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。这个项目结合了这个经典的线程与最近的发展来自等变代数拓扑。“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归因于解决方案的PI，霍普金斯，和Ravenel的Kervaire不变的一个问题，一个最古老的突出问题的代数拓扑。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，本项目中的问题集中在解包这些新构造中的一些，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。PI目前正在开发工具，帮助其他人为研究生建立系列会议，并为年轻研究人员开发自己的会议，以吸引他们进入某个领域。PI还与HBCU讨论在学生和PI机构之间建立更直接的联系，首先是电子研讨会，向学生介绍代数拓扑学的活跃研究人员。这些合作的目标是有更多的学生从代表性不足的群体进入并成功地在研究生课程代数拓扑。最后，PI已经制定并继续完善关于“数学中的妇女”的第一年研讨会。“研讨会将学生与女性数学家联系起来，让学生有机会听到她们的研究和经验。现代稳定同伦理论大量利用了稳定同伦范畴表现得像模的派生范畴这一事实。这里的基环是球谱，计算它的同伦群是这个主题的首要主题之一。这个问题可以通过首先寻找p-局部来解决，我们可以传递到p-局部稳定同伦范畴。在这里，代数几何通过形式群理论提供了进一步的改进，这是代数拓扑的基石。 目前理解单色同伦的方法是通过某些同伦不动点计算。计算同伦不动点群和同伦不动点群一般是很困难的。为解决Kervaire问题而开发的最令人兴奋的新工具之一是一般切片过滤，这是一种直接计算不动点的同伦群的方法。对于真实的Landweber精确理论，这是一个极其有效的工具。对于更大的群体，计算仍然是容易处理的，但更加神秘。在所有情况下，许多概念工具，从非等变同伦是不可用的。许多为稳定的等变同伦而开发的技术也可以不稳定地应用。这给出了一个自然的和几何的“甚至”的概念，它细化了普通的一个非等变，并涵盖了空间有关的真实的bordism及其规范。 PI期望看到不稳定的均匀空间、真实的边界规范的各种方向和代数几何中的麦基函子对象之间的密切联系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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

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.

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.