Quantitative, Computational, and Stochastic Aspects of 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和高维数据分析等领域得到了应用。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。该项目将通过证明三种类型的结果来丰富我们对几何拓扑思想的理解：定量结果，在各种意义上测量通过代数已知存在的物体的大小和复杂性；算法结果，表明某些问题可以通过算法解决，而其他问题则不能；随机结果，描述随机物体的性质。人们可以想到许多这样的结果，如回答关于函数和模空间的几何问题。密切相关的问题在几何群论，结论，展开图的组合学，和拓扑数据分析理论研究。然而，高维单连通流形和复合体的研究需要一套不同的工具，而这些工具才刚刚开始发展。由于函数空间在数学中无处不在，这个项目提供了从理论计算机科学到流体动力学的许多数学领域的联系。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 and analysis of high-dimensional data. 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.The project will enrich our understanding of the ideas of geometric topology by proving results of three types: quantitative results, measuring the size and complexity, in various senses, of objects whose existence is known via algebra; algorithmic results, showing that certain problems can be resolved algorithmically while others cannot; and stochastic results, describing the properties of random objects. One can think of many such results as answering questions about the geometry of function and moduli spaces. Closely related questions are studied in geometric group theory, knot theory, the combinatorics of expander graphs, and the theory of topological data analysis. However, the study of high-dimensional and simply connected manifolds and complexes requires a different set of tools which have only begun to be developed. Since function spaces are found all over mathematics, this project offers connections to a number of mathematical fields ranging from theoretical computer science to fluid dynamics.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.