Equivariant Approaches to Chromatic Homotopy

该项目直接解决了代数拓扑学的核心：计算不变量，如数字、群和环来理解空间。代数拓扑学的目标是系统地在像数这样的代数对象和像空间这样的几何对象之间建立联系。这种联系允许信息的双向流动，用代数不变量来区分空间，用拓扑方法来通知代数问题。从Quillen的基础工作出发，代数和代数几何数据就像形式群一样，产生了具有显著性质的空间的新的不变量。这个项目将这一经典线索与来自等变代数拓扑的更新发展结合在一起。“等变代数拓扑学”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，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微的计算，提供了有关经典描述的不变量在对称下如何变化的更多信息。等变代数拓扑学最近经历了一次复兴，这是由于PI、Hopkins和Ravenel解决了Kervaire不变一问题，这是代数拓扑学中最古老的突出问题之一。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经典和等变代数拓扑中具有显著的分支，本项目专注于拆解其中一些新结构，探索它们在经典研究计算中的分支，并描述它们对代数拓扑的一般意义。许多项目侧重于STEM的多样性。在国际数学联合会前一年关于女性参与数学的研讨会的基础上，国际数学联合会将创建一个多元化驱动的课程，将数学内容和教学与数学中的表征和包容的讨论结合起来。与此同时，PI打算为那些不认为自己是“数学爱好者”的学生创造更多机会，让他们利用加州大学洛杉矶分校的“Maker Spaces”与代数和几何概念联系起来，让学生设计和构建具体的模型。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腾出空间的会议，以此作为联系学生与稳定同伦的想法和研究人员的方式。PI将使用新开发的等变稳定同伦工具，研究超实谱的某些色意义商的切片谱序列。它们与使用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，这里的计算包含所有先前已知的高实K理论计算。该项目主要关注具体的计算(既有超现实边界主义的色意义商，也有更传统的对象，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，探索了所有这些机制在向量丛的定向性这一经典问题上的应用。这一奖项反映了NSF的法定使命，并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，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.