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

[9423719]库恩同伦理论和表示理论有着截然不同的起源，前者源于理解几何物体（如流形）的整体结构的愿望，而后者则源于对代数结构如何在向量空间上作为对称性的研究。库恩教授正在继续研究这些学科之间的共同点，以他发展的“一般表示理论”为代表，该理论最初受到拓扑学家对Steenrod代数的工作的启发，但它似乎为完全理解表示理论主题提供了新的强大工具，例如有限一般线性群的模表示和代数k -理论家的MacLane同质。正在研究的具体问题包括处理“在大素数处”同伦的问题，拓扑实现问题以及稳定k理论与MacLane同伦的关系。同伦理论和表示理论都是数学学科，其中人们试图发现并最终分类结构的基本“构建块”：同伦处理几何对象的变形，如高维曲面，而表示理论涉及离散代数对象的对称性，如线和面的配置。库恩教授正在继续研究它们之间的一些共同点，使用他开发的代数工具，这些工具受到同伦理论经验的启发。＊＊＊

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. ***