Mathematical Sciences: Homotopy Theory and its Applications

数学科学：同伦理论及其应用

• 批准号: 9204291
• 负责人: Douglas Ravenel
• 金额: $ 25万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-01 至 1996-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204291&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Homotopy; Theory; its

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.

Ravenel将在其反示例对望远镜猜想提出的稳定同义理论中调查几个问题。 他还制定了一个计划，以计算Eilenberg-Maclane空间的复杂COBORDISM，以及有关计算Lambda代数的共同体学的一些想法。 Neisendorfer计划追求一个事实，即在完成时，可以从其连接的封面中回收有限的综合体。 他还打算调查覆盖空间中同源有限的保存和保存。 该项目与将几何信息减少到主题进行计算的工具有关。 涉及的几何信息的性质是问题的关键。 虽然关于长度，区域，角度，体积等的问题实际上呼唤到计算中，但与几何对象的拓扑特性相差很大。 这些都是连接性（一件一件），打结，没有孔等的属性。 例如，所有对此类属性的系统研究，例如，如何分辨两个几何对象在这些属性之一方面是否确实有所不同，或者仅在表面上是不同的，或者如何分类可能发生的差异的种类，只有在降低到计算问题的情况下，所有这些才真正理解和掌握。 现代代数提供了许多工具和对所需工具的态度，而代数与拓扑之间的相互作用仍然是一个引人入胜的主题。