Mathematical Sciences: Finite Groups and Complex Oriented Theories in Homotopy

群论结构传统上是 代数拓扑学家的武器库 在最基本的层面上， 从（有限）群理论的工具已被用来构建 拓扑工具：Steenrod平方，特征类， 具有A（n）-free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决定的 理论结构-例如，无限循环空间结构 由对称群的组合学决定， 计算K（BG）的Atiyah定理说， 决定了这枚戒指。 Nicholas J. Kuhn最近和当前的研究继续了这一点。 用群论指导研究的传统 拓扑问题 特别是，在他的持续 与M合作。霍普金斯和D. Ravenel，结合了 群论、等变同伦和数论 用于更好地理解“几何学” 复上同调理论 其中包括，在 特别是摩拉瓦K理论--以E. Devinatz、霍普金斯和J.史密斯是至关重要的， 稳定同伦理论

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.