Stable Homotopy Groups: Theory and Computation

球体是拓扑学的基本构建块。通过将这些球体拟合在一起，可以构造更复杂的拓扑对象。不同尺寸的球体只能以几种不同的方式组合在一起。列举这些球体的组合是一个叫做同伦理论的数学分支的基本问题之一。本课题主要研究球面稳定同伦群的计算，采用理论、人工计算和机器计算三大类技术。这些类别是相辅相成的。理论建立了新的计算技术；人工计算揭示了指向理论进步的新结构；机器计算提供了对人工计算的独立验证，并将其发扬光大。该项目的更广泛影响包括电子计算同伦理论（eCHT）虚拟研究社区的发展和成长。为了实现自我可持续发展的长期目标，eCHT社区将尝试不同类型的在线参与，例如面向早期职业研究生的基础课程、重点主题的系列研讨会、职业发展活动、在线会议，以及与邻近学科的在线网络建立联系。这些新的互动模式旨在克服传统的进入障碍，特别是对偏远地区和具有非传统背景的人来说。该项目的主要研究目标是计算经典、c -动机、r -动机和c2等变环境下的稳定同伦群。它分为三个主要的主题：亚当斯谱序列、有效切片谱序列和稳定同伦理论的变形。有效谱序列是r -动机和c2等变情况下Adams-Novikov谱序列的替代。前两个主题本质上是计算性的，尽管它们指向了一些更理论性的问题。这些计算的许多可能结果中的两个是：关于Mahowald（根）不变量的新信息；解决了最后一个未知的柯维尔不变量问题。第三个主题是最近通过计算探索发现的理论视角。变形将在计算稳定同伦理论中扮演越来越重要的角色。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Spheres are the basic building blocks of topology. More complicated topological objects can be constructed by fitting these spheres together. Spheres of different dimensions can fit together in only a few different ways. Enumerating these combinations of spheres is one of the fundamental questions of a branch of mathematics called homotopy theory. This project focuses on the computation of the stable homotopy groups of spheres, using three general categories of techniques: theory, manual computation, and machine computation. These categories are mutually reinforcing. Theory establishes new techniques of computation; manual computation reveals new structure that points toward theoretical advances; machine computation provides independent verification of manual computation and carries it even further. Broader impacts of the project include development and growth of the electronic Computational Homotopy Theory (eCHT) virtual research community. With a long-term goal of self-sustainability, the eCHT community will experiment with different types of online engagement, such as foundational courses for early-career graduate students, seminar series on focused topics, career development events, online conferences, and building connections to online networks in neighboring subjects. These new modes of interaction aim to overcome traditional barriers to entry, especially for people in remote geographical locations and for those with non-traditional backgrounds. The project's main research goal is to compute stable homotopy groups of spheres in the classical, C-motivic, R-motivic, and C2-equivariant contexts. It breaks into three main topics: the Adams spectral sequence, the effective slice spectral sequence, and deformations of stable homotopy theory. The effective spectral sequence is a replacement for the Adams-Novikov spectral sequence in the R-motivic and C2-equivariant situations. The first two topics are essentially computational, although they point towards a number of more theoretical questions. Two of the many possible outcomes of these computations are: new information about Mahowald (root) invariants; and settling the last unknown case of the Kervaire invariant one problem. The third topic is a theoretical perspective that was recently uncovered by computational exploration. Deformations will play a growing role in computational stable homotopy theory.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.