Directions in Quantitative Topology

Weinberger计划发展已知的代数拓扑学和几何拓扑学的定量版本，将受控拓扑学的基本思想扩展到更光滑和更不光滑的方向。在某种程度上，这与逻辑和计算机科学相互作用，如在Dehn函数的分析中，以及Kolmogorov复杂性论点在变分问题中的应用，但其目标是获得特定(可解)拓扑问题的更现实的上下界。辅助任务是理解几何问题的“基于信息的复杂性”，并通过古德威利演算的哲学来理解分类问题中出现的各种集合的增长率。这样做的动机来自许多方向，既有应用方面的，也有理论上的。在将拓扑学应用于实验科学时，人们很少有明确的空间可供研究：一个人有大量的数据集，研究人员必须推断出一个潜在的空间。在数学定理中经常作为假设出现的特征不能简单地看到：它们必须被检测到。一个需要研究的问题是如何确定，比方说，这样一个底层空间的维度。紧随其后的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。另一类问题来自应用数学和几何的许多方程中奇点的存在。由于拓扑学通常局限于对连续映射的研究，各种不连续性的含义可能会有潜在的用处。最后，在许多纯几何问题中，只被证明存在，但从未被“看到”的解的详细结构将是非常有价值的。用定量估计来反驳它们的存在，应该会导致更好地理解它们的性质。

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.