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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.

球面是拓扑学中简单而又重要的研究对象。代数拓扑学的核心问题之一是将高维球面到低维球面的所有可能映射进行分类。事实证明，球体映射的这种分类在几何学和物理学中产生了广泛的影响。最近，这个问题在其他方面引起了人们的注意：当球面被考虑在代数几何领域时，或者当球面指定了必须由所讨论的映射保持的对称性时。最近，人们对这些不同的背景如何相互影响有了更多的了解。该奖项支持的研究将利用这些新发现的联系来扩大理解这些问题的范围，特别是在具有双重对称性的球体的背景下。这个项目为研究生提供并资助研究培训。首席研究员将继续与Dan Isaksen合作计算球面的Motivic和C2-等变稳定同伦群。R-Motivic计算更接近实际，从而确定了C2-等变稳定同伦群的一部分。主要工具将是Rho-Bockstein谱序列和Adams谱序列。将使用各种技术来运行这些光谱序列，包括使用Massey产品。PI和合作者还将研究R-Motivic和C2-等变设置中的v1-周期性，产生支持周期性算子的有限复形。这将导致在这些上下文中球面的稳定同伦群中的元素的周期族。在另一个方向上，另一个合作将分析等变上同调理论的加法运算。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。