Uniformization of Surfaces and Mapping Problems in Metric Spaces

古典几何和微积分在很大程度上涉及到平滑变化的函数和空间。然而，现实世界中的对象通常并不平滑。现代数学的洞见之一是，许多非光滑物体可以像光滑物体一样被彻底研究和理解。更重要的是，这样的研究往往会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学研究中，通常都将曲率作为一个基本概念进行研究。曲率是衡量空间“弯曲”程度的一种指标。这个项目的目的是最大限度地了解几何空间的结构，而不依赖于曲率和其他标准假设。这样的尝试有内在的利益，也受到邻近学科的推动，如复杂动力学和几何群论，这些空间自然地出现在这些学科中。非光滑几何还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为本科生研究提供机会的问题。这个项目植根于Klein，Poincaré和Koebe等人发展的经典的均匀化定理，该定理指出任何光滑的曲面都可以共形映射到常曲率的曲面上。这一定理给出了一幅简单而全面的曲面几何图景，是19世纪大部分数学的结晶。该项目有两个主要目标：第一个是发展潜在非光滑度量空间的一致化定理的版本：确定一个空间在具有良好几何性质的映射下何时可以映射到另一个空间，例如拟共形、拟对称或双Lipschitz映射。这延续了主要研究人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。这种近似方案有更多的潜在应用，有待探索。第二个目标是研究与不同几何映射类有关的各种附加问题。其中包括双Lipschitz映射的因式分解，以及Lipschitz映射和拟对称映射的扩张。这些问题捕捉到了公制空间和它们之间的地图的基本方面。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.