SGER: The Algebraic Topology of Random Fields and its Applications

该项目的核心是建立一个研究网络，最初由三个机构的四名研究人员组成，支持他们（Technion的Robert Adler、斯坦福大学的Jonathan Taylor、芝加哥的Shmuel Weinberger和基思沃斯利）之间的交流，更重要的是，支持他们的学生之间的交流。他们将研究与随机结构的几何特征有关的问题，例如随机场的偏移（节点）集的拓扑不变量，并在存在噪声的情况下学习流形的同伦和同胚类型，并开发量化所得估计的统计准确性的技术。鉴于以前的工作随机场几何集中在来自微分拓扑的特点，如Lipshitz-Killing曲率（其中包括欧拉特征），这个项目将集中在来自代数拓扑的特点，涉及同源不变的概念，如连接组件的数量，贝蒂数，和其他相关的不变量。此外，虽然以前在这些领域的工作是在黎曼流形的设置，计划将其扩展到允许非黎曼度量，即使在光滑流形上。在流形学习中，这一点很重要，因为在某些应用中，数据集背景下的自然贴近度度量很可能是非黎曼的。在随机场的背景下，它可能会打开纯非高斯场的研究新的大门，如稳定场，其中由场诱导的参数流形上的自然几何是非黎曼的。这个项目的动机，它连接代数拓扑，概率，纯统计和应用统计来自其内在的数学兴趣和丰富的应用，从功能磁共振成像脑图像的统计分析，计算机科学中的高维数据分析，以及宇宙学项目，如COBE和WMAP实验和斯隆数字巡天等广泛的领域。这四位PI来自不同的知识背景，是具有不同但互补的技术技能的专家。该项目不仅将加入这些技能，而且在未来两年内，还将增加一些其他研究人员，加强其国际组成部分，并扩展到美国年轻科学家的培训。

The core of this project is the setting up of a research network, initially between four researchers, in three institutions, supporting interchange between them (Robert Adler, Technion, Jonathan Taylor, Stanford, Shmuel Weinberger and Keith Worsley, Chicago) and, more importantly, their students. They will study problems related to the geometric characterization of random structures, such as topological invariants of excursion (nodal) sets of random fields and learning the homotopy and homeomorphism types of manifolds in the presence of noise and developing techniques for quantifying the statistical accuracy of the resulting estimates. Whereas previous work on random field geometry has concentrated on characteristics coming from differential topology, such as Lipshitz-Killing curvatures (which include the Euler characteristic) this project will concentrate on characteristics coming from algebraic topology, involving homology-invariant concepts such as the number of connected components, Betti numbers, and other related invariants. Furthermore, while previous work in these areas was in the setting of Riemannian manifolds, it is planned to extend this to allowing non-Riemannian metrics even on smooth manifolds. In manifold learning this is important due to the fact that in some applications natural measures of closeness underlying the context of a data set can well be non-Riemannian. In the context of random fields, it may open new doors in the study of purely non-Gaussian fields, such as stable fields, where the natural geometry on the parameter manifold induced by the field is non-Riemannian.The motivation for this project, which links algebraic topology, probability, and pure and applied statistics comes from both its intrinsic mathematical interest and from a wealth of applications, from areas as broadly spaced as the statistical analysis of fMRI brain images, high dimensional data analysis in computer science, and cosmological projects such as the COBE and WMAP experiments and the Sloan digital sky survey. The four PIs come from a variety of intellectual backgrounds and are specialists with different, but complementary technical skills. Not only will this project join these skills, but over the next two years a number of other researchers will be added to the project, strengthening, among other aspects, its international component and extending to the training of young scientists in the US.