Equivariant and Chromatic Stable Homotopy Theory

等变和色稳定同伦理论

• 批准号: 1606623
• 负责人: Douglas Ravenel
• 金额: $ 20.13万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1606623&HistoricalAwards=false
• 关键词: Equivariant; Chromatic; Stable; Homotopy; Theory

This research project explores questions in algebraic topology, a branch of mathematics that concerns shapes in higher dimensions. Despite its abstract nature, algebraic topology has proven to be useful in a variety of applications, including theoretical physics, where the subject arises naturally in attempts to reconcile general relativity with quantum mechanics, and data science, where the subject has had considerable impact in the analysis of large data sets. A fifty-year-old question in the field known as the Kervaire invariant problem was solved in 2009; the answer to the question at the heart of the problem was the opposite of what most experts had expected, and surprising new techniques, potentially useful in other areas, were required in the proof. The research project aims to amplify this discovery and adapt it to further applications.The principal investigator plans to follow up on this advance in two ways. First, a book in progress is designed to make the solution methods accessible to graduate students and other interested non-experts in the field, amplifying the mathematical infrastructure in equivariant homotopy theory and category theory for the Kervaire invariant problem with illustrative examples and explanations. Second, the tools developed to solve the Kervaire invariant problem are being adapted to further applications. In particular, there is a counterpart to the problem for each prime number. The recent solution of the original, geometrically motivated, problem was for the prime 2. The algebraic analog for primes 5 and larger had been solved in the late 1970s. The algebraic problem remains open for the prime 3, and the principal investigator has a plan for solving it. In addition, the surprising nature of the solution to the 2-primary problem raises more questions than it answers, implying in particular that certain predicted patterns in the homotopy groups of spheres cannot occur. The question of what might replace them is wide open.

该研究项目探讨了代数拓扑的问题，代数拓扑是数学的一个分支，涉及较高维度的形状。尽管具有抽象的性质，但代数拓扑已被证明在包括理论物理学在内的各种应用中很有用，在这些应用中，该主题自然而然地试图将一般相对论与量子力学和数据科学调和，在该应用程序中，该受试者在对大型数据集的分析中产生了相当大的影响。 2009年解决了一个名为“克尔维尔不变问题”的五十岁问题。问题核心的问题的答案与大多数专家所期望的相反，并且在证明中需要在其他领域有可能有用的新技术。 研究项目旨在扩大这一发现并将其适应进一步的应用程序。主要研究者计划通过两种方式跟进这一进步。首先，一本正在进行的书旨在使研究生和该领域的其他感兴趣的非专家可以访问解决方案方法，从而在近似同型理论中放大了数学基础架构和类别理论，用于与说明性示例和解释的Kervaire不变性问题。其次，为解决Kervaire不变问题而开发的工具正在适应进一步的应用程序。特别是，每个素数的问题都与问题有关。原始的，几何动机的解决方案是针对Prime 2。代数问题对于Prime 3仍然开放，主要研究人员制定了解决该问题的计划。此外，对2个主要问题的解决方案的惊人性质提出的问题比它的回答更多，这特别意味着不能发生某些同型球体中某些预测模式。什么可能取代它们的问题是开放的。