Mathematical Sciences: Finite Groups and Complex Oriented Theories in Homotopy

传统上，群论构造一直是代数拓扑学家的武器库的一部分。在最基本的层面上，来自(有限)群论的工具已被用来构造拓扑工具：Steenrod正方形、特征类、具有A(N)自由上同调的空间等。在更微妙的层面上，许多基本的拓扑结构似乎是由群论结构决定的--例如，无限循环空间结构由对称群的组合学确定，而计算K(BG)的Atiyah定理说，群论决定了这个环。尼古拉斯·J·库恩最近和现在的研究延续了使用群论来指导一个人研究拓扑问题的传统。特别是，在他与M.Hopkins和D.Ravenel正在进行的合作中，群论、等变同伦和数论的组合被用来更好地理解以复数为导向的上同调理论背后的“几何”。这些理论尤其包括Morava K-理论--E.Devinatz、Hopkins和J.Smith的工作已知它在稳定同伦理论中具有核心重要性。

Group theoretic constructions have traditionally been part of the arsenal of an algebraic topologist . On the most basic level, tools from (finite) group theory have been used to construct topological tools: Steenrod squares, characteristic classes, spaces with A(n)-free cohomology, etc. On a more subtle level, many basic topological structures seem to be determined by group theoretic structures - for example, infinite loop space structures are determined by the combinatorics of the symmetric groups, and the theorem of Atiyah that calculates K(BG), says that group theory determines this ring. Nicholas J. Kuhn's recent and current research continues this tradition of using group theory to guide one's study of topological questions. In particular, in his ongoing collaboration with M. Hopkins and D. Ravenel, a combination of group theory, equivariant homotopy, and number theory is being used to gain greater understanding of the "geometry" underlying complex oriented cohomology theories. These include, in particular, the Morava K-theories - known by the work of E. Devinatz, Hopkins, and J. Smith to be of central importance in stable homotopy theory.