Uniformization of Surfaces and Mapping Problems in Metric Spaces

经典几何和微积分主要关注平滑变化的函数和空间。然而，现实世界中的物体通常不是光滑的。现代数学的一个洞见是，许多非光滑的物体可以像它们的光滑对应物一样被彻底地研究和理解。更重要的是，这种研究倾向于澄清和简化先前已知的经典理论。在光滑和非光滑环境下的几何研究，通常都将曲率——一种空间“弯曲”的度量——作为一个基本概念。该项目的目的是在不依赖曲率和其他标准假设的情况下，最大限度地理解几何空间的结构。这样的工作具有内在的兴趣，也受到这些空间自然产生的复杂动力学和几何群论等邻近学科的推动。非光滑几何也出现在各种应用领域，包括理论计算机科学和数据科学。这个项目还包含了一系列的问题，将为本科生的研究提供机会。该项目源于Klein、poincar<s:1>和Koebe等人提出的经典均匀化定理，该定理指出，任何光滑表面都可以共形映射到恒定曲率的表面上。这个定理给出了一个简单而全面的曲面几何图像，是19世纪大部分数学的巅峰之作。该项目有两个主要目标：第一是为潜在的非光滑度量空间开发统一化定理的版本：确定一个空间何时可以在具有良好几何性质的映射下映射到另一个空间，例如拟共形、拟对称或双lipschitz映射。这继续了首席研究员使用一种新的多面体近似方案作为主要方法的早期工作。这个近似方案有进一步的潜在应用，将被探讨。第二个目标是研究与不同几何类型的地图相关的各种附加问题。这包括双Lipschitz映射的分解以及Lipschitz映射和拟对称映射的扩展。这些问题抓住了度量空间和它们之间的映射的基本方面。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。

Classical geometry and calculus largely concern functions and spaces that change smoothly. However, objects in the real world are usually not smooth. One of the insights of modern mathematics is that many non-smooth objects can be studied and understood just as thoroughly as their smooth counterparts. Even more, such study tends to clarify and simplify previously known classical theory. Research in geometry, in both the smooth and non-smooth settings, typically involves curvature—a measure of the “bending” of a space—as a fundamental notion. The aim of this project is to understand the structure of geometric spaces in maximum generality, without relying on curvature and other standard assumptions. Such an undertaking has intrinsic interest and is also motivated by neighboring subjects such as complex dynamics and geometric group theory where these spaces naturally arise. Non-smooth geometry also arises in a variety of applied fields, including theoretical computer science and data science. This project also incorporates a range of questions that will provide opportunities for undergraduate research.This project is rooted in the classical uniformization theorem developed by Klein, Poincaré and Koebe, among others, which states that any smooth surface can be mapped conformally onto a surface of constant curvature. This theorem gives a simple yet comprehensive picture of the geometry of surfaces and is the culmination of a large portion of 19th century mathematics. The project has two main goals: first is to develop versions of the uniformization theorem for potentially non-smooth metric spaces: to determine when one space can be mapped to another under a map with good geometric properties, such as a quasiconformal, quasisymmetric or bi-Lipschitz map. This continues earlier work of the principal investigator using a novel polyhedral approximation scheme as the main method. This approximation scheme has further potential applications which will be explored. The second goal is to investigate a variety of additional problems related to the different geometric classes of maps. These include the factorization of bi-Lipschitz maps and extensions of Lipschitz and quasisymmetric maps. These questions capture fundamental aspects of metric spaces and maps between them.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.