Mathematical Sciences: Some Problems in Algebraic and Geometric Topology

研究人员计划研究代数和几何拓扑以及代数K-理论中的问题。其中包括研究度量空间参数化的K-理论，研究作用在球交欧氏空间上的无限群，研究某些无限群的L群，以及研究受控类型的外科理论。拓扑学最初是作为研究微分方程的一种方法出现的。在其间的一个世纪里，它发生了惊人的演变，往往看起来完全有自己的生命，独立于其来源。然而，它已经越来越多地开始偿还债务。问题的关键在于，例如，知道一个方程的解是否为闭合环或是否打结，往往比准确和数值地知道其所有点的坐标更为重要。此外，虽然前者的定性信息隐含在后者的完整描述中，但如何提取它却一点也不明显。由拓扑学家开发的复杂机械，例如这个项目中正在研究的外科手术小组和代数K理论，现在已经发展到了通常允许以自然方式处理这些自然问题的程度。

The investigator plans to study problems in algebraic and geometric topology and algebraic K-theory. These include studying K-theory parameterized by metric spaces, studying infinite groups acting on a sphere cross euclidean space, studying the L-groups of certain infinite groups, and studying controlled-type surgery theories. Topology arose originally as an approach to the study of differential equations. Over the intervening century it has evolved prodigiously, often seeming to have a life entirely its own and independent of its source. Increasingly, however, it has begun to repay its debt. The point is that it is often more important to know of the solution of an equation whether it is a closed loop or whether it is knotted, for example, than to know precisely and numerically what are the coordinates of all its points. Furthermore, although the former qualitative information is implicitly contained in the latter complete description, it is not at all obvious how to extract it. The sophisticated machinery that has been developed by topologists, such as the surgery groups and algebraic K-theory which are under investigation in this project, has now been developed to a degree that often permits these natural problems to be handled in a natural way.