Mathematical Sciences: Controlled Topology, Group Actions and Stratified Spaces

数学科学：受控拓扑、群作用和分层空间

• 批准号: 9504759
• 负责人: C. Bruce Hughes
• 金额: $ 14万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-09-15 至 1999-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504759&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Controlled; Topology; Group

9504759 Hughes This project applies techniques from controlled topology to solve problems about manifolds. The manifolds in question are of two types. The first are manifolds equipped with a natural map to the circle. Special properties of these manifolds have been investigated for the past three decades, but finally controlled topology (in a parametrized incarnation) reveals the possibility of a grand unified theory for all the previously investigated phenomena. The point of view of this project is to assemble all manifolds which map to a circle into one large space. The controlled topology can be used to factor this space into various geometrically and algebraically significant components. The techniques suggest that a study of the space of all manifolds which map to a fixed manifold of nonpositive curvature would lead to new phenomena. The point here is that the circle is the simplest manifold of nonpositive curvature. The second type of manifolds under consideration are those equipped with a group of symmetries. When this group acts on the manifold, there results an "orbit space" which is, in fact, a stratified space. The stratification presents a decomposition of the orbit space into manifold pieces. Since the orbit space is not itself a manifold, geometric methods cannot be applied until it is understood how the manifold pieces fit together. One of the most important aspects of the project is the uncovering of the geometric structure used to join together the manifold pieces. This structure is not only subject to classification and quantification, but can also be applied for the development of new geometric tools for studying stratified spaces. The structure on the neighborhoods of the strata is a bundle in the category of controlled topology. Thus, controlled topology becomes the vehicle for transforming and applying classical manifold techniques to study these more intricate manifolds with singularities. Manifolds are geometric objects that locally resemble balls in a suitable Euc lidean space. A sphere or a doughnut can serve as a familiar example of a low dimensional manifold. High dimensional manifolds are not nearly so esoteric and unphysical as one might at first believe, since dimensions correspond to degrees of freedom. Accordingly, solutions to ordinary and/or differential equations describing physical systems of many particles have three dimensions for each particle, and frequently relevant qualitative behavior of the system (as opposed to detailed quantitative behavior) is captured by the topology. Such properties of planetary orbits as being closed, instead of receding to infinity like those of some comets, are topological properties. Portions of this project deal with decompositions or stratifications of manifolds obtained by considering orbits and the reconstruction of the manifold from manifold pieces. ***

9504759休斯这个项目应用了从受控拓扑的技术来解决有关歧管的问题。所讨论的流形有两种类型。第一个是装有自然图到圆的歧管。在过去的三十年中，已经研究了这些歧管的特殊特性，但最终控制拓扑（在参数化的化身中）揭示了所有先前研究的现象的大统一理论的可能性。该项目的观点是将所有歧管组装成一个圆形成一个大空间。受控拓扑可用于将该空间分为各种几何和代数显着的组成部分。这些技术表明，对所有歧管的空间进行的研究将映射到非阳性曲率的固定歧管将导致新现象。这里的一点是，圆是非阳性曲率的最简单歧管。所考虑的第二种歧管是配备一组对称性的歧管。当该组作用于歧管时，结果“轨道空间”实际上是一个分层空间。该分层将轨道空间的分解为歧管碎片。由于轨道空间本身不是一种歧管，因此在理解歧管碎片如何拟合在一起之前，几何方法无法应用。该项目最重要的方面之一是揭开用于将歧管碎片结合在一起的几何结构。这种结构不仅需要分类和定量，而且还可以用于开发用于研究分层空间的新几何工具。地层附近的结构是受控拓扑类别的捆绑包。因此，受控拓扑成为转换和应用经典的歧管技术来研究这些更复杂的奇异歧管的工具。 歧管是在合适的EUC LIDEAN空间中局部类似球的几何对象。球体或甜甜圈可以作为低维歧管的熟悉例子。高维流形并不像最初认为的那样深奥和非物理，因为尺寸与自由度相对应。 因此，描述许多粒子物理系统的普通和/或微分方程的解决方案对于每个粒子都有三个维度，并且拓扑捕获了系统的经常相关的定性行为（与详细的定量行为相反）。 行星轨道的这种特性是被封闭的，而不是像某些彗​​星那样向无限恢复到无穷大，是拓扑特性。 该项目的一部分涉及通过考虑轨道和从歧管碎片重建歧管获得的歧管的分解或分层。 ***