Mathematical Sciences: Some Problems on the Interface Between Geometry and Topology

数学科学：几何与拓扑接口的一些问题

• 批准号: 9401058
• 负责人: F. Thomas Farrell
• 金额: $ 12.9万
• 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: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401058&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Problems; Interface

9401058 Farrell F.T. Farrell in collaboration with L.E. Jones (SUNY at StonyBrook) will attempt to verify their conjectured calculation of the algebraic K- and L-theories of the integral group ring of a given group in terms of the K- and L-theories of the integral group rings of a class of subgroups of that group. This class consists of all the virtually cyclic subgroups of the given group. They envision applications of this work to differential geometry, in particular, to questions of rigidity. The objective is to classify geometric objects by numerical invariants. The geometric objects are Riemannian manifolds, spaces which locally resemble Euclidean space except that the notion of distance may be warped, and they arise naturally and ubiquitously in the mathematical formulation of physical theories. Thus, as is familiar from the general theory of relativity, a space can be curved; e.g., it could be a sphere or a torus (doughnut surface). Some basic numerical invariants of such a space are its fundamental group and its sectional curvatures. For example, the curvature of the torus changes continuously from point to point, varying between positive and negative numbers, whereas the curvature of the sphere of radius 1 is identically 1. Farrell and Jones have shown that two closed negatively curved Riemannian manifolds with the same fundamental group invariant are homeomorphic; i.e., there is a continuous one-to-one mapping between their points. (The case of 3- and 4-dimensional manifolds is still open.) But they gave examples where the mapping can never be smooth. A search will be made for additional numerical invariants guaranteeing a smooth mapping, which is highly significant for any physical applications. ***

9401058 Farrell F.T.法雷尔与L.E.合作琼斯（Stonybrook的SUNY）将试图根据给定组的积分组环的代数K和L理论的猜想计算，以一类的积分组的K-和L理论计算。该组的子组。 该类由给定组的所有实际循环子组组成。 他们将这项工作应用于差异几何形状，特别是将其应用于僵化问题。 目的是通过数值不变性对几何对象进行分类。 几何对象是Riemannian歧管，除了可以扭曲距离的概念，并且它们在物理理论的数学表述中自然而无处不在。 因此，从相对论的一般理论中熟悉，空间可以弯曲。例如，它可能是一个球体或圆环（甜甜圈表面）。 这样一个空间的一些基本数值不变是其基本组及其截面曲率。 例如，圆环的曲率从点到点不断变化，在正数和负数之间有所不同，而半径1的球的曲率​​相同1。Farrell和Jones表明，两个闭合的负面弯曲的Riemannian歧管，同一弯曲的Riemannian歧管相同基本组不变是同型的；即，他们的点之间有一个连续的一对一映射。 （3维和4维流形的情况仍然是开放的。）但是它们给出了映射永远无法平滑的示例。 将进行搜索，以确保平滑映射的其他数值不变性，这对于任何物理应用都非常重要。 ***