Rational and equivariant phenomena in chromatic homotopy theory

色同伦理论中的有理和等变现象

• 批准号: 2304781
• 负责人: Nathaniel Stapleton
• 金额: $ 29.76万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304781&HistoricalAwards=false
• 关键词: Rational; equivariant; phenomena; chromatic; homotopy

Many of the most productive themes in modern mathematics sit at the intersection of several different fields. This project involves problems in between algebraic topology, number theory, and group theory with some applications to differential geometry and input from algebraic geometry. One of the primary goals of this project is to solve a problem suggested by the work of Morava in the 1970s. A solution to this problem will be an important step toward understanding how spheres of different dimensions can wrap around each other -- a problem that is central to modern algebraic topology. The research will be integrated with the PI's educational efforts at the undergraduate and graduate level.The goal of this project is to make use of recently developed tools to attack several open problems in chromatic homotopy theory. The PI will work with collaborators to show that the rationalization of the monochromatic layers of the sphere spectrum are exterior algebras over the p-adic rationals on certain generators. This problem dates back to the 1970s and is one of the central open problems in chromatic homotopy theory. The PI will also address a question concerning the kernel of the canonical map from the Burnside ring of a finite group to its monochromatic cohomotopy. This question is bound up in the theory of power operations and the theory of fusion systems. Together with graduate students, the PI will develop tools to help produce a universal exponential relationship between multiplicative and additive power operations. Finally, the PI will work with collaborators to build on previous work and advance understanding of the multiplicative properties of global equivariant complexified elliptic genera.This project is jointly funded by the Topology program, and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

现代数学中许多最富有成效的主题都处于几个不同领域的交叉点。这个项目涉及代数拓扑学，数论和群论之间的问题，以及微分几何和代数几何输入的一些应用。该项目的主要目标之一是解决Morava在20世纪70年代的工作中提出的问题。这个问题的解决方案将是理解不同维度的球体如何相互缠绕的重要一步-这是现代代数拓扑学的核心问题。该研究将与PI在本科和研究生阶段的教育工作相结合。该项目的目标是利用最近开发的工具来解决色同伦理论中的几个开放问题。PI将与合作者合作，证明球谱的单色层的合理化是某些生成元上的p-adic有理数上的外代数。这个问题可以追溯到20世纪70年代，是色同伦理论中的中心开放问题之一。PI还将解决一个问题的核心规范的映射从伯恩赛德环的有限群的单色同伦。这一问题涉及到电力运营理论和融合系统理论。PI将与研究生一起开发工具，以帮助产生乘法和加法幂运算之间的通用指数关系。 最后，PI将与合作者合作，以以前的工作为基础，推进对全局等变复杂椭圆属的乘法性质的理解。该项目由拓扑计划共同资助，以及刺激竞争研究的既定计划（EPSCoR）该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的评估被认为值得支持。影响审查标准。