Quantitative problems in Topology

这是一个提案，旨在研究（族）拓扑问题的解决方案的大小相对于各种规范。这方面的纯度量方面被称为“控制拓扑”，并在拓扑刚性、分层空间（特别是轨道）和同调流形方面有许多应用，这些将继续被研究。然而，涉及较少或更多导数的其他规范对于分析（特别是变分问题）和微分几何（例如系统）的应用很重要-这些将与它们的应用同时研究。一般来说，传统的拓扑学应用主要是作为一种定性工具。然而，可以将拓扑思想与分析思想混合在一起，并设计出不仅可以衡量事情是可能还是不可能的工具，还可以衡量它们的难度。在此过程中，拓扑学与数学的其他分支有了联系：特别是分析和概率论。希望这将因此丰富所有这些领域。

This is a proposal that aims to study the sizes of solutions to (families of) topological problems with respect to various norms. The purely metric aspect of this goes under the name of "controlled topology" and has had numerous applications to topological rigidity, stratified spaces (especially orbifolds), and homology manifolds and these will continue to be studied. However, other norms involving fewer or more derivatives are important for applications to analysis (especially variational problems) and differential geometry (e.g. systoles) --- these will be studied simultaneously with their applications.In general, topology, when applied traditionally, has been used mainly as a qualitative tool. However, it is possible to mix topological ideas with analytic ones, and fashion tools that measure not only whether things are possible or impossible, but rather how difficult they are. In so doing, topology makes contact with other branches ofmathematics: notably analysis and probability. It is hoped that this will thereby enrich all these fields.