Power operations in equivariant cohomology

等变上同调中的幂运算

• 批准号: 1406121
• 负责人: Charles Rezk
• 金额: $ 19.47万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406121&HistoricalAwards=false
• 关键词: Power; operations; equivariant; cohomology

Homotopy theory is a branch of topology; it arose as the study of certain invariant properties of spaces, namely those left unchanged by continuous deformations. The most powerful tools for studying such properties are what are called "cohomology theories". Cohomology theories are illuminated by the theory of formal groups, which in turn are closely related to problems in number theory. The aim of this project is to understand aspects of this relationship, with the prospect of creating new computational tools in homotopy theory. This project concerns the theory of power operations in equivariant cohomology theories. Several interrelated projects are proposed, which aim to draw connections between the theory of ultracommutative ring spectra, which generalize commutative ring spectra to the equivariant setting, and the algebraic geometry of isogenies of elliptic curves and formal groups. These connections will advance understanding of topological invariants such as elliptic cohomology.

同伦论是拓扑学的一个分支，它是研究空间的某些不变性质，即那些因连续变形而保持不变的性质。研究这些性质最有力的工具是所谓的“上同调理论”。上同调理论是由形式群理论阐明的，而形式群理论又与数论中的问题密切相关。这个项目的目的是了解这种关系的各个方面，并展望在同伦理论中创造新的计算工具的前景。本课题涉及等变上同调理论中的幂运算理论。提出了几个相关的方案，旨在建立超交换环谱理论与椭圆曲线和形式群的同构的代数几何之间的联系。超交换环谱理论将交换环谱推广到等变环境。这些联系将促进对诸如椭圆上同调之类的拓扑不变量的理解。