Stratifications and Ends of Spaces

拓扑学作为几何学的一个分支，致力于对被称为流形的抽象空间进行分类、刻画和探索。流形局部类似于普通的欧几里德空间(直线、平面等)；然而，它们被允许具有全局扭曲、弯曲和孔(例如，圆、球面、环面)。它们出现在许多物理和生物现象的模型中。具有奇点的流形或分层空间更为普遍，因为它们表现为代数方程和微分方程式的解空间，以及流形的极限、退化和紧化。休斯通过建立奇点邻域理论，在理解拓扑分层空间方面取得了突破性进展。这使得人们可以使用几何技术，几乎就像奇点不存在一样。目前的项目主要致力于利用这一技术来进一步理解具有奇点的流形。作为一个例子，Hughes将研究奇异集附近的周期性程度。建议的研究涉及流形、分层空间、树和度量空间的拓扑和几何。研究领域包括分层空间、树、超度量和非对易几何以及代数K-理论的基本定理。主要工具有控制拓扑学、外科理论和群胚的C*-代数。具体问题涉及流形分层空间奇异集的邻域周期性，分层h-余弦和分层伪同余的分类，以及非局部平坦拓扑嵌入。此外，非对易几何的技巧将被应用于研究无穷大树和其他在无穷远处的非紧空间的几何。

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.