Stratifications, Ends, and Controlled Topology

该项目涉及流形、分层空间、无限树和超尺度空间的拓扑和几何。主要的工具是控制拓扑学、外科理论、非交换几何和群拟的C*-代数。主要的焦点是无限树和其他非紧流形分层空间在无穷远处的几何。将讨论的具体问题包括由树的末端空间产生的局部相似群的Baum-Connes猜想，流形分层空间奇异集邻域的周期结构，分层h-协群的分类，以及非正弯曲流形上的控制拓扑。最后一个问题与诺维科夫猜想有关。拓扑学试图对那些被称为流形的抽象空间进行分类、表征和探索。流形局部类似于普通的欧几里得空间（直线、平面、三维空间等）；然而，它们被允许具有全局扭曲，弯曲和孔（例如，圆，球，环面）。流形出现在许多物理和生物现象的模型中。具有奇点的流形，或者说是分层空间，当它们作为代数方程和微分方程的解空间，以及流形的极限和紧化出现时，就更加普遍了。数学树是分层空间的一维情况，用于模拟分支过程。有许多不对称无限树，然而它们有许多相似的无限子树。这个项目的目标之一是使用一个相当新的数学分支——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，我们将研究无穷远处分层空间的对称性和周期性结构。

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.