Computations in Stable and Unstable Equivariant Chromatic Homotopy

稳定和不稳定等变色同伦的计算

• 批准号: 1811189
• 负责人: Michael Hill
• 金额: $ 41.89万
• 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: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811189&HistoricalAwards=false
• 关键词: Computations; Stable; Unstable; Equivariant; Chromatic

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 experience a renaissance recently dues 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 the problems in this project focus on unpacking some of these new constructions and describing what they mean for algebraic topology in general. Many of the projects focus on diversity in STEM. The PI is currently developing tools to help others build a conference series for graduate students and develop their own conferences for younger researchers to attract them to a field. The PI is also in discussions with an HBCU about building more direct connections between their students and the PI's institutions, starting with electronic seminars to introduce students to active researchers in algebraic topology. The goal of these collaborations is to have more students from underrepresented groups enter and succeed in graduate programs in algebraic topology. Finally, the PI has developed and continues to refine a First Year seminar on "Women in Math." The seminar connects students with female mathematicians, allowing the students the opportunity to hear about their research and experience.Modern stable homotopy theory heavily utilizes the fact that the stable homotopy category behaves like a derived category of modules. The ground ring here is the sphere spectrum, and computing its homotopy groups is one of the overarching themes in the subject. The problem can be approached by first looking p-locally, and we can pass to the p-local stable homotopy category. Here algebraic geometry provides a further refinement via the theory of formal groups, a cornerstone of algebraic topology. The current approach to understanding monochromatic homotopy is via certain homotopy fixed points computations. Computing the homotopy groups of fixed points and homotopy fixed points is very difficult in general. One of the most exciting new tools developed to solve the Kervaire problem is a general slice filtration, a method which directly computes homotopy groups of fixed points. For Real Landweber exact theories, this is an extremely efficient tool. For larger groups, computations are still tractable but much more mysterious. In all cases, many of the conceptual tools from non-equivariant homotopy are not available. Many of the techniques developed for stable equivariant homotopy can also be applied unstably. This gives a natural and geometric notion of "even" which refines the ordinary one non-equivariantly and which encompasses spaces related to Real bordism and its norms. The PI expects to see a close connection between unstable even spaces, various orientations by norms of Real bordism, and Mackey functor objects in algebraic geometry.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。 Equivariant代数拓扑已经历了PI，Hopkins和Ravenel的解决方案的文艺复兴时期，这是Kervaire不变的一个问题，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具有引人注目的后果，并且该项目的问题着重于解开这些新结构，并描述它们对代数拓扑的意义。许多项目着重于STEM的多样性。 PI目前正在开发工具，以帮助其他人为研究生建立会议系列，并为年轻的研究人员开发自己的会议，以吸引他们进入领域。 PI还在与HBCU讨论有关学生与PI机构之间建立更直接联系的讨论，从电子研讨会开始，向学生介绍代数拓扑的活跃研究人员。这些合作的目的是让来自代表性不足的小组的更多学生进入代数拓扑的研究生课程。最后，PI已经发展并继续完善了关于“数学女性”的第一年研讨会。 研讨会将学生与女性数学家联系起来，使学生有机会听到他们的研究和经验。现代稳定的同性恋理论大量利用了稳定同型类别的行为就像派生的模块类别。这里的接地环是球形频谱，计算其同型组是该受试者的总体主题之一。可以通过首先查看p-local来解决问题，我们可以传递给P-Local稳定同型类别。在这里，代数几何形状通过代数拓扑基石的形式群体的理论提供了进一步的完善。 当前理解单色同型的方法是通过某些同质固定点计算。一般而言，计算固定点和​​同轴固定点的同质组非常困难。为解决Kervaire问题而开发的最令人兴奋的新工具之一是一种通用切片过滤，该方法可以直接计算固定点的同型组。对于真正的Landweber精确理论，这是一种极其有效的工具。对于较大的群体，计算仍然可以进行操作，但更神秘。在所有情况下，都不可用的许多非等级同型概念工具。为稳定的均匀同型开发的许多技术也可以不稳定地应用。这给出了“偶数”的自然和几何概念，该概念可以优化普通的概念，并涵盖了与真实的bordism及其规范相关的空间。 PI希望看到不稳定的空间之间的联系，真实的王权规范的各种取向以及代数几何形状中的Mackey Founctor对象之间的联系。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的审查标准来通过评估来通过评估来获得支持的。

项目成果

Equivariant stable categories for incomplete systems of transfers

不完全转移系统的等变稳定类别

• DOI: 10.1112/jlms.12441
• 发表时间: 2021
• 期刊: Journal of the London Mathematical Society
• 作者: Blumberg, Andrew J.;Hill, Michael A.
• 通讯作者: Hill, Michael A.

Detecting exotic spheres in low dimensions using coker J

• DOI: 10.1112/jlms.12301
• 发表时间: 2017-08
• 期刊: Journal of the London Mathematical Society
• 作者: Mark Behrens;Michael Hill;Michael J. Hopkins;M. Mahowald

An equivariant tensor product on Mackey functors

Mackey 函子上的等变张量积

• DOI: 10.1016/j.jpaa.2019.04.001
• 发表时间: 2019
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Hill, Michael A.;Mazur, Kristen
• 通讯作者: Mazur, Kristen

Bi-incomplete Tambara functors

双不完全 Tambara 函子

• 发表时间: 2022
• 期刊: London Mathematical Society lecture note series
• 作者: Blumberg, Andrew J;Hill, Michael A.
• 通讯作者: Hill, Michael A.

Free Incomplete Tambara Functors are Almost Never Flat

自由不完全 Tambara 函子几乎从不平坦

• DOI: 10.1093/imrn/rnab361
• 发表时间: 2022
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Hill, Michael A.;Mehrle, David;Quigley, James D.
• 通讯作者: Quigley, James D.