Mathematical Sciences: Some Problems in Algebraic and Geometric Topology

研究人员计划研究代数和 几何拓扑学和代数K理论。 这些包括 研究度量空间参数化的K-理论，研究 作用在球面上的无限群穿过欧氏空间， 研究某些无限群的L-群， 控制型手术理论 拓扑学最初是作为一种研究 微分方程 在其后的世纪里， 进化惊人，往往似乎有一个完全属于自己的生活， 独立于自己的来源。 然而， 开始偿还债务。 关键是， 一个方程的解是否是一个 封闭的循环或它是否打结，例如，比知道 精确地和数字地， 点 此外，虽然前者的质量信息 隐含地包含在后面的完整描述中， 一点也不清楚如何提取。复杂的 机器已经开发的拓扑学家，如 外科手术组和代数K理论， 在这个项目的调查，现在已经发展到一定程度， 这通常允许这些自然问题被处理在一个 自然的方式。

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.