Equivariant Derived Algebraic Geometry

该项目直接解决了代数拓扑的核心问题：计算数字，群和环等不变量来理解空间。代数拓扑的目标是系统地建立代数对象（如数字）和几何对象（如空间）之间的联系。这种连接允许信息的双向流动，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拓扑模形式的高斯-霍普金斯-米勒理论，这是一种用拓扑语言编码椭圆曲线（代数几何中的一个基本对象）的方法。这构建了新类型的椭圆曲线，并突出了通过普通代数看不到的共性。“等变代数拓扑”将空间中固有的对称性的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。记住额外的结构使计算更丰富，但更复杂，它允许人们梳理其他相互关联的问题。例如，使用等变方法，PI，霍普金斯和Ravenel解决了Kervaire Invariant One问题，这是代数拓扑学中最古老的突出问题，其根源可以追溯到20世纪30年代。这反过来又提供了关于如何从更简单的空间（如球体）中构建空间的信息。该项目旨在建立在解决方案中开发的技术基础上，解决代数和拓扑中的其他计算问题。特别是，该项目旨在探索在设置中的可见的等方差之间的相互作用，如由底层代数数据产生的拓扑模形式和由拓扑放置的约束。现代稳定同伦理论大量利用了这样一个事实，即稳定同伦范畴表现得像一个派生的模块范畴。这里的基环不是普通的环，而是一个环谱，即球谱。在过去的二十年里，人们描述了如何直接用交换环谱来做代数几何：导出代数几何的理论。许多自然发生的例子都是作为有限群作用的交换环谱出现的，所以人们会问，什么时候存在一个基础的等变交换环谱是计算上可访问的。这是这个项目的主要重点，一个新的研究领域称为“等变导出代数几何”。从计算的角度来看，我们的目标是了解同伦群的不动点的频谱上的群作用和频谱的同伦群之间的相互作用。一般来说，这是一个非常困难的问题。为解决Kervaire问题而开发的最令人兴奋的新工具之一是一般切片过滤，这是一种直接计算不动点的同伦群的方法。对于真实的Landweber精确理论，以及扎根于基本代数几何的理论，这是一个非常有效的工具。对于较大的群体，计算是容易处理的，但要神秘得多。该项目的目标之一是确定当从等变导出的代数几何中产生的各种光谱具有像真实的Landweber精确理论那样好的切片时。等变同伦也是代数K-理论的同伦方法的核心。代数K群的计算也非常困难，即使知道它们是否为零也会解决长期存在的数论难题。同伦的主要方法是通过一个谱塔，TR塔，归纳建立了拓扑Hochschild同调的不动点谱。新的等变机制提供了替代的，更简单的拓扑Hochschild同源性的建设，使我们能够评估它的Thom光谱，并建立相对的版本。

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.