Applications of equivariant stable homotopy theory

代数拓扑学是通过在空间连续变形时不改变的代数不变量来研究拓扑对象，如拓扑空间。稳定同伦理论是代数拓扑的一个核心部分，它从空间的研究扩展到谱的研究，谱是“稳定的空间”，它也代表了广义上同伦理论，或者是将代数不变量赋给空间或其他谱的方法。等变稳定同伦理论进一步给谱增加了群的作用，群的作用可以被认为是将谱以可组合和可逆的方式映射到自身的方法。等变稳定同伦理论已经成为研究代数拓扑学中许多深层次问题的重要工具。等变稳定同伦理论的技术在数学的其他领域也有应用，包括代数几何和数论。该项目更广泛的影响方面包括指导研究生和本科生进行数学研究。首席研究员（PI）也将继续努力，努力使她的研究领域向公众开放。本课题包括等变稳定同伦理论中的一圈思想。PI将继续她正在进行的等变复协谱的工作，特别是将她以前对初级p群的这种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将追求的项目的另一部分。PI还将与她的合作者一起研究她最近计算的奇素数等变麦基·斯蒂恩罗德代数的应用。具体来说，PI正在研究的一个这样的应用是构建真实布朗-彼得森谱的奇初级版本。一个密切相关的问题是构造和理解等变椭圆和Barsotti-Tate上同，以及与这些谱相关的形式群律。PI也将继续她正在进行的项目，与她的合作者一起，在计算自共轭和双实共体光谱。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.