Mathematical Sciences: Homotopy Theory and its Applications

Ravenel将研究稳定同伦中的几个问题 他的反例提出的理论望远镜猜想。 他还有一个程序可以计算 Eilenberg-MacLane空间和计算空间的一些想法 Lambda代数的上同调 Neisendorfer计划追求 发现的事实，直到完成，有限复形可能 从他们相连的封面上找到 他还打算 调查保存和缺乏保存 复盖空间中的同调有限性 这个项目涉及的工具，减少几何 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， 对可能发生的各种差异进行分类，所有这些 只有当他们真正理解和掌握， 减少到计算的问题。 现代代数使许多 以及对所需工具的态度， 代数和拓扑之间的相互作用 很有趣的话题

Ravenel will investigate several questions in stable homotopy theory raised by his counter-example to the telescope conjecture. He also has a program to compute the complex cobordism of Eilenberg-MacLane spaces and some ideas about computing the cohomology of the Lambda algebra. Neisendorfer plans to pursue the discovery of the fact that, up to completion, finite complexes may be recovered from their connected covers. He also intends to investigate the preservation and lack of preservation of homological finiteness in covering spaces. This project is concerned with tools for 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, and the interplay between the algebra and the topology remains a fascinating subject.