Quantitative problems in Topology

这是一个旨在研究关于各种范数的拓扑问题(族)的解的大小的建议。它的纯度量方面被称为“受控拓扑学”，并在拓扑刚性、分层空间(特别是奥比流形)和同调流形方面有许多应用，这些将继续被研究。然而，其他涉及更少或更多导数的范数对于应用于分析(特别是变分问题)和微分几何(例如，系统微商)是重要的-这些将与它们的应用同时被研究。一般而言，拓扑学在传统上应用时，主要被用作定性工具。然而，可以将拓扑学思想与分析思想混合在一起，并使用时尚工具来衡量事情是可能的还是不可能的，而且还可以衡量事情的难度有多大。在这样做的过程中，拓扑学接触到了数学的其他分支：特别是分析和概率。人们希望这将从而丰富所有这些领域。

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.