Power operations in equivariant cohomology

同伦理论是拓扑学的一个分支；它起源于对空间某些不变性质的研究，即那些连续变形所留下的不变性质。研究这些性质最有力的工具是所谓的“上同调理论”。上同调理论是由形式群理论阐释的，而形式群理论又与数论中的问题密切相关。该项目的目的是了解这种关系的各个方面，以期在同伦理论中创造新的计算工具。本课题研究等变上同调理论中的幂运算理论。将交换环谱推广到等变环境的超交换环谱理论与椭圆曲线和形式群的等同构代数几何之间建立了联系。这些联系将促进对椭圆上同调等拓扑不变量的理解。

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.