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Stable Homotopy Theory in Algebra, Topology, and Geometry

代数、拓扑和几何中的稳定同伦理论

基本信息

批准号：
2314082
负责人：
James Quigley
金额：
$ 21.2万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-02-01 至 2024-02-29
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2314082&HistoricalAwards=false
关键词：
Stable Homotopy Theory Algebra Topology

项目摘要

Stable homotopy theory was developed throughout the twentieth century to study high-dimensional topological spaces. Since spheres are the fundamental building blocks of topological spaces, the stable stems, which encode the possible relations between high-dimensional spheres up to continuous deformation, are a central object of study. Beyond topology, the stable stems have surprisingly broad applications throughout mathematics, ranging from geometric problems, such as classifying differentiable structures on spheres, to algebraic problems, such as classifying projective modules over rings. This project will explore further applications of stable homotopy theory in algebra, topology, and geometry. Broader impacts center on online community building. The PI will continue co-organizing the Electronic Computational Homotopy Theory Online Research Community, which aims to increase inclusion at the undergraduate, graduate, and senior levels by organizing undergraduate research opportunities, graduate courses, online seminars, mini-courses, and networking events. To address inequality at the K-12 level, the PI will develop and manage a program pairing undergraduates from his home institution with students from local after-school programs for online tutoring. This program would circumvent certain barriers to participation, such as lack of access to transportation and facilities, which are common in traditional outreach.Specific research projects include the study of the stable stems and their applications in geometric topology, algebro-geometric analogues of the stable stems and their connections to number theory, and equivariant analogues of algebraic K-theory and their applications in algebra and geometry. More specifically, building on previous work, the PI will study the stable stems using topological modular forms and the Mahowald invariant, aiming to deduce the existence of exotic spheres in new dimensions. In a related direction, the PI will use the kq-resolution introduced in previous work to study the motivic stable stems, an algebro-geometric analogues of the stable stems. The main goal is to apply the kq-resolution to relate the motivic stable stems to arithmetic invariants like Hermitian K-theory. Real algebraic K-theory, which encodes classical invariants like algebraic K-theory, Hermitian K-theory, and L-theory, will also be studied using the trace methods developed in previous work. The overarching goal is extending results from algebraic K-theory to real algebraic K-theory, thereby obtaining results for Hermitian K-theory and L-theory that will have applications in algebra and geometry.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.

稳定同伦理论是在整个世纪发展起来的，用于研究高维拓扑空间。由于球面是拓扑空间的基本组成部分，稳定的茎，它编码高维球面之间的可能关系，直到连续变形，是一个中心的研究对象。除了拓扑学之外，稳定茎在整个数学中有着令人惊讶的广泛应用，从几何问题（如分类球面上的可微结构）到代数问题（如分类环上的投射模）。本专题将探讨稳定同伦理论在代数、拓扑学和几何学中的进一步应用。更广泛的影响集中在在线社区建设上。PI将继续共同组织电子计算同伦理论在线研究社区，旨在通过组织本科生研究机会，研究生课程，在线研讨会，迷你课程和网络活动来增加本科生，研究生和高级水平的包容性。为了解决K-12水平的不平等问题，PI将开发和管理一个项目，将他所在机构的本科生与当地课后项目的学生配对进行在线辅导。该计划将避免参与的某些障碍，如缺乏交通和设施，这是在传统的推广，具体的研究项目包括稳定的茎及其在几何拓扑学中的应用，代数几何模拟的稳定茎及其连接到数论，代数K-理论的等变模拟及其在代数和几何中的应用的研究。更具体地说，在以前工作的基础上，PI将使用拓扑模形式和Mahowald不变量研究稳定的茎，旨在推断新维度中奇异球体的存在。在一个相关的方向，PI将使用kq-分辨率在以前的工作中引入到研究motivic稳定的茎，代数几何类似的稳定的茎。主要目标是应用kq-分辨率将motivic稳定茎与像Hermitian K-理论这样的算术不变量联系起来。真实的代数K-理论，它编码经典的不变量，如代数K-理论，埃尔米特K-理论和L-理论，也将研究使用的跟踪方法在以前的工作。总体目标是将结果从代数K理论扩展到真实的代数K理论，从而获得埃尔米特K理论和L理论的结果，这些结果将在代数和几何中得到应用。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。