Parametrization, Embedding and Extension Problems in Metric Spaces

度量空间中的参数化、嵌入和扩展问题

基本信息

批准号：
1952510
负责人：
Vyron Vellis
金额：
$ 8.57万
依托单位：
University of Tennessee Knoxville
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1952510&HistoricalAwards=false
关键词：
Parametrization Embedding Extension Problems Metric

项目摘要

Geometric function theory is a field of mathematics that was developed starting in the 1920s in order to study analytic functions from a geometric point of view, and was later developed to what is known today as analysis of metric spaces. The advantage of a geometric approach, is that first order differential calculus and geometric measure theory can be extended from the classical Euclidean or Riemannian settings to the realm of spaces without a priori smooth structure (such as fractal spaces). Results and techniques in geometric function theory have recently found important applications in geometric group theory, structure of manifolds and analysis on fractals. Furthermore, besides their mathematical importance, physical applications of these theories include reconstruction theory, study of thin films, control theory, graphic imaging and analysis of large data sets.This project features new approaches to three long-standing problems in the realm of geometric function theory that bring together several fields in analysis and geometry including geometric topology, sub-Riemannian geometry, PL geometry and geometric measure theory. The first problem aims at recognizing the intrinsic qualities of a metric space, from which a "nice" parametrization (e.g. quasisymmetric, Holder, bi-Lipschitz) by the Euclidean unit sphere or the Euclidean space can be recovered. The principal investigator proposes to relate forms of discrete curvature with global parametrizations in high dimensions. The second problem asks for conditions for which an embedding of a set into a Euclidean space with some desired properties (e.g. quasisymmetric, bi-Lipschitz) can be extended to the whole Euclidean space with the same properties. Finally, the third problem concerns the bi-Lipschitz embedability of big sets of a sub-Riemannian manifolds (such as the Heisenberg group) into some Euclidean space. Results in this direction will shed new light on the structure of the space and will improve our understanding of its geometry.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.

几何函数理论是从 20 世纪 20 年代开始发展的一个数学领域，旨在从几何角度研究解析函数，后来发展到今天所谓的度量空间分析。几何方法的优点是，一阶微分微积分和几何测度理论可以从经典的欧几里得或黎曼设置扩展到没有先验平滑结构的空间领域（例如分形空间）。几何函数理论的结果和技术最近在几何群论、流形结构和分形分析中得到了重要的应用。此外，除了数学重要性之外，这些理论的物理应用还包括重构理论、薄膜研究、控制理论、图形成像和大数据集分析。该项目的特色是解决几何函数领域三个长期存在的问题的新方法该理论汇集了分析和几何中的多个领域，包括几何拓扑、亚黎曼几何、PL 几何和几何测度理论。第一个问题旨在认识度量空间的内在品质，从中可以恢复欧几里得单位球面或欧几里得空间的“良好”参数化（例如拟对称、霍尔德、双利普希茨）。首席研究员建议将离散曲率的形式与高维度的全局参数化联系起来。第二个问题要求将具有某些所需属性（例如拟对称、双李普希茨）的集合嵌入到欧几里德空间中可以扩展到具有相同属性的整个欧几里德空间的条件。最后，第三个问题涉及亚黎曼流形（例如海森堡群）的大集合到某些欧几里得空间的双利普希茨嵌入性。这个方向的结果将为空间结构提供新的线索，并提高我们对其几何形状的理解。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。