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：计算数字，组和戒指等不变性以了解空间。代数拓扑的目的是系统地在数字和几何对象（如空格）之类的代数对象之间建立连接。该连接允许双向信息流，而代数不变式区分空间和拓扑方法，为代数问题提供了信息。从Quillen，代数和代数几何数据（如正式组）的基础工作开始，为具有惊人特性的空间带来了新的不变性。该项目结合了这个古典线程与来自近似代数拓扑的最新进展相结合。 “ Eproiriant代数拓扑”记得数据中固有的对称性的集合，作为数据的一部分，系统地将空间与相同的对称性分组，并且产生的数字和不变性必须反映这一点。这种额外的结构提供了更多细微的计算，提供了有关在对称下如何变化的经典描述的更多信息。由于Pi，Hopkins和Ravenel解决了Kervaire Infortiant One问题，因此最近的代数拓扑已经经历了文艺复兴，这是代数拓扑中最古老的杰出问题之一。该解决方案引入了许多新的结构和技术，这些新结构和技术在古典和模棱两可的代数拓扑中产生了惊人的后果，该项目的重点是解开这些新结构中的一些，在经典研究的计算中探索了他们的后果，并描述了它们对代数拓扑的意义。许多项目着重于STEM的多样性。 PI的基础是PI先前的数学女性第一年研讨会，将创建以多样性为导向的类别，将数学内容和教学法与数学中的表示和包容性的讨论相结合。同时，PI打算为那些不将自己视为“数学人”的学生创造更多机会，以使用UCLA的“制造商空间”与代数和几何概念建立联系，以使学生设计和建立具体的模型。 PI将继续组织会议组织，尤其是专注于为早期职业数学家和高级本科生腾出空间的会议，以一种将学生与稳定同型的想法和研究人员联系起来的方式。使用Equivariant稳定同型的新工具，PI将在某些具有有意义有意义的倍增量的slice selice sequence slice Spectral sek selice secors seke selice sequence。这些与使用霍普金斯（Hopkins）更高的真实K理论光谱研究K（n） - 局部现象的经典方法紧密相关，在Prime 2中，此处的计算都包含以前已知的较高的实际K理论计算。该项目主要集中于具体计算（既有有意义的高核心狂物的色度有意义的商），以及诸如双重steenrod代数（等传统对象），同时还研究了我们可以看到哪种乘法结构的更多抽象问题。最后，探讨了所有这些机械在经典问题上的应用问题。该奖项反映了NSF的法定使命，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来提供支持。