Problems in Geometric and Quantitative Topology

该方案以一种跟踪问题解的大小的方式处理拓扑问题。其中包括位移(有界拓扑)、Lipschitz常数和扭曲的测量，以及相关的度量不变量。这些思想的应用出现在纯拓扑学、分层空间、同调流形、黎曼几何、几何群论和理论计算机科学中。该提案涉及来自这些领域的一系列问题，其中一些与奇点和群体行动有关，涉及改进和应用先前开发的工具，而另一些问题则受到刚性理论的推动，以及似乎需要新方法的格罗莫夫猜想。拓扑学通常被认为是本质上是定性的数学领域。当前一些核心问题的进展需要对相关问题的解决方案进行更精确的量化控制，而这些问题在前几代人中是通过强大但不明确的代数方法解决的。这种控制使拓扑学与其他数学领域以及计算机科学和工程中的应用程序更紧密地交互。这项提议意味着要解决其中的一些问题。

This proposal deals with topological problems in a way that keeps track of sizes of solutions to the problems. These include measurements of displacement (bounded topology), of Lipschitz constants and distortions, and related metric invariants. Applications of such ideas arise within pure topology, stratified spaces, homology manifolds, Riemannian geometry, geometric group theory, and theoretical computer science. The proposal involves a mix of problems from these areas, some related to singularities and group actions which involve refining and applying previously developed tools, while others are motivated by rigidity theory, and conjectures of Gromov that seem to require new methods.Topology is ordinarily thought of as an area of mathematics that is qualitative in nature. Progress on a number of current central problems requires more precise quantitative control of the solutions of related problems that were solved in earlier generations by powerful but inexplicit algebraic methods. Such control brings topology in closer interaction with applications in other fields of mathematics as well as computer science and engineering. This proposal suggests an attack on some of these problems.