Equivariant Approaches to Chromatic Homotopy

色同伦的等变方法

• 批准号: 2105019
• 负责人: Michael Hill
• 金额: $ 30.28万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105019&HistoricalAwards=false
• 关键词: Equivariant; Approaches; Chromatic; Homotopy

The project addresses directly the heart of algebraic topology: computing invariants like numbers, groups, and rings to understand spaces. The goal of algebraic topology is to systematically build a connection between algebraic objects like numbers and geometric objects like spaces. This connection allows a two-way flow of information, with algebraic invariants distinguishing spaces and topological methods informing algebraic problems. Starting from foundational work of Quillen, algebraic and algebraic geometry data like formal groups gives rise to new invariants for spaces with striking properties. This project combines this classical thread with much more recent developments coming from equivariant algebraic topology. "Equivariant algebraic topology" remembers a collection of symmetries inherent in a space as part of the data, systematically grouping spaces with the same symmetries, and the numbers and invariants produced must reflect this. This extra structure provides more nuanced computations, giving more information about how the classically described invariants change under symmetries. Equivariant algebraic topology has experienced a renaissance recently due to the solution by the PI, Hopkins, and Ravenel to the Kervaire Invariant One problem, one of the oldest outstanding problems in algebraic topology. The solution introduced a host of new constructions and techniques that have striking ramifications in classical and equivariant algebraic topology, and this project focuses on unpacking some of these new constructions, exploring their ramifications in classically studied computations, and describing what they mean for algebraic topology in general. Many of the projects focus on diversity in STEM. Building on the PI's prior First Year seminar on Women in Math, the PI will create a diversity-driven class, combining mathematical content and pedagogy with discussions of representation and inclusion in mathematics. At the same time, the PI intends to create more opportunities for students who do not see themselves as "math people" to connect with algebra and geometry concepts using UCLA's "Maker Spaces" to have students design and build concrete models. The PI will continue conference organizing, especially conferences focusing on making space for early career mathematicians and for advanced undergraduates, using these as a way to connect students with the ideas and researchers in stable homotopy.Using newly developed tools in equivariant stable homotopy, the PI will study the slice spectral sequences for certain chromatically meaningful quotients of hyperreal spectra. These are closely connected to the classical approaches to studying K(n)-local phenomena using the Hopkins--Miller higher real K-theory spectra, and at the prime 2, computations here subsume all previously known higher real K-theory computations. The project focuses mainly on concrete computations (both of chromatically meaningful quotients of hyperreal bordism and of more traditional objects like the dual Steenrod algebra), while also studying more abstract questions of what kinds of multiplicative structures we can see. Finally, an application of all of this machinery to the classical questions of orientability of vector bundles is explored.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.

该项目直接解决了代数拓扑的核心问题：计算数字、群和环等不变量来理解空间。代数拓扑的目标是系统地建立代数对象（如数字）和几何对象（如空间）之间的联系。这种连接允许信息的双向流动，用代数不变量来区分空间，用拓扑方法来解决代数问题。从 Quillen 的基础工作开始，代数和代数几何数据（如形式群）为具有惊人属性的空间带来了新的不变量。该项目将这一经典思路与来自等变代数拓扑的最新发展结合起来。 “等变代数拓扑”将空间中固有的对称性集合作为数据的一部分，系统地对具有相同对称性的空间进行分组，并且产生的数字和不变量必须反映这一点。这种额外的结构提供了更细致的计算，提供了有关经典描述的不变量在对称性下如何变化的更多信息。由于 PI、Hopkins 和 Ravenel 对 Kervaire 不变一问题（代数拓扑中最古老的突出问题之一）的解决，等变代数拓扑最近经历了复兴。该解决方案引入了许多新的结构和技术，这些结构和技术在经典和等变代数拓扑中具有显着的影响，该项目的重点是解开其中一些新结构，探索它们在经典研究计算中的影响，并描述它们对代数拓扑的一般意义。许多项目都注重 STEM 的多样性。在 PI 之前举办的关于数学领域女性的第一年研讨会的基础上，PI 将创建一个多元化驱动的课程，将数学内容和教学法与数学中的代表性和包容性讨论相结合。与此同时，PI 打算为那些不认为自己是“数学人”的学生创造更多机会，利用加州大学洛杉矶分校的“创客空间”来连接代数和几何概念，让学生设计和构建具体模型。 PI 将继续组织会议，特别是专注于为早期职业数学家和高年级本科生腾出空间的会议，利用这些会议作为将学生与稳定同伦的想法和研究人员联系起来的一种方式。使用新开发的等变稳定同伦工具，PI 将研究切片光谱序列，以获得超真实光谱的某些色度有意义的商。这些与使用 Hopkins-Miller 高实 K 理论谱研究 K(n) 局域现象的经典方法密切相关，并且在素数 2 处，此处的计算包含所有先前已知的高实 K 理论计算。该项目主要侧重于具体计算（超实数的色商和更传统的对象（如对偶斯廷罗德代数）），同时也研究更抽象的问题，即我们可以看到什么样的乘法结构。最后，探讨了所有这些机制在向量束定向性的经典问题上的应用。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。