Mathematical Sciences: Manifolds and Algebraic K-Theory

本研究项目分为三个部分。 第一个涉及 的同伦型的一些想法 流形以外的稳定范围，其中这些都涉及到 代数K理论 第二个是理解的程序 用循环子群表示的群环的K-理论。 的 第三是关于分圆迹在 K（Z）和A（S1）。 这三个部分的细节各不相同，但都是有关的 或者通过减少对象的几何信息 计算或完善一个主要的代数 用于此目的的工具。 几何的本质 信息化是难点所在。 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。 代数k-理论 已发展成为实现这一目标的主要工具， 所涉及的代数和拓扑之间的相互作用仍然是 很有趣的话题

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.