Metric, computational, and stochastic questions in topology

拓扑学是研究几何物体在变形下保持的全局性质的学科；它最近在蛋白质折叠、材料科学、高维数据分析和机器人等领域得到了应用。一个特别成功的方法，自20世纪50年代以来产生了无数的结果，是计算代数不变量，然后通过代数方法研究。然而，在许多情况下，这种转换问题的方式隐藏了一些固有的几何复杂性——例如，可以将一个对象变形为另一个对象，但只能在中间的某个地方使其变得非常复杂。在这种情况下，从物理的、面向应用的角度来看，变形的存在可能不是特别有意义。相反，在其他情况下，人们总能找到一种相当直接的变形，从而验证了代数方法在应用中的使用。这个项目的目的就是调查这些现象。本项目的更广泛影响包括为研究生提供研究训练机会。过去几年在定量同伦理论领域取得了重大进展：理解对象的复杂性，通常由Lipschitz常数测量，例如映射和同伦，其存在是由代数拓扑保证的。这种进步在单连通空间中尤为明显。目前的项目将主要通过两种方式利用这一进展。第一个目标是获得几何拓扑主题的定量结果，如浸入、嵌入和手术理论。由于这些主题通常是通过约简到同伦理论来研究的，这也需要对这些约简有更多的几何理解。第二个目标是应用定量同伦理论的思想来研究幂零群的几何，一个几何群论、亚黎曼几何和度量空间分析的交叉区域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Topology is the study of global properties of geometric objects which are preserved under deformation; it has recently found applications in areas such as protein folding, materials science, analysis of high-dimensional data, and robotics. A particularly successful approach, which has produced innumerable results since the 1950's, is computing algebraic invariants which are then studied through algebraic means. However, in many situations this way of transforming the problem hides some inherent geometric complexity - for example, one can deform one object to another, but only by making it very complicated somewhere in the middle. In such a case, the existence of a deformation may not be particularly meaningful from a physical, application-oriented point of view. In other cases, in contrast, one can always find a reasonably straightforward deformation, validating the use of algebraic methods for applications. The purpose of this project is to investigate these phenomena. Broader impacts of this project include research training opportunities for graduate students.The past few years have seen significant progress in the area of quantitative homotopy theory: understanding the complexity, typically measured by the Lipschitz constant, of objects, such as maps and homotopies, whose existence is guaranteed by algebraic topology. This progress has been particularly pronounced for simply connected spaces. The current project will seek to harness this progress mainly in two ways. The first goal is to obtain quantitative results about topics in geometric topology such as immersions, embeddings, and surgery theory. Since these topics are usually studied via reduction to homotopy theory, this will also require a more geometric understanding of these reductions. The second goal to apply the ideas of quantitative homotopy theory to studying the geometry of nilpotent groups, an area at the intersection of geometric group theory, sub-Riemannian geometry, and analysis on metric spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.