Stable Homotopy Theory in Algebra, Topology, and Geometry

代数、拓扑和几何中的稳定同伦理论

• 批准号: 2203785
• 负责人: James Quigley
• 金额: $ 21.2万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-12-15 至 2023-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203785&HistoricalAwards=false
• 关键词: Stable; Homotopy; Theory; Algebra; Topology

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.

稳定的同型理论是在整个20世纪开发的，用于研究高维拓扑空间。由于球体是拓扑空间的基本构建块，因此稳定的茎，它编码高维球之间可能的关系到连续变形，这是研究的核心对象。除拓扑之外，稳定的茎在整个数学过程中都具有出人意料的广泛应用，从几何问题（例如对球体上的可区分结构进行分类，到代数问题），例如对环上的投影模块进行分类。该项目将探讨稳定同义理论在代数，拓扑和几何形状中的进一步应用。在线社区建设中更广泛的影响中心。 PI将通过组织本科研究机会，研究生课程，在线研讨会，小型课程和网络活动来继续共同组织电子计算同义理论在线研究社区。为了解决K-12级别的不平等问题，PI将开发和管理他的家庭机构与本地课程课程的学生在线补习的学生的计划配对的本科生。 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.更具体地说，在以前的工作上，PI将使用拓扑模块化形式和Mahowald不变性研究稳定的茎，旨在推断出新维度中的异国球体的存在。在相关的方向上，PI将使用先前工作中引入的KQ分辨率来研究动机稳定茎，即稳定茎的代数几何类似物。主要目标是应用KQ分辨率将动机稳定茎与Hermitian K理论等算术不变性联系起来。真正的代数K理论，该理论编码诸如代数K理论，Hermitian K理论和L理论等古典不变性，也将使用以前的工作中开发的痕量方法进行研究。总体目标是将代数K理论的结果扩展到真实的代数K理论，从而获得了Hermitian K理论和L理论的结果，这些结果将在代数和几何学中申请。该奖项反映了NSF的法规使命，并认为通过基金会的知识优点和广泛的crietia crietia criter criter criter criteria criteria criteria criter criteria criteria criteria criteria criteria criteria criteria criteria crietia crietia crietia crietia crietia均值得一提。