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.

现代数学中许多最有效的主题位于几个不同领域的交集。该项目涉及代数拓扑，数量理论和群体理论之间的问题，其中一些应用于代数几何形状的差异几何和输入。该项目的主要目标之一是解决1970年代Morava的工作提出的问题。解决这个问题的解决方案将是了解不同维度的球体如何相互缠绕的重要步骤 - 这是现代代数拓扑核心的问题。这项研究将与PI在本科和研究生层面的教育努力融合。该项目的目的是利用最近开发的工具来攻击色度均匀理论中的几个开放问题。 PI将与合作者合作，以表明球体光谱的单色层的合理化是某些发电机对P-ADIC理性的外部代数。这个问题可以追溯到1970年代，并且是色素同拷贝理​​论中的核心开放问题之一。 PI还将解决一个有关从有限组的伯恩赛环（Burnside）到其单色共同体的规范图的内核的问题。这个问题与权力运营理论和融合系统理论结合在一起。 PI将与研究生一起开发工具，以帮助建立乘法和添加功率操作之间的普遍指数关系。 最后，PI将与合作者合作以先前的工作为基础，并提高对全球层次化的椭圆形的综合特性的理解。该项目由拓扑计划共同资助，既定的计划刺激了竞争性研究（EPSCOR）（EPSCOR）。这一奖项反映了NSF的法定任务，并反映了通过评估的范围的构成者，其构成师的范围是构成的，构成了构成的范围。