ravenel solved the Kervaire Invariant One problem, the oldest outstanding problem in algebraic topology with roots dating back to the 1930s. This in turn gave information about how one can build spaces out of simpler ones like spheres. This project aims to build on the techniques developed in the solution, tackling other computational problems in algebra and topology. In particular, the project seeks to explore the interaction between the visible equivariance in settings like topological modular forms arising from underlying algebraic data and the constraints placed by the topology.Modern stable homotopy theory heavily utilizes the fact that the stable homotopy category behaves like a derived category of modules. Here the ground ring is not an ordinary ring but rather a ring spectrum, the sphere spectrum. Work over the last twenty years has described how to do algebraic geometry directly with commutative ring spectra: the theory of derived algebraic geometry. Many of the naturally occurring examples arise as commutative ring spectra with an action of a finite group, so one asks when there is an underlying equivariant commutative ring spectrum which is computationally accessible. This is the main focus of this project, a new area of research called "equivariant derived algebraic geometry". From a computational perspective, the goal is to understand the interplay between the homotopy groups of fixed points of a group action on a spectrum and the underlying homotopy groups of the spectrum. In general, this is a very difficult problem. One of the most exciting new tools developed to solve the Kervaire problem is a general slice filtration, a method which directly computes homotopy groups of fixed points. For Real Landweber exact theories, theories well-rooted in the underlying algebraic geometry, this is an extremely efficient tool. For larger groups, computations are tractable but much more mysterious. One of the goals of the project is to determine when the kinds of spectra arising from equivariant derived algebraic geometry have slices as nice as those for Real Landweber exact theories. Equivariant homotopy is also central to the homotopical approach to algebraic K-theory. Algebraic K groups are also exceedingly difficult to compute, and even knowing whether or not they are zero would settle long-standing number theory conjectures. The primary approach in homotopy is via a tower of spectra, the TR tower, built inductively out of fixed point spectra for topological Hochschild homology. The new equivariant machinery provides alternate, simpler construction of topological Hochschild homology, allowing us to evaluate it on Thom spectra and to build relative versions.