Mathematical Sciences: Topics in Topology

本项目涉及拓扑和代数、代数k理论和代数几何相关领域的广泛主题。Weinberger使用解析和拓扑技术研究分层空间的外科理论。Rothenberg研究流形上离散群作用的等变手术理论和其他方面。来访者Madsen正在合作这个项目。May研究了等变代数拓扑的各个领域，并与Kriz一起开始研究代数几何中的动机。Swan研究与k理论有关的问题，以及代数和拓扑向量束的研究，Kriz已经从组合学的一般领域转移到代数拓扑的各种问题，包括模p同伦理论的代数方法。这些不同项目的细节各不相同，但都以这样或那样的方式将几何信息简化为计算的主题。所涉及的几何信息的性质是问题的关键。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。现代代数提供了许多工具和对所需工具的态度，但代数和拓扑之间的相互作用仍然是一个迷人的主题。它反过来又揭示了现代物理学中的一些微妙问题，最近，物理学提供了一种新颖而富有成效的方法来研究拓扑学，这一点得到了回报。

This project deals with a wide range of topics in topology and related areas of algebra, algebraic K-thoery and algebraic geometry. Weinberger studies the surgery theory of stratified spaces, using analytic as well as topological techniques. Rothenberg studies equivariant surgery theory and other aspects of discrete group actions on manifolds. The visitor, Madsen, is collaborating on this project. May studies various areas of equivariant algebraic topology and, with Kriz, has begun work on motives in algebraic geometry. Swan works on problems connected with K-thoery and the study of algebraic and topological vector bundles, Kriz has moved from the general areas of combinatorics and is now studying various problems in algebraic topology, including an algebraic approach to mod p homotopy theory. The details of these various projects vary, but all are concerned in one way or another with reducing geometric information to a subject for calculation. The nature of the geometric information involved is the crux of the problem. 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. Modern algebra furnishes many of the tools and the attitudes toward the tools that are needed, but the interplay between the algebra and the topology remains a fascinating subject. It has in turn shed light on subtle problems in modern physics, and this has recently been repaid when physics provided a novel and fruitful way to look at the topology.