Mathematical Sciences: Manifolds, Stratified Spaces, and Controlled Topology

该项目涉及受控的 拓扑学到流形理论和分层空间。 的 特别重要的是一个程序，用于发现几何 分层空间的结构 问题是要理解 奇异集的邻域 据了解，这样一个 邻里有“泪滴结构”。“一个积极的解决方案 猜想将使人们能够扩展到分层空间许多 基本结果以前只知道流形或 尤其是严格的分层。 其他应用包括 广义Novikov拓扑，拓扑作用于 流形，以及流形间映射的改进。 的 在这项研究中使用的中心工具是流形理论 近似纤维化，以前由主要分类 调查员和他的同事。 这个项目的一个方面是 将这项工作扩展到分层设置。 流形是拓扑学的主要研究对象。 他们是 这些空间是局部欧几里得空间。 最终目标 是对流形及其对称性进行分类。 受控拓扑 已成为高分子研究中最有力的新技术 自外科理论发展以来，三维流形。 更戏剧性的是，受控拓扑正在充分展示其 在研究奇异流形中的力量。 这些 分层空间包括多面体，复代数簇， 流形上群作用的轨道空间，以及 黎曼流形

This project is concerned with applications of controlled topology to the theory of manifolds and stratified spaces. Of particular significance is a program for discovering the geometric structure of stratified spaces. The problem is to understand the neighborhood of the singular set. It is conjectured that such a neighborhood has "teardrop structure." A positive solution to this conjecture would enable one to extend to stratified spaces many fundamental results previously known only for manifolds or for especially rigid stratifications. Other applications include generalized Novikov conjectures, groups acting topologically on manifolds, and the improvement of maps between manifolds. The central tool to be used in this research is the theory of manifold approximate fibrations, previously classified by the principal investigator and his collaborators. One aspect of this project is to extend that work to the stratified setting. Manifolds are the main object of study in topology. They are the spaces which are locally euclidean spaces. The ultimate goal is to classify manifolds and their symmetries. Controlled topology has become the most powerful new technique in the study of high- dimensional manifolds since the development of surgery theory. Even more dramatically, controlled topology is showing its full force in the study of manifolds with singularities. These stratified spaces include polyhedra, complex algebraic varieties, orbit spaces of group actions on manifolds, and certain limits of Riemannian manifolds.