Stratifications, Ends, and Controlled Topology

该项目涉及流形，分层空间，无限树和超度量空间的拓扑和几何。 主要的工具是控制拓扑，外科手术理论，非交换几何，和C*-代数的广群。主要的重点是几何的无限树和其他非紧流形分层空间在无穷远。具体的问题，将解决包括鲍姆-康纳斯猜想的局部相似广群所产生的终端空间的树木，周期性结构的奇异集附近的一个流形分层空间，分层h-cobordisms的分类，并控制拓扑非正弯曲的流形。最后一个问题与诺维科夫式拓扑有关。拓扑学试图对那些被称为流形的抽象空间进行分类、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、三维空间等）;但是允许它们具有整体扭曲、弯曲和孔（例如，圆、球体、环面）。流形出现在物理和生物现象的许多模型中。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，以及流形的极限和紧化中，更是无处不在。数学树是分层空间的一维情况，用于建模分支过程。有许多不对称的无限树，但有许多类似的无限子树。这个项目的目标之一是使用一个相当新的数学分支，称为非交换几何，来研究无限树的这些非全局对称性。更一般地说，分层空间在无穷远处的对称性和周期性结构将被研究。

The project concerns the topology and geometry of manifolds, stratified spaces, infinite trees, and ultrametric spaces. The main tools are controlled topology, surgery theory, noncommutative geometry, and C*-algebras of groupoids. The main focus is on the geometry of infinite trees and other non-compact manifold stratified spaces at infinity. Specific problems that will be addressed include the Baum-Connes conjecture for local similarity groupoids arising from the end spaces of trees, periodic structure in the neighborhood of the singular set of a manifold stratified space, the classification of stratified h-cobordisms, and controlled topology over nonpositively curved manifolds. This last problem is related to Novikov-like conjectures.Topology seeks to classify, characterize, and explore those abstract spaces known as manifolds. Manifolds are locally like ordinary euclidean spaces (the line, the plane, three-dimensional space, etc.); however, they are allowed to have global twisting, curving and holes (e.g., circles, spheres, tori). Manifolds arise in many models of physical and biological phenomena. Manifolds with singularities, or stratified spaces, are even more ubiquitous as they appear as solution spaces of algebraic and differential equations, and as limits and compactifications of manifolds. Mathematical trees are the one-dimensional case of stratified spaces and are used to model branching processes. There are many asymmetric infinite trees that nevertheless have many similar infinite subtrees. One of the goals of this project is to study these non-global symmetries of infinite trees using a fairly new branch of mathematics called noncommutative geometry. More generally, the symmetry and periodic structure of stratified spaces at infinity will be investigated.