Power operations in equivariant cohomology

同伦理论是拓扑学的一个分支;它是研究空间的某些不变性质而产生的，也就是那些连续变形而保持不变的性质。 研究这些性质的最有力的工具是所谓的“上同调理论”。 上同调理论由形式群理论所阐明，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，在同伦理论创造新的计算工具的前景。 这个项目涉及等变上同调理论中的幂运算理论。 提出了几个相互关联的项目，其目的是绘制超交换环谱理论之间的联系，其中广义交换环谱的等变设置，和代数几何的椭圆曲线和形式群的同构。 这些联系将促进对拓扑不变量如椭圆上同调的理解。

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.