Equivariant and Chromatic Stable Homotopy Theory

该研究项目探讨代数拓扑中的问题，代数拓扑是涉及高维形状的数学分支。尽管具有抽象性质，代数拓扑已被证明在各种应用中都很有用，包括理论物理学（该主题在试图协调广义相对论与量子力学）和数据科学（该主题在大型数据集的分析中产生了相当大的影响）中自然出现。该领域一个有 50 年历史的问题，即 Kervaire 不变量问题，于 2009 年得到解决；问题核心问题的答案与大多数专家的预期相反，证明中需要令人惊讶的新技术，这些新技术可能在其他领域有用。 该研究项目旨在扩大这一发现并使其适应进一步的应用。首席研究员计划通过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该领域其他感兴趣的非专家能够使用解决方法，通过说明性示例和解释来放大 Kervaire 不变问题的等变同伦理论和范畴论的数学基础设施。其次，为解决 Kervaire 不变问题而开发的工具正在适应进一步的应用。特别是，每个素数都有一个对应的问题。最初的几何驱动问题的最新解决方案是素数 2。素数 5 及更大素数的代数模拟已在 20 世纪 70 年代末得到解决。素数 3 的代数问题仍然悬而未决，首席研究员已经制定了解决该问题的计划。此外，2-原问题的解的令人惊讶的性质提出的问题多于它所回答的问题，特别意味着球体同伦群中的某些预测模式不会发生。什么可以取代它们的问题是悬而未决的。

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.