Equivariant and Chromatic Stable Homotopy Theory

这个研究项目探讨了代数拓扑中的问题，代数拓扑是数学的一个分支，关注高维的形状。尽管代数拓扑具有抽象的性质，但它已被证明在各种应用中都是有用的，包括理论物理学，其中该主题自然出现，试图调和广义相对论与量子力学，以及数据科学，其中该主题在大型数据集的分析中具有相当大的影响。一个在该领域存在了50年的问题，被称为Kervaire不变量问题，在2009年得到了解决；这个问题的核心问题的答案与大多数专家的预期相反，在证明过程中需要出人意料的新技术，这些新技术在其他领域可能很有用。该研究项目旨在扩大这一发现，并使其适应进一步的应用。首席研究员计划以两种方式跟进这一进展。首先，正在编写的一本书旨在使研究生和其他对该领域感兴趣的非专家能够访问解决方法，通过举例说明和解释，扩大了Kervaire不变量问题的等变同伦理论和范畴理论的数学基础。其次，为解决Kervaire不变量问题而开发的工具正在适应进一步的应用。特别地，对于每个素数都有对应的问题。原来的几何问题的最新解是质数2。5及更大质数的代数模拟在20世纪70年代末已经解决。这个代数问题对素数3仍然是开放的，首席研究员有一个解决它的计划。此外，对于二原色问题的令人惊讶的解答所提出的问题比它所回答的问题要多，这尤其意味着某些在球的同伦群中预测的模式是不可能发生的。什么可能取代它们的问题还没有定论。

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.