Mathematical Sciences: Algebraic Topology and Its Interactions With Representation Theory

虽然历史上不同，代数拓扑，群论和表示理论日益相互作用。代数k理论和对有限群的分类空间的研究就是很好的例子。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，他将本着这种精神学习各种课题。一个项目包括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。另一个包括使用这些相同的工具来研究代数k理论中最先进的问题。第三个是将代数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简化为一个计算主题，或者完善用于此目的的主要代数工具之一。所涉及的几何信息的性质是难点的关键。虽然关于长度、面积、角度、体积等问题实际上迫切需要简化为计算，但这与几何对象的拓扑特性大不相同。这些属性包括连通性（整体）、打结性、无孔等等。所有对这些性质的系统研究，例如，如何判断两个几何物体在其中一个性质上是否真的不同，或者只是表面上的不同，或者如何对可能出现的各种差异进行分类，所有这些只有在它们被简化为计算问题时才真正被理解和掌握。代数k理论已经发展成为这一目的的主要工具，代数和拓扑之间的相互作用仍然是一个迷人的主题。

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.