Mathematical Sciences: Some Problems in Algebraic and Geometric Topology

数学科学：代数和几何拓扑中的一些问题

• 批准号: 9104026
• 负责人: Erik Pedersen
• 金额: $ 10.14万
• 依托单位: SUNY at Binghamton
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-06-01 至 1995-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9104026&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Problems; Algebraic

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.

研究者计划研究代数和几何拓扑学以及代数k理论中的问题。这包括研究度量空间参数化的k理论，研究作用于球面的无穷群，研究某些无穷群的l群，以及研究受控型外科理论。拓扑学最初是作为研究微分方程的一种方法而出现的。在这中间的一个世纪里，它发生了惊人的进化，似乎经常有一个完全独立于它的来源的生命。然而，它已经开始越来越多地偿还债务。关键是，知道一个方程的解，比如它是一个闭环还是一个打结，往往比精确地、数值地知道它所有点的坐标更重要。此外，尽管前一种定性信息隐含在后一种完整的描述中，但如何提取它却并不明显。由拓扑学家开发的复杂机制，如本项目中正在研究的外科小组和代数k理论，现在已经发展到一定程度，通常允许以自然的方式处理这些自然问题。