Mathematical Sciences: Algebraic Topology and Its Interactions With Representation Theory

虽然在历史上是截然不同的，但代数拓扑学、群论和表示论之间的互动越来越多。代数K-理论和关于有限群的分类空间的工作就是很好的例子。库恩教授长期以来一直对这种重叠感兴趣，最近他的工作是将Steenrod代数技术与群表示理论联系起来。在这里，他将本着这种精神研究各种课题。其中一个项目是使用他的表示理论工具来研究一些非常经典的拓扑实现问题。另一种是使用同样的工具来研究代数K理论中的最新问题。第三种是将代数K-理论方法应用于群环同构问题。这三个部分的细节各不相同，但都涉及将几何信息归结为用于计算的主题，或完善用于此目的的主要代数工具之一。涉及的几何信息的性质是困难的症结所在。虽然关于长度、面积、角度、体积等的问题实际上需要归结为计算，但它与几何对象的拓扑属性有很大的不同。这些属性包括连通性(完好无损)、多节、无洞等。所有对这些性质的系统研究，例如，如何区分两个几何物体在这些性质中的一个是否真的不同，或者仅仅是表面上的不同，或者如何对可能发生的各种不同进行分类，所有这些只有在归结为计算的问题时才被真正理解和掌握。代数K-理论已经发展成为实现这一目的的主要工具，代数和所涉及的拓扑学之间的相互作用仍然是一个迷人的课题。

Although historically distinct, algebraic topology, group theory, and representation theory have increasingly interacted. Algebraic K-theory and work on classifying spaces of finite groups are good examples of this. Professor Kuhn has long been interested in such overlaps, most recently in his work relating Steenrod algebra technology to group representation theory. Here he will study various topics in this spirit. One project consists of using his representation theoretic tools to study some very classical types of topological realization questions. Another consists of using these same tools to study state-of-the-art questions in algebraic K-theory. A third involves applying algebraic K- theoretic methods to the group ring isomorphism problem. The details of these three parts vary, but all are concerned either with reducing geometric information to a subject for calculation or to perfecting one of the principal algebraic tools used for this purpose. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation. Algebraic K-theory has been developed into a major tool for this purpose, and the interplay between the algebra and the topology involved remains a fascinating subject.