Equivariant and Chromatic Stable Homotopy Theory

这项研究项目探索代数拓扑学中的问题，代数拓扑学是数学的一个分支，与更高维度的形状有关。尽管代数拓扑学具有抽象的性质，但它已被证明在各种应用中都很有用，包括理论物理和数据科学，前者是在试图协调广义相对论和量子力学的过程中自然产生的，后者在大型数据集的分析中产生了相当大的影响。2009年，该领域一个已有50年历史的被称为科维埃不变量问题的问题得到了解决；这个问题的核心问题的答案与大多数专家的预期相反，证明中需要令人惊讶的新技术，这些技术可能在其他领域有用。该研究项目旨在扩大这一发现，并将其应用于进一步的应用。首席研究员计划通过两种方式跟进这一进展。首先，正在设计一本书，以使研究生和其他感兴趣的非专家在该领域的解决方法，扩大等变同伦理论和范畴理论的数学基础，通过说明性的例子和解释来解决Kervaire不变问题。其次，为解决Kervaire不变量问题而开发的工具正在适应更多的应用。具体地说，对于每个素数，都有一个对应的问题。最初的几何激励问题最近的解决方案是素数2。素数5和更大素数的代数类比在20世纪70年代末就解决了。这个代数问题对于素数3仍然是开放的，首席研究者有一个解决它的计划。此外，2-素数问题的解的令人惊讶的性质引发的问题比它所回答的问题更多，这特别意味着球面同伦群中的某些预测模式不可能发生。什么可能会取代他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.