Rational and equivariant phenomena in chromatic homotopy theory

现代数学中许多最有成效的主题都是在几个不同领域的交叉点上进行的。本课题涉及代数拓扑学、数论和群论之间的问题，并应用于微分几何和代数几何的输入。该项目的主要目标之一是解决Morava在20世纪70年代提出的问题。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。本计画的目标是利用最近发展的工具来解决色同伦理论中的几个开放问题。PI将与合作者一起证明球面谱的单色层的合理化是在某些发生器上的p进理性上的外代数。这个问题可以追溯到20世纪70年代，是色同伦理论的中心开放问题之一。本文还讨论了有限群的Burnside环到它的单色上同伦的正则映射的核问题。这个问题与动力运行理论和核聚变系统理论密切相关。与研究生一起，PI将开发工具，以帮助产生乘法和加性幂运算之间的普遍指数关系。最后，PI将与合作者在之前的工作基础上，进一步了解全局等变复化椭圆属的乘法性质。该项目由拓扑学计划和促进竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.