Computations in Stable and Unstable Equivariant Chromatic Homotopy

稳定和不稳定等变色同伦的计算

基本信息

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

项目摘要

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，用代数不变量来区分空间，用拓扑方法来解决代数问题。从 Quillen 的基础工作开始，代数和代数几何数据（如形式群）为具有惊人属性的空间带来了新的不变量。该项目将这一经典思路与来自等变代数拓扑的最新发展结合起来。 “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，系统地对具有相同对称性的空间进行分组，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，提供了有关经典描述的不变量在对称性下如何变化的更多信息。由于 PI、Hopkins 和 Ravenel 对 Kervaire 不变一问题（代数拓扑中最古老的突出问题之一）的解决，等变代数拓扑最近经历了复兴。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着的影响，并且该项目中的问题集中于解开其中一些新结构并描述它们对代数拓扑的一般意义。许多项目都注重 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会议系列，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，首先通过电子研讨会向学生介绍代数拓扑领域的活跃研究人员。这些合作的目标是让更多来自代表性不足群体的学生进入代数拓扑研究生课程并取得成功。最后，PI 制定并继续完善了“数学领域的女性”第一年研讨会。该研讨会将学生与女性数学家联系起来，让学生有机会了解她们的研究和经验。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。这里的底环是球面谱，计算其同伦群是该主题的首要主题之一。该问题可以通过首先查看 p-局部来解决，然后我们可以传递到 p-局部稳定同伦类别。在这里，代数几何通过形式群理论（代数拓扑的基石）提供了进一步的细化。当前理解单色同伦的方法是通过某些同伦不动点计算。一般来说，计算不动点的同伦群和同伦不动点是非常困难的。为解决 Kervaire 问题而开发的最令人兴奋的新工具之一是通用切片过滤，这是一种直接计算不动点同伦群的方法。对于 Real Landweber 精确理论来说，这是一个极其有效的工具。对于较大的群体，计算仍然容易处理，但更加神秘。在所有情况下，许多非等变同伦的概念工具都不可用。许多为稳定等变同伦而开发的技术也可以不稳定地应用。这给出了一种自然和几何的“均匀”概念，它非等价地细化了普通的概念，并且包含了与真实边界及其规范相关的空间。 PI 期望看到不稳定均匀空间、实数几何规范的各种方向以及代数几何中的 Mackey 函子对象之间的密切联系。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和能力进行评估，被认为值得支持。更广泛的影响审查标准。