Homotopy Theory and Ring Spectra

本课题的目的是研究严格交换环谱与形式群理论之间的关系。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科祖尔的猜想。该项目的另一部分是研究交换环谱的e无穷方向，扩展了Ando，Hopkins和PI在拓扑模形式的弦方向上的工作。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，即连续变形留下的不变性质。研究这类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，这是一个令人惊讶的事实，而形式群理论又与代数数论中的问题密切相关。这个项目的目标是理解这种关系。

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.