Lipschitz Analysis in Normed and Metric Spaces

距离或物体的不相似性的概念在几何、数据分析、图像和信号处理以及其他学科中无处不在。自然地，与之相联系的是变换的概念，它使距离至多扭曲一个固定的量。例如，为了直观地表示高维数据，需要减少维数，同时保持距离的低失真。另一个例子是图像恢复，其目标是生成几何上自然的图像（具有清晰的线条和低噪声），与给定的噪声图像具有很小的差异。纠错码的信号恢复也依赖于距离最小化的概念：损坏的比特序列被最近的符合编码规范的序列所取代。首席研究员将向研究生和本科生介绍这些概念和相关的研究问题。该项目旨在推进对度量空间的李普希茨几何的理解，重点是它们的嵌入、撤回和双李普希茨等价。它的目的是解决有关有限子集空间相互缩回的可能性的开放问题；线性控制双lipschitz映射的对称和扩展过程，向量从其框架系数的绝对值鲁棒恢复的可行性，可移动集和拟凸集的几何，度量空间的低维表示与其紧跨几何的关系。一系列的方法从分析，几何，几何测量理论将被采用，如梯度流的度量空间和支持它们的概率测度，曲率边界的空间的比较估计，几何映射理论的构造方法，以及变分法应用于几何动机的问题。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。

The notion of distance, or dissimilarity of objects, is ubiquitous in geometry, data analysis, image and signal processing, and other disciplines. Naturally associated to it is the notion of transformations that distort the distances by at most a fixed amount. For example, in order to represent high-dimensional data visually, one needs to reduce the number of dimensions while keeping the distortion of distances low. Another example is image recovery, where the goal is to produce a geometrically natural image (with crisp lines and low noise) that has a small measure of dissimilarity from a given noisy image. Signal recovery by error-correcting codes depends on the concept of distance minimization as well: a corrupted sequence of bits is replaced by the nearest sequence that conforms to the specifications of the encoding. The principal investigator will introduce graduate and undergraduate students to these concepts and associated research problems.The project aims to advance the understanding of the Lipschitz geometry of metric spaces, focusing on their embeddings, retractions, and bi-Lipschitz equivalence. It aims to solve open problems concerning the possibility of retracting finite subset spaces onto each other; the process of symmetrizing and extending bi-Lipschitz maps with linear control of their distortion, the feasibility of robust recovery of a vector from the absolute values of its frame coefficients, the geometry of removable sets and quasi-convex sets, and the relation of low-dimensional representations of a metric space with the geometry of its tight span. An array of methods from analysis, geometry, and geometric measure theory will be employed, such as gradient flows on metric spaces and probability measures supported on them, comparison estimates on spaces with curvature bounds, constructive methods of geometric mapping theory, and the calculus of variations as applied to geometrically motivated problems.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.