Mathematical Sciences: Topics in Topology

这个项目涉及拓扑学和代数、代数K理论和代数几何的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研究等变外科理论和流形上离散群作用的其他方面。参观者马德森正在合作这个项目。梅研究等变代数拓扑的各个领域，并与克里兹一起开始研究代数几何中的动机。Swan致力于与K-定理相关的问题以及代数和拓扑向量丛的研究，Kriz已经从组合数学的一般领域转移到现在研究代数拓扑中的各种问题，包括mod 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涉及到将几何信息减少到用于计算的主题。涉及的几何信息的性质是问题的症结所在。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。现代代数提供了许多工具和对所需工具的态度，但代数和拓扑学之间的相互作用仍然是一个令人着迷的主题。它反过来揭示了现代物理学中的微妙问题，最近当物理学提供了一种新颖而富有成效的方法来观察拓扑学时，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.