Equivariant Derived Algebraic Geometry

等变导出的代数几何

• 批准号: 1509652
• 负责人: Michael Hill
• 金额: $ 21.53万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1509652&HistoricalAwards=false
• 关键词: Equivariant; Derived; Algebraic; Geometry

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.

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，用代数不变量来区分空间，用拓扑方法来解决代数问题。后者的一个很好的例子是拓扑模形式的戈尔斯-霍普金斯-米勒理论，这是一种用拓扑语言编码椭圆曲线（代数几何中的基本对象）的方法。这构建了新型椭圆曲线，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集合作为数据的一部分，系统地对具有相同对称性的空间进行分组，并且产生的数字和不变量必须反映这一点。记住额外的结构可以使计算更丰富，但也更复杂，并且它允许人们梳理其他相互关联的问题。例如，PI、Hopkins 和 Ravenel 使用等变方法解决了 Kervaire 不变一问题，这是代数拓扑中最古老的突出问题，其根源可以追溯到 20 世纪 30 年代。这反过来又提供了有关如何用球体等更简单的物体构建空间的信息。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，该项目旨在探索由基础代数数据产生的拓扑模形式等设置中的可见等变性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这样一个事实，即稳定同伦范畴的行为就像派生的模块类别。这里的地环不是普通的环，而是环光谱，球光谱。过去二十年的工作描述了如何直接用交换环谱进行代数几何：派生代数几何理论。许多自然发生的例子都是作为具有有限群作用的交换环谱出现的，因此人们会问何时存在可通过计算访问的潜在等变交换环谱。这是这个项目的主要焦点，一个名为“等变导出代数几何”的新研究领域。从计算的角度来看，目标是理解谱上群作用的固定点的同伦群与谱的基础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。为解决 Kervaire 问题而开发的最令人兴奋的新工具之一是通用切片过滤，这是一种直接计算不动点同伦群的方法。对于 Real Landweber 精确理论，即植根于基础代数几何的理论，这是一个极其有效的工具。对于较大的群体，计算很容易处理，但更加神秘。该项目的目标之一是确定等变派生代数几何产生的光谱类型何时具有与 Real Landweber 精确理论的切片一样好的切片。等变同伦也是代数 K 理论同伦方法的核心。代数 K 群也非常难以计算，甚至知道它们是否为零也能解决长期以来的数论猜想。同伦的主要方法是通过光谱塔，即 TR 塔，它是根据拓扑 Hochschild 同调的定点光谱归纳构建的。新的等变机制提供了拓扑 Hochschild 同调的替代、更简单的构造，使我们能够在 Thom 谱上对其进行评估并构建相对版本。

项目成果

Equivariant chromatic localizations and commutativity

等变色定位和交换性

• DOI: 10.1007/s40062-018-0226-2
• 发表时间: 2019-09
• 期刊: Journal of Homotopy and Related Structures
• 影响因子: 0.5
• 作者: Hill; Michael A.
• 通讯作者: Michael A.

On the André–Quillen homology of Tambara functors

论 Tambara 函子的 AndréQuillen 同调

• DOI: 10.1016/j.jalgebra.2017.06.029
• 发表时间: 2017-11
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Hill; Michael A.
• 通讯作者: Michael A.

Detecting exotic spheres in low dimensions using coker J

使用 coker J 检测低维中的奇异球体

• DOI: 10.1112/jlms.12301
• 发表时间: 2020-02
• 期刊: Journal of the London Mathematical Society
• 作者: Behrens, M.;Hill, M.;Hopkins, M. J.;Mahowald, M.
• 通讯作者: Mahowald, M.

The cohomology of C2-equivariant 𝒜(1) and thehomotopy of koC2

C2-等变式 ?(1) 的上同调和 koC2 的同伦

• DOI: 10.2140/tunis.2020.2.567
• 发表时间: 2020-01
• 期刊: Tunisian Journal of Mathematics
• 影响因子: 0.9
• 作者: Guillou, Bertrand J.;Hill, Michael A.;Isaksen, Daniel C.;Ravenel, Douglas Conner
• 通讯作者: Ravenel, Douglas Conner

The right adjoint to the equivariant operadic forgetful functor on incomplete Tambara functors

不完全 Tambara 函子上等变歌剧健忘函子的右伴随

• DOI: 10.1090/conm/729/14691
• 发表时间: 2019-01
• 期刊: Contemporary mathematics
• 作者: Blumberg, Andrew J.;Hill, Michael A.
• 通讯作者: Hill, Michael A.