Directions in Quantitative Topology

温伯格计划开发代数和几何拓扑已知定理的定量版本，将受控拓扑的基本思想扩展到或多或少平滑的方向。 在某种程度上，这与逻辑和计算机科学相互作用，如德恩函数的分析以及柯尔莫哥洛夫复杂性论证在变分问题中的应用，但目标是在特定的（可解的）拓扑问题上获得更现实的上限和下限。 附属任务是理解几何问题的“基于信息的复杂性”，并通过古德威利微积分的哲学理解分类问题中出现的各种集合的增长率。 这样做的动机来自多个方向，包括应用和理论。 在拓扑学在实验科学中的应用中，人们很少有明确的研究空间：拥有大量数据集，研究人员必须推断出潜在的空间。 数学定理中经常作为假设出现的特征不能简单地被看到：它们必须被检测到。 需要研究的一个问题是如何确定这样一个底层空间的尺寸。 接下来是实施软件和了解需要多少数据才能做出可靠估计的问题。 另一类问题来自于许多应用数学和几何方程中存在奇点。 由于拓扑通常仅限于连续映射的研究，因此各种不连续性的含义可能会有用。 最后，在许多纯几何问题中，仅被证明存在但从未“见过”的解的详细结构将具有很大的价值。 通过定量估计来证明它们的存在应该可以更好地理解它们的本质。

Weinberger plans to develop both quantitative versions of known theorems of algebraic and geometric topology, extending the basic ideas of controlled topology into directions of both more and less smoothness. In part, this interacts with logic and computer science, as in the analysis of Dehn functions, and applications of Kolmogorov complexity arguments to variational problems, but the goal is to get more realistic upper and lower bounds on specific (solvable) topological problems. A subsidiary task is to understand the "information based complexity" of geometric problems, and understand the rates of growth of various sets arising in classification problems via the philosophy of the Goodwillie calculus. The motivation for this comes from a number of directions, both applied and theoretical. In applications of topology to experimental sciences, one rarely has as explicit space to study: one has large data sets, and the investigator must infer an underlying space. The features that often arise as hypotheses in mathematical theorems cannot simply be seen: they must be detected. One problem to be studied is how to determine, say, the dimension of such an underlying space. After that come problems of implementing software and understanding how much data one needs to make reliable estimates. Another sort of problem comes from the existence of singularities in many equations of applied mathematics and geometry. As topology usually confines itself to the study of continuous maps, the implication of discontinuities of various sorts could potentially be of use. Finally, in many problems of pure geometry, the detailed structure of solutions which have only been proved to exist, but which have never been "seen" would be of great value. Reproving their existence with quantitative estimates should lead to a better understanding of their nature.