Metric, computational, and stochastic questions in topology

拓扑学是研究几何对象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，这已经产生了无数的结果，自20世纪50年代以来，是计算代数不变量，然后通过研究代数手段。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、面向应用的观点来看，变形的存在可能不是特别有意义的。相反，在其他情况下，人们总是可以找到一个相当简单的变形，验证使用代数方法的应用。本项目的目的就是研究这些现象。在过去的几年里，在定量同伦理论领域取得了重大进展：理解对象的复杂性，通常用Lipschitz常数来衡量，例如映射和同伦，它们的存在性由代数拓扑保证。 这一进展在单连通空间中尤为明显。本项目将主要通过两种方式来利用这一进展。 第一个目标是获得有关几何拓扑学主题的定量结果，如浸入，嵌入和手术理论。 由于这些主题通常通过同伦理论的简化来研究，这也需要对这些简化有更多的几何理解。 第二个目标是将定量同伦理论的思想应用于幂零群的几何研究，这是几何群论、亚黎曼几何和度量空间分析的交叉领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.