Stratifications and Ends of Spaces

拓扑学作为几何学的一个分支，其目的是对流形这种抽象空间进行分类、刻画和探索。流形在局部上类似于普通的欧几里得空间（直线、平面等）;但是允许它们具有整体扭曲、弯曲和孔（例如，圆、球体、环面）。它们出现在许多物理和生物现象的模型中。具有奇点的流形，或称分层空间，在代数和微分方程的解空间以及流形的极限、退化和紧化中更是无处不在。Hughes建立了奇点邻域理论，在拓扑分层空间的理解上取得了突破。这使得人们可以使用几何技术，几乎就像奇点不存在一样。目前的项目主要是利用这种技术，以进一步了解流形的奇点。 作为一个例子，Hughes将研究奇异集附近的周期性的程度。拟议的研究涉及流形，分层空间，树和度量空间的拓扑和几何。调查的领域是分层空间，树，超度量和非交换几何和基本定理的代数K理论。 主要工具是控制拓扑学、外科理论和广群的C*-代数。具体问题涉及周期性的奇异集附近的一个流形分层空间，分层h-cobordisms和分层pseudositopies的分类，和非局部平坦的拓扑嵌入。此外，从非交换几何的技术将应用于研究几何的无限树和其他非紧空间在无穷远。

Topology, as a branch of geometry, seeks to classify, characterize and explore those abstract spaces known as manifolds. Manifolds are locally like ordinary euclidean spaces(the line, the plane, etc.); however, they are allowed to have global twisting, curving and holes (e.g., circles, spheres, tori). They arise in many models of physical and biologicalphenomena. Manifolds with singularities, or stratified spaces, are even more ubiquitous as they appear as solution spaces of algebraic and differential equations and as limits, degenerations and compactifications of manifolds. Hughes has made a breakthrough in the understanding of topologically stratified spaces by establishing a theory of theneighborhoods of the singularities. This allows one to use geometric techniques almost as if the singularities were not present. The current project is largely concerned with exploiting this technique to further the understanding of manifolds with singularities. As an example, Hughes will investigate the extent of periodicity near the singular sets.The proposed research concerns the topology and geometry of manifolds, stratified spaces, trees and metric spaces. The investigations are in the areas of stratified spaces, trees, ultrametrics and noncommutative geometry and the fundamental theorem of algebraic K-theory. The main tools are controlled topology, surgery theory and C*-algebras of groupoids. Specific questions concern periodicity in the neighborhood of the singular set of a manifold stratified space, the classification of stratified h-cobordisms and stratified pseudoisotopies, and non-locally flat topological embeddings. In addition, techniques from noncommutative geometry will be applied to study the geometry of infinite trees and other non-compact spaces at infinity.