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.

代数拓扑是对拓扑对象（例如拓扑空间）通过代数不变的研究，而当空间不断变形时不会改变。代数拓扑的一个核心稳定同义理论从对空间的研究扩展到光谱的研究，这些空间是“稳定空间”，它也代表了一般的同胞理论，或者是将代数不变性分配给空间或其他光谱的方法。模棱两可的稳定同义理论进一步增加了群体的作用，可以将其视为以合并和可逆的方式将光谱映射到自身的方法。模棱两可的稳定同义理论已成为一个重要的工具，可以洞悉代数拓扑中许多深层问题。模化稳定同质理论的技术还发现了在其他数学领域，包括代数几何和数字理论。该项目的更广泛的影响包括指导数学研究的研究生和本科生。首席调查员（PI）还将通过努力使她的研究领域可通往公众。该项目包括一系列稳定的同型理论。 PI将继续她在持续进行的对等效的复合物谱光谱，特别是她以前对此类频谱的系数的计算扩展到了原代P组的系数到更一般的组。这对对模棱两可的正式群体法律的研究具有重要意义，这是PI将追求的项目的另一部分。 PI还将调查她最近计算的申请，以及她的合作者Mackey Steenrod代数的奇数。具体而言，PI所追求的一种应用是真正的眉毛谱系的奇数版本的构建。一个密切相关的问题是对椭圆形和barsotti-tate共同体的构建和理解，以及与这些光谱相关的正式群体定律。 PI还将在计算自我轭和双重COBORDISP SPECTRA的计算中继续她正在进行的项目以及她的合作者。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评估来评估的。影响审查标准。