Equivariant Stable Stems

等变稳定茎

基本信息

批准号：
2003204
负责人：
Bertrand Guillou
金额：
$ 22.12万
依托单位：
University of Kentucky Research Foundation
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2003204&HistoricalAwards=false
关键词：
Equivariant Stable Stems

项目摘要

Spheres are simple yet important objects of study in topology. One of the central questions of algebraic topology is the classification of all possible mappings of a high-dimensional sphere onto a sphere of lower dimension. It turns out that this classification of mappings of spheres has wide-ranging repercussions in geometry and in physics. Recently, this question has received attention in other contexts: when the spheres are considered in the realm of algebraic geometry, or when the spheres have specified symmetries which must be preserved by the mappings in question. More recently, greater understanding of how these various contexts impact each other has emerged. The research supported by this award will employ these newfound connections to expand the range in which these questions are understood, especially in the setting of spheres with a twofold symmetry. This project provides and funds research training for graduate students.The principal investigator will continue joint work with Dan Isaksen on computations of the motivic and C2-equivariant stable homotopy groups of spheres. The R-motivic computations are more approachable, and these determine a portion of the C2-equivariant stable homotopy groups. The main tools will be the rho-Bockstein spectral sequence and the Adams spectral sequence. Various techniques will be employed to run these spectral sequences, including the use of Massey products. The PI and collaborators will also investigate v1-periodicity in the R-motivic and C2-equivariant settings, producing finite complexes that support periodicity operators. This will lead to periodic families of elements in the stable homotopy groups of spheres in these contexts. In another direction, another collaboration will analyze additive power operations for equivariant cohomology theories.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.

领域是拓扑中简单而重要的研究对象。代数拓扑的中心问题之一是将高维球体所有可能映射的所有可能映射分类到较低维度的球体上。事实证明，这种球体映射的分类在几何和物理学方面具有广泛的影响。最近，这个问题在其他情况下引起了人们的注意：当在代数几何形状的领域中考虑球时，或者当球体已指定的对称性时，这些符号必须由所讨论的映射保留。最近，对这些各种环境如何相互影响的更了解。该奖项支持的研究将采用这些新发现的联系来扩大这些问题的理解范围，尤其是在具有双重对称性的领域的环境中。该项目为研究生提供并为研究培训提供了研究培训。 R-动力计算更容易平易近人，这些计算决定了C2-均衡稳定同型组的一部分。主要工具将是Rho-Bockstein光谱序列和ADAMS光谱序列。将采用各种技术来运行这些光谱序列，包括使用Massey产品。 PI和合作者还将调查R-Motivic和C2-均衡设置中的V1 - 周期性，从而产生支持周期性运营商的有限复合物。这将导致在这种情况下稳定的球体稳定同型组中的元素家族。在另一个方向上，另一项合作将分析对等效的共同体学理论的添加功率操作。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优点和更广泛的影响评估标准通过评估来获得支持的。