Mathematical Sciences: Topics in Topology

这个项目涉及拓扑学的广泛主题 代数及相关领域，代数K理论和代数 几何Weinberger研究了分层的外科理论， 空间，使用分析以及拓扑技术。 Rothenberg研究等变手术理论等方面 流形上的离散群作用 来访者马德森是 在这个项目上合作。 5月研究的各个领域 等变代数拓扑，并与Kriz，已经开始工作， 代数几何中的动机 斯旺致力于解决 与代数和拓扑向量的研究 bundles，Kriz已经从组合学的一般领域 现在正在研究代数拓扑学中的各种问题， 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，但都是 在某种程度上， 信息传递给受试者进行计算。 的性质 其中涉及的几何信息是问题的关键。 而 关于长度、面积、角度、体积等的问题 实际上哭出来被减少到计算，这是远远 与已知的拓扑性质不同， 几何物体 这些属性比如连通性 （都在一块），打结，没有洞，等等 向前。 所有系统的研究，例如，如何 来判断两个几何对象是否真的在 这些属性之一，或者只是表面上的不同，或者 如何对可能发生的各种差异进行分类， 只有当他们真正理解和掌握这些东西时， 已经被简化为计算问题。 近世代数 它描述了许多工具和对工具的态度 但是代数和数学之间的相互作用 拓扑学仍然是一个迷人的主题。它反过来揭示了 现代物理学中的一些微妙问题， 当物理学提供了一种新颖而富有成效的方式来看待 拓扑结构。

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.