Mathematical Sciences: Homotopy Theory and its Applications

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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.