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: 2414922
• 负责人: James Quigley
• 金额: $ 21.2万
• 依托单位: University of Virginia Main Campus
• 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: 2024-01-15 至 2025-11-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2414922&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.

稳定同伦理论是在整个二十世纪发展起来的，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（编码高维球体之间的可能关系直至连续变形）是研究的中心对象。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（例如对球体上的可微结构进行分类）到代数问题（例如对环上的射影模进行分类）。该项目将探索稳定同伦理论在代数、拓扑和几何中的进一步应用。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，该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、迷你课程和社交活动来增加本科生、研究生和高年级学生的包容性。为了解决 K-12 年级的不平等问题，PI 将开发和管理一个项目，将其所在机构的本科生与当地课外项目的学生配对进行在线辅导。该计划将规避传统推广中常见的某些参与障碍，例如缺乏交通和设施。具体的研究项目包括稳定茎的研究及其在几何拓扑、稳定的代几何类似物中的应用词干及其与数论的联系，以及代数 K 理论的等变类似物及其在代数和几何中的应用。更具体地说，在之前工作的基础上，PI将使用拓扑模形式和Mahowald不变量来研究稳定茎，旨在推断新维度中奇异球体的存在。在相关方向上，PI 将使用之前工作中引入的 kq 分辨率来研究动机稳定干，这是稳定干的代数几何类似物。主要目标是应用 kq 分辨率将动机稳定词干与算术不变量（如 Hermitian K 理论）联系起来。实代数 K 理论，编码代数 K 理论、埃尔米特 K 理论和 L 理论等经典不变量，也将使用先前工作中开发的迹方法进行研究。总体目标是将代数 K 理论的结果扩展到实代数 K 理论，从而获得厄米特 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，并被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来获得支持。