Parametrization, Embedding and Extension Problems in Metric Spaces

几何函数论（英语：Geometric function theory）是数学的一个领域，从20世纪20年代开始发展，目的是从几何的角度研究解析函数，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光滑结构的空间（如分形空间）。几何函数论的结果和技巧在几何群论、流形结构和分形分析中有着重要的应用。此外，除了数学上的重要性，这些理论的物理应用包括重建理论，薄膜研究，控制理论，图形成像和大数据集的分析。这个项目的特点是对几何函数理论领域中三个长期存在的问题的新方法，这些问题汇集了分析和几何中的几个领域，包括几何拓扑学，亚黎曼几何，PL几何与几何测度理论。第一个问题的目的是认识度量空间的内在性质，从中可以恢复由欧几里得单位球或欧几里得空间的“好”参数化（例如，准对称，保持器，双Lipschitz）。主要研究者建议将离散曲率的形式与高维的全局参数化联系起来。第二个问题要求的条件下，嵌入到一个欧几里德空间的一组与一些所需的性质（如拟对称，双Lipschitz）可以扩展到整个欧几里德空间具有相同的性质。最后，第三个问题是关于次黎曼流形（如海森堡群）的大集到某个欧氏空间的双Lipschitz嵌入性。这一方向的成果将为空间的结构提供新的线索，并将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。

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.