Rational and equivariant phenomena in chromatic homotopy theory

现代数学中许多最多产的主题位于几个不同领域的交叉点上。这个项目涉及代数拓扑、数论和群论之间的问题，以及在微分几何和代数几何输入方面的一些应用。这个项目的主要目标之一是解决莫拉瓦在20世纪70年代的工作提出的一个问题。这个问题的解决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。这项研究将与PI在本科生和研究生阶段的教育努力相结合。这个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公开问题。PI将与合作者合作，证明球谱的单色层的有理化为某些生成器上的p-进有理数上的外代数。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公开问题之一。PI还将讨论从有限群的Burnside环到其单色上同伦的标准映射的核的问题。这个问题与权力运作理论和聚变系统理论有关。PI将与研究生一起开发工具，帮助产生乘法和加法幂运算之间的普遍指数关系。最后，PI将与合作者合作，在以前工作的基础上，促进对全球等变复杂椭圆世代的乘法性质的理解。该项目由拓扑学计划和既定的刺激竞争研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.