Mathematical Sciences: Problems in Homotopy Theory, K-theory, and Representation Theory

小行星9423719 同伦理论和表示论有着非常不同的起源，前者是出于理解几何对象（如流形）的全局结构的愿望，而后者则是从研究代数结构如何作为向量空间上的对称性发展而来的。 库恩教授正在继续研究这些学科之间的共同点，这一点以他的“一般表示理论”的发展为代表，“一般表示理论”最初是受到拓扑学家在Steenrod代数上的工作的启发，但它似乎提供了新的强大的工具，为理解完全表示理论的主题，如有限的一般线性群的模表示和MacLane同调的代数K理论家 其中正在研究的具体问题是那些处理同伦“在一个大素数”，拓扑实现问题，以及稳定的K理论和麦克莱恩同源之间的关系。 同伦理论和表示论都是数学学科，其中一个试图发现并最终分类结构的基本“积木”：同伦处理几何对象的变形，如高维表面，表示论关注离散代数对象的对称性，如线和平面的配置。 库恩教授正在继续他的研究之间的一些共同点，这些，使用代数工具开发的启发，他的同伦理论的经验。 ***

9423719 Kuhn Homotopy theory and representation theory have quite distinct origins, with the former being motivated by the desire to understand the global structure of geometric objects such as manifolds, and the latter evolving out of the study of how algebraic structures act as symmetries on vector spaces. Professor Kuhn is continuing his study of the common ground between these subjects, as typified by his development of "generic representation theory," which was originally inspired by work by topologists on the Steenrod algebra, but which seems to offer new powerful tools for understanding completely representation theoretic topics such as the modular representations of the finite general linear groups and the MacLane homology of algebraic K-theorists. Among the specific questions being studied are ones dealing with homotopy "at a large prime," topological realization questions, and the relation between stable K-theory and MacLane homology. Both homotopy theory and representation theory are mathematical subjects in which one is trying to discover, and ultimately classify, fundamental "building blocks" of structure: homotopy dealing with deformations of geometric objects such as higher dimensional surfaces, and representation theory being concerned with symmetries of discrete algebraic objects such as configurations of lines and planes. Professor Kuhn is continuing his study of some of the common ground between these, using algebraic tools developed by him that were inspired by homotopy theoretic experiences. ***