Applications of equivariant stable homotopy theory

等变稳定同伦理论的应用

• 批准号: 2301520
• 负责人: Po Hu
• 金额: $ 15.38万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301520&HistoricalAwards=false
• 关键词: Applications; equivariant; stable; homotopy; theory

Algebraic topology is the study of topological objects, such as topological spaces, through algebraic invariants that do not change when spaces are continuously deformed. Stable homotopy theory, a central part of algebraic topology, expands from the study of spaces to that of spectra, which are "stabilized spaces" that also represent generalized cohomology theories, or ways to assign algebraic invariants to spaces or other spectra. Equivariant stable homotopy theory further adds to spectra the actions of groups, which can be thought of as ways to map a spectrum to itself in composable and invertible ways. Equivariant stable homotopy theory has grown to be an important tool that offers insights into many deep questions in algebraic topology. The techniques of equivariant stable homotopy theory have also found applications in other areas of mathematics, including algebraic geometry and number theory. The broader impact aspect of the project includes mentoring of graduate and undergraduate students in mathematical research. The principal investigator (PI) will also continue outreach efforts by working to make her research area accessible to the public.This project includes a circle of ideas in equivariant stable homotopy theory. The PI will continue her ongoing work on equivariant complex cobordism spectra, in particular the extension of her previous calculation of the coefficients of such spectra for primary p-groups to more general groups. This has important implications to the study of equivariant formal group laws, another part of the project that the PI will pursue. The PI will also investigate applications to her recent calculation, together with her collaborators, of the equivariant Mackey Steenrod algebra for odd primes. Specifically, one such application the PI is pursing is the construction of odd-primary versions of the Real Brow-Peterson spectrum. A closely related question is the construction and understanding of equivariant elliptic and Barsotti-Tate cohomologies, as well as the formal group laws associated with these spectra. The PI will also continue her ongoing project, along with her collaborators, in the calculation of self-conjugate and double-real cobordism spectra.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.

代数拓扑是通过空间连续变形时不改变的代数不变量来研究拓扑对象（例如拓扑空间）。稳定同伦理论是代数拓扑的核心部分，它从空间研究扩展到谱研究，谱是“稳定空间”，也代表广义上同调理论，或者将代数不变量分配给空间或其他谱的方法。等变稳定同伦理论进一步将群的作用添加到谱中，这可以被认为是以可组合和可逆的方式将谱映射到其自身的方法。等变稳定同伦理论已经发展成为一个重要的工具，为代数拓扑中许多深刻的问题提供了见解。等变稳定同伦理论的技术也在其他数学领域得到了应用，包括代数几何和数论。该项目更广泛的影响包括指导研究生和本科生进行数学研究。首席研究员 (PI) 还将继续努力向公众开放她的研究领域。该项目包括等变稳定同伦理论的一系列思想。 PI 将继续她正在进行的等变复配边谱的工作，特别是将她之前对初级 p 群的谱系数的计算扩展到更一般的群。这对等变形式群定律的研究具有重要意义，这是 PI 将进行的项目的另一部分。 PI 还将与她的合作者一起研究她最近计算的奇素数等变 Mackey Steenrod 代数的应用。具体来说，PI 正在寻求的一个这样的应用是构建 Real Brow-Peterson 谱的奇原色版本。一个密切相关的问题是等变椭圆和 Barsotti-Tate 上同调的构造和理解，以及与这些谱相关的形式群定律。 PI 还将与她的合作者一起继续她正在进行的项目，计算自共轭谱和双实配边谱。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的评估进行评估，被认为值得支持。影响审查标准。