Equivariant and Chromatic Stable Homotopy Theory

本研究计画探讨代数拓扑学中的问题，代数拓扑学是数学的一个分支，关注更高维度的形状。尽管它的抽象性质，代数拓扑已被证明是有用的各种应用，包括理论物理学，其中该主题自然出现在试图调和广义相对论与量子力学，和数据科学，其中该主题在大型数据集的分析中产生了相当大的影响。2009年，一个在该领域存在了50年的问题--科威尔不变量问题得到了解决;这个问题的核心答案与大多数专家的预期相反，证明过程中需要使用可能在其他领域有用的令人惊讶的新技术。 该研究项目旨在扩大这一发现并使其适用于进一步的应用。首席研究员计划通过两种方式跟进这一进展。首先，一本正在编写的书旨在使研究生和该领域其他感兴趣的非专家能够使用解决方法，放大等变同伦理论和范畴理论中的数学基础设施，为Kervaire不变量问题提供说明性的例子和解释。其次，开发的工具，以解决Kervaire不变的问题正在适应进一步的应用。特别是，对于每个素数，都有一个对应的问题。最新的解决方案的原始，几何动机，问题是为素数2。素数5和更大素数的代数类比在20世纪70年代末已经解决。这个代数问题对于素数3仍然是开放的，并且主要研究者有一个解决它的计划。此外，2-素数问题的解决方案的令人惊讶的性质引起了比它回答的更多的问题，特别是暗示某些预测的同伦群的模式不可能发生。什么可以取代他们的问题是完全开放的。

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.