Stratifications, Ends, and Controlled Topology

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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.