Mathematical Sciences: Controlled Topology, Group Actions and Stratified Spaces

9504759休斯这个项目应用受控拓扑学的技术来解决关于流形的问题。所讨论的流形有两种类型。第一种是配备了通向圆圈的自然地图的歧管。在过去的三十年里，人们一直在研究这些流形的特殊性质，但最终受控拓扑学(以一种参数化身)揭示了对所有先前研究的现象建立大统一理论的可能性。这个项目的观点是将映射到一个圆圈的所有流形组装成一个大空间。受控拓扑可用于将该空间分解为各种几何和代数上有意义的分量。这些技巧表明，研究映射到非正曲率的固定流形的所有流形的空间将导致新的现象。这里的要点是，圆是最简单的非正曲率流形。正在考虑的第二种类型的流形是那些配备了一组对称的流形。当这个群作用在流形上时，就会产生一个“轨道空间”，它实际上是一个分层空间。分层将轨道空间分解成多个部分。由于轨道空间本身并不是一个流形，在了解流形部分如何组合在一起之前，几何方法是不能应用的。该项目最重要的方面之一是揭示了用于连接歧管部件的几何结构。这种结构不仅可以进行分类和量化，而且还可以用于研究分层空间的新几何工具的开发。地层邻域上的结构是受控拓扑学范畴中的一个丛。因此，受控拓扑学成为变换和应用经典流形技术来研究这些更复杂的奇点流形的工具。流形是在适当的EUC李德空间中局部类似于球的几何对象。球体或甜甜圈可以作为低维流形的常见例子。高维流形并不像人们最初认为的那样深奥和非物理，因为维度对应于自由度。因此，描述许多粒子的物理系统的普通和/或微分方程式的解对于每个粒子具有三个维度，并且系统的相关定性行为(而不是详细的定量行为)经常被拓扑捕获。行星轨道的这种性质，如闭合，而不是像某些彗星那样退缩到无穷远，是拓扑性质。这个项目的部分内容涉及通过考虑轨道而获得的流形的分解或分层，以及从流形片段重建流形。***

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. ***