Applications of equivariant stable homotopy theory

代数拓扑学是通过代数不变量来研究拓扑对象，如拓扑空间，当空间连续变形时，代数不变量不会改变。稳定同伦理论是代数拓扑学的核心部分，它从空间的研究扩展到谱的研究，谱是“稳定空间”，也代表广义上同调理论，或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合成和可逆的方式将谱映射到自身的方法。等变稳定同伦理论已经发展成为一个重要的工具，它为代数拓扑学中的许多深层次问题提供了深刻的见解。等变稳定同伦理论的技巧也在数学的其他领域得到应用，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的数学研究。首席研究员（PI）也将继续努力，使她的研究领域向公众开放。这个项目包括一个等变稳定同伦理论的思想圈。PI将继续她正在进行的等变复配边光谱的工作，特别是她以前的计算系数的扩展，这种光谱的主要p-群更一般的群体。这对等变形式群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还将调查应用到她最近的计算，连同她的合作者，等变麦基Steenrod代数奇素数。具体来说，PI正在追求的一个这样的应用是构造真实的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正式群律。PI还将继续她正在进行的项目，沿着与她的合作者，在自共轭和双实配边光谱的计算。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.