Computations in Classical and Motivic Stable Homotopy Theory

经典和动机稳定同伦理论的计算

• 批准号: 2427220
• 负责人: Eva Belmont
• 金额: $ 12.51万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-04-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2427220&HistoricalAwards=false
• 关键词: Computations; Classical; Motivic; Stable; Homotopy

Algebraic topology is a field of mathematics that involves using algebra and category theory to study properties of geometric objects that do not change when those objects are deformed. A central challenge is to classify all maps from spheres to other spheres, where two maps are considered equivalent if one can be deformed to the other. The equivalence classes of these maps are called the homotopy groups of spheres, and collectively they form one of the deepest and most central objects in the field. Historically, much important theory has arisen out of attempts to compute more homotopy groups of spheres and understand patterns within them. This project involves furthering knowledge of the homotopy groups of spheres, using old and new techniques as well as computer calculations. The project also involves studying an analogue of these groups in algebraic geometry; this falls under a relatively new and actively developed area called motivic homotopy theory, which applies techniques in algebraic topology to study algebraic geometry. The broader impacts of this project center around supporting the local mathematics community through mentoring and promoting diversity. The principal investigator will help build the nascent homotopy theory community at the university and promote women and minorities in the subject through seminar organization and mentoring. One of the main planned projects is a large-scale effort to compute the homotopy groups of spheres at the prime 3 in a range, using old and new techniques such as the Adams-Novikov spectral sequence as well as infinite descent machinery. This work will be aided by computer calculations, which short-circuits some of the technical difficulties encountered in previous attempts. Another main group of projects concerns computing the analogue of the stable homotopy groups of spheres in the world of R-motivic homotopy theory. This represents a continuation of prior work of the PI and collaborator; the plan is to supplement the techniques used in that work with computer calculations and a new tool, the slice spectral sequence. A third project concerns theory and spectral sequence computations aimed at computing the cohomology of profinite groups such as special linear groups and Morava stabilizer groups.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.

代数拓扑学是一个数学领域，它涉及使用代数和范畴论来研究几何物体在变形时不改变的性质。一个核心挑战是将所有地图从球体分类到其他球体，其中两个地图被认为是等效的，如果一个可以变形为另一个。这些映射的等价类被称为球的同伦群，它们共同构成了场中最深和最中心的对象之一。从历史上看，许多重要的理论都是在试图计算更多的同伦球群并理解其中的模式时产生的。该项目涉及进一步了解球体的同伦群，使用新旧技术以及计算机计算。该项目还包括在代数几何中研究这些群的类比；这属于一个相对较新的和积极发展的领域，称为动力同伦理论，它应用代数拓扑中的技术来研究代数几何。该项目更广泛的影响集中在通过指导和促进多样性来支持当地数学社区。首席研究员将帮助在大学建立新生的同伦理论社区，并通过组织研讨会和指导来促进妇女和少数民族在这一学科上的发展。计划中的主要项目之一是大规模地计算一个范围内质数为3的球体的同伦群，使用新旧技术，如亚当斯-诺维科夫谱序列以及无限下降机器。这项工作将得到计算机计算的帮助，它可以缩短以前尝试中遇到的一些技术困难。另一组主要项目涉及计算r -动机同伦理论世界中球的稳定同伦群的模拟。这代表了PI和合作者先前工作的延续；该计划是用计算机计算和一种新工具——切片光谱序列——来补充这项工作中使用的技术。第三个项目涉及理论和谱序列计算，旨在计算无限群（如特殊线性群和Morava稳定群）的上同调。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。