Equivariant Approaches to Chromatic Homotopy

该项目直接解决了代数拓扑的核心问题：计算不变量，如数字、群和环，以理解空间。代数拓扑的目标是系统地建立代数对象（如数）和几何对象（如空间）之间的联系。这种连接允许双向信息流，代数不变量区分空间和拓扑方法通知代数问题。从Quillen的基础工作开始，代数和代数几何数据（如形式群）为具有显著性质的空间提供了新的不变量。这个项目结合了这个经典的线程和更多来自等变代数拓扑的最新发展。“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，系统地将具有相同对称性的空间分组，并且产生的数字和不变量必须反映这一点。这个额外的结构提供了更细致的计算，提供了更多关于经典描述的不变量在对称下如何变化的信息。由于PI、Hopkins和Ravenel解决了代数拓扑中最古老的突出问题之一——Kervaire不变量1问题，等变代数拓扑最近经历了一次复兴。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著的分支，本项目侧重于解开这些新结构中的一些，探索它们在经典计算中的分支，并描述它们对代数拓扑的一般意义。许多项目关注STEM的多样性。在PI之前的“数学中的女性”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的讨论结合起来。与此同时，PI打算为那些不认为自己是“数学人”的学生创造更多的机会，让他们利用加州大学洛杉矶分校的“创客空间”，将代数和几何概念联系起来，让学生设计和建造具体的模型。PI将继续组织会议，特别是为早期职业数学家和高级本科生提供空间的会议，利用这些会议将学生与稳定同伦的思想和研究人员联系起来。利用新开发的等变稳定同伦工具，PI将研究超实数光谱中某些色有意义商的切片光谱序列。这些与使用Hopkins- Miller高实K理论谱研究K(n)局部现象的经典方法密切相关，并且在素数2处，这里的计算包含了所有先前已知的高实K理论计算。该项目主要侧重于具体的计算（包括超实数bordism的色彩意义商和更传统的对象，如对偶Steenrod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可定向性的经典问题中的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.