Mathematical Sciences: Manifolds and Algebraic K-Theory

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This research project has three parts. The first involves some ideas about the homotopy type of diffeomorphisms of manifolds beyond the stable range, in which these are related to algebraic K-theory. The second is a program for understanding the K-theory of group rings in terms of cyclic subgroups. The third is concerned with applications of the cyclotomic trace to K(Z) and to A(S1). The details of these three parts vary, but all are concerned either with reducing geometric information to a subject for calculation or to perfecting one of the principal algebraic tools used for this purpose. The nature of the geometric information involved is the crux of the difficulty. 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. Algebraic K-theory has been developed into a major tool for this purpose, and the interplay between the algebra and the topology involved remains a fascinating subject.