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.

球体是所有几何对象的基本构建块。通过将这些球体装配在一起可以构造更复杂的几何物体，但是不同尺寸的球体只能以某些组合方式装配在一起。枚举这些球面组合是稳定同伦理论的基本问题之一。这个问题被称为球面同伦群的计算。该项目使用光谱序列来进行这些计算。它们微妙、错综、微妙、复杂，但只要有足够的洞察力和耐心，也是可以理解的。每次了解机器的新部分时，就可以进一步研究另一层复杂性。计算机计算起着很大的支撑作用。该项目提倡使用视频会议与同行合作、为研究生提供建议以及举办在线研讨会。这些努力旨在建立一个自我维持的虚拟数学研究社区。这些新的互动模式的一大好处是它们消除了传统的进入壁垒。这对于偏远地区的人们以及那些通常无法进入传统数学系的非传统或非著名背景的人尤其有利。该项目将计算经典、C-动机、R-动机和 C2 -等变稳定同伦群。关键工具是 Adams 谱序列、Adams-Novikov 谱序列和有效切片谱序列。该项目由一系列相互关联的代数问题和同伦问题组成。许多问题提出了获得稳定同伦群计算数据的具体方法。其他问题涉及相关的结构问题，例如奇异周期性和马霍瓦尔德不变量。一个关键思想是使用 C 动机计算来更深入地了解经典结构。初步结果表明，这是一种令人惊讶的强大技术，可以计算一定范围内的新稳定同伦群。该项目还包括一系列计算 R 动机和 C2 等变稳定同伦群的技术。这是在计算层面上首次认真努力解决 Adams 谱序列等工具的等变版本。关键点是通过 C-motivic 和 R-motivic 中间步骤逐渐建立 C2-equivariant 计算。这项工作的回报的一个例子是计算众所周知困难的马霍瓦尔德不变量的新值。最后，该项目将研究有效切片谱序列，特别是在 R 动机同伦理论中，而且也在任意域上。该谱序列是 Adams-Novikov 谱序列的动机替代。 R 动机计算更容易理解，而任意域涉及有趣的算术。受动机有效切片过滤的启发，该项目还将尝试研究一种新的 C2 等变过滤，该过滤应该对研究 C2 等变稳定同伦群有用。该奖项反映了 NSF 的法定使命，并通过评估认为值得支持利用基金会的智力优势和更广泛的影响审查标准。

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