Motivic and Equivariant Stable Homotopy Groups

动机和等变稳定同伦群

基本信息

批准号：
1904241
负责人：
Daniel Isaksen
金额：
$ 17.52万
依托单位：
Wayne State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1904241&HistoricalAwards=false
关键词：
Motivic Equivariant Stable Homotopy Groups

项目摘要

Spheres are the basic building blocks of all geometric objects. More complicated geometric objects can be constructed by fitting these spheres together, but spheres of different dimensions can fit together in only certain combinations. Enumerating these combinations of spheres is one of the fundamental questions of stable homotopy theory. This problem is known as the computation of homotopy groups of spheres. The project uses spectral sequences to carry out these computations. They are delicate, intricate, subtle, and complicated, but they are also understandable with enough insight and patience. Each time a new part of the machinery is understood, another layer of complexities becomes accessible for further study. Computer calculations play a large supporting role. The project promotes the use of videoconferencing to collaborate with peers, to advise graduate students, and to host online seminars. These efforts build towards a self-sustaining virtual mathematical research community. One major benefit of these new modes of interaction is that they erode traditional barriers to entry. This is especially beneficial for people in remote geographical locations and for those with non-traditional or non-prestigious backgrounds who are not typically afforded access to traditional departments of mathematics.The project will compute classical, C-motivic, R-motivic, and C2-equivariant stable homotopy groups. The key tools are the Adams spectral sequence, the Adams-Novikov spectral sequence, and the effective slice spectral sequence. The project consists of a series of interlocking problems, both algebraic and homotopical. Many of the problems suggest specific methods for obtaining calculational data about stable homotopy groups. Other problems address related structural issues, such as exotic periodicity and Mahowald invariants. A key idea is to use C-motivic calculations to see deeper into classical structure. Preliminary results suggest that this is a surprisingly powerful technique that allows for the computation of new stable homotopy groups in a range. The project also includes a series of techniques for computing R-motivic and C2-equivariant stable homotopy groups. This represents the first serious effort to grapple with equivariant versions of tools like the Adams spectral sequence at a computational level. The key point is to build up to the C2-equivariant computations gradually, through C-motivic and R-motivic intermediate steps. One example of the kind of payoff for this work are computations of new values of notoriously difficult Mahowald invariants. Finally, the project will study the effective slice spectral sequence, especially in R-motivic homotopy theory but also over arbitrary fields. This spectral sequence is a motivic replacement for the Adams-Novikov spectral sequence. The R-motivic calculation is more accessible, while the arbitrary fields involve interesting arithmetic. Inspired by the motivic effective slice filtration, the project will also attempt to study a new C2-equivariant filtration that ought to be useful for studying C2-equivariant stable homotopy groups.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.

球是所有几何对象的基本构建块。可以通过将这些球拟合在一起来构建更复杂的几何对象，但是不同尺寸的球体只能在某些组合中结合在一起。列举这些球体的组合是稳定同义理论的基本问题之一。该问题称为球体同拷贝组的计算。该项目使用光谱序列来执行这些计算。它们精致，复杂，微妙且复杂，但也可以通过足够的洞察力和耐心来理解。每次了解机械的新部分时，都可以访问另一层复杂性以进行进一步研究。计算机计算起着很大的支持作用。该项目促进了使用视频会议与同龄人合作，为研究生提供建议并主持在线研讨会。这些努力旨在建立自我维持的虚拟数学研究界。这些新互动方式的主要好处之一是它们侵蚀了传统的进入障碍。这对偏远地理位置的人们以及那些通常无法访问传统数学部门的非传统或非遗产背景的人特别有益。关键工具是ADAMS光谱序列，Adams-Novikov光谱序列和有效的片段序列。该项目包括一系列互锁问题，包括代数和同位素。许多问题提出了获得有关稳定同型组的计算数据的特定方法。其他问题解决了相关的结构问题，例如异国周期性和Mahowald不变性。一个关键的想法是使用C-动力计算以更深入地了解经典结构。初步结果表明，这是一项令人惊讶的功能强大的技术，可以在范围内计算新的稳定同型组。该项目还包括一系列用于计算R-动力和C2均衡稳定同型组的技术。这代表了在计算级别上努力努力应对诸如ADAMS光谱序列之类工具的首次认真努力。关键点是通过C-动力和R-动力中间步骤逐渐逐步构建C2-量化计算。这项工作的回报的一个例子是计算臭名昭著的Mahowald不变性的新价值。最后，该项目将研究有效的切片频谱序列，尤其是在R-动力同型理论中，但在任意领域上也是如此。该频谱序列是亚当斯诺维科夫光谱序列的动机替代。 R-动力计算更容易访问，而任意字段涉及有趣的算术。受动机有效的切片过滤的启发，该项目还将尝试研究一种新的C2 equivariant过滤，该过滤对于研究C2- equivariant稳定同型组应该很有用。该奖项反映了NSF的法定任务，并通过该基金会的知识分子优点和广泛的影响来评估NSF的法定任务，并被认为是值得的。

项目成果

期刊论文数量（7）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Stable homotopy groups of spheres: from dimension 0 to 90

DOI：
10.1007/s10240-023-00139-1
发表时间：
2020-01
期刊：
Publications mathématiques de l'IHÉS
影响因子：
0
作者：
Daniel Isaksen;Guozhen Wang;Zhouli Xu
通讯作者：
Daniel Isaksen;Guozhen Wang;Zhouli Xu

Stable homotopy groups of spheres

球体的稳定同伦群

DOI：
10.1073/pnas.2012335117
发表时间：
2020-01
期刊：
Proceedings of the National Academy of Sciences
影响因子：
0
作者：
Isaksen Daniel C.;Wang Guozhen;Xu Zhouli
通讯作者：
Xu Zhouli

$\mathbb{C}$-motivic modular forms

$mathbb{C}$-动机模块化形式

DOI：
10.4171/jems/1171
发表时间：
2022
期刊：
Journal of the European Mathematical Society
影响因子：
2.6
作者：
Gheorghe, Bogdan;Isaksen, Daniel C.;Krause, Achim;Ricka, Nicolas
通讯作者：
Ricka, Nicolas

The cohomology of C2-equivariant ?(1) and thehomotopy of koC2

C2-等变式 ?(1) 的上同调和 koC2 的同伦

DOI：
10.2140/tunis.2020.2.567
发表时间：
2020
期刊：
Tunisian Journal of Mathematics
影响因子：
0.9
作者：
Guillou, Bertrand J.;Hill, Michael A.;Isaksen, Daniel C.;Ravenel, Douglas Conner
通讯作者：
Ravenel, Douglas Conner

The Mahowald operator in the cohomology of the Steenrod algebra

Steenrod 代数上同调中的 Mahowald 算子