Mathematical Sciences: Algebraic Topology and Its Interactions With Representation Theory

虽然历史上不同，代数拓扑，群 理论和表征理论越来越多地相互作用。 代数K-理论与有限群空间的分类工作 都是很好的例子 库恩教授一直对 在这种重叠，最近在他的工作有关斯廷罗德 代数技术到群表示理论。 在这里，他将 本着这种精神研究各种课题。 一个项目包括使用 他的表示论工具来研究一些非常经典的 拓扑实现问题的类型。 另一个由 使用同样的工具来研究最先进的问题， 代数K理论 第三个是应用代数K- 群环同构问题的理论方法。 这三个部分的细节各不相同，但都是有关的 或者通过减少对象的几何信息， 计算或完善一个主要的代数工具 用于此目的。 几何信息的本质 问题的症结在于。 虽然关于 长度、面积、角度、体积等等，几乎都在呼喊着， 如果我们把它简化为计算，它与我们所知道的大不相同。 几何物体的拓扑性质。 这些是 诸如连通性（所有的都是一体的）之类的属性， 结，没有洞，等等。 所有系统研究 例如，如何判断两个几何 对象在这些属性之一方面确实不同，或者 只是表面上的不同，或者如何分类的品种， 可能发生的差异，所有这些都只是真正的 当它们被简化为 计算. 代数学K-理论已经发展成为一个主要的 工具，以及代数和 所涉及的拓扑结构仍然是一个令人着迷的主题。

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.