Homotopy Theory and Ring Spectra

这个项目的目标是研究严格交换环谱与形式群理论之间的关系。 本项目的一部分是阐明Morava E-理论的幂运算代数的结构，特别是证明了这个幂运算代数是Koszul的猜想，另一部分是研究交换环谱的E-无限定向，推广了Ando，霍普金斯和PI关于拓扑模形式的弦定向的工作。 同伦理论是拓扑学的一个分支，它起源于对空间的某些不变性质的研究，即那些不被连续变形改变的性质。 研究这类性质的最有力的工具是所谓的“上同调理论”。 一个令人惊讶的事实是，上同调理论是由形式群的理论所阐明的，而形式群又与代数数论中的问题密切相关。 这个项目的目标是了解这种关系。

The goal of this project is to study phenomena relating strictlycommutative ring spectra to the theory of formal groups. Onepart of the project is to elucidation of the structure of thealgebra of power operations for Morava E-theory; and in particular aproof of the conjecture that this power-operation algebra is Koszul.Another part of the project is the study of E-infinityorientations of commutative ring spectra, extending the work of Ando,Hopkins, and the PI on the string orientation of topological modularforms. Homotopy theory is a branch of topology; it arose as the study ofcertain invariant properties of spaces, namely those left unchanged bycontinuous deformations. The most powerful tools for studying suchproperties are what are called "cohomology theories". It is asurprising fact that cohomology theories are illuminated by the theoryof formal groups, which in turn are closely related to problems inalgebraic number theory. The goal of this project is to understandthis relationship.