Equivariant Approaches to Chromatic Homotopy

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，代数不变量区分空间和拓扑方法通知代数问题。从基础工作的奎伦，代数和代数几何数据，如正式团体产生了新的不变量的空间与惊人的性质。该项目将这一经典线索与来自等变代数拓扑的最新发展相结合。“等变代数拓扑”将空间中固有的对称性的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。这种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性下如何变化的信息。等变代数拓扑经历了一个复兴最近由于解决方案的PI，霍普金斯，和Ravenel的Kervaire不变一个问题，其中一个最古老的突出问题，代数拓扑。该解决方案引入了许多新的构造和技术，这些构造和技术在经典和等变代数拓扑中具有显著的分支，本项目的重点是解开这些新构造中的一些，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意义。许多项目侧重于STEM的多样性。在PI之前关于数学中的妇女的第一年研讨会的基础上，PI将创建一个多样化驱动的班级，将数学内容和教学法与数学中的代表性和包容性的讨论相结合。与此同时，PI打算为那些不认为自己是“数学人”的学生创造更多的机会，让他们使用加州大学洛杉矶分校的“创客空间”来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年级本科生腾出空间的会议，将这些会议作为学生与稳定同伦思想和研究人员联系的一种方式。使用等变稳定同伦的新开发工具，PI将研究超实光谱的某些色有意义的子的切片光谱序列。这些是密切相关的经典方法来研究K（n）-本地现象使用霍普金斯-米勒高真实的K-理论的频谱，并在素数2，计算在这里submanders所有以前已知的高真实的K-理论计算。该项目主要集中在具体的计算（包括超实边数的色有意义的代数和更传统的对象，如对偶Steenrod代数），同时也研究我们可以看到什么样的乘法结构的更抽象的问题。最后，所有这些机器的可定向性的向量bundles的经典问题的应用explored.This奖反映了NSF的法定使命，并已被认为是值得支持的，通过使用该基金会的智力价值和更广泛的影响审查标准进行评估。

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.