Uniformization of Surfaces and Mapping Problems in Metric Spaces

度量空间中曲面的均匀化和映射问题

• 批准号: 2413156
• 负责人: Matthew Romney
• 金额: $ 14.81万
• 依托单位: Stevens Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-11-15 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2413156&HistoricalAwards=false
• 关键词: Uniformization; Surfaces; Mapping; Problems; Metric

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.

经典几何学和微积分主要关注函数和空间的平滑变化。然而，真实的世界中的物体通常不是光滑的。现代数学的见解之一是，许多非光滑物体可以像它们的光滑对应物一样被彻底研究和理解。更重要的是，这样的研究往往澄清和简化以前已知的经典理论。几何学的研究，无论是在光滑还是非光滑的情况下，通常都涉及到曲率--一种空间“弯曲”的度量--作为一个基本概念。这个项目的目的是了解几何空间的结构在最大的一般性，而不依赖于曲率和其他标准的假设。这样的事业有内在的兴趣，也是由邻近的主题，如复杂动力学和几何群论，这些空间自然产生的动机。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。 这个项目也包含了一系列的问题，将提供机会，本科生的研究。这个项目是植根于经典的一致化定理开发的克莱因，庞加莱和Koebe，其中指出，任何光滑的表面可以共形映射到一个表面的常曲率。这个定理给出了一个简单而全面的图片几何曲面，是顶点的大部分19世纪数学。该项目有两个主要目标：第一个是为潜在的非光滑度量空间开发一致化定理的版本：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，例如拟共形映射，拟对称映射或双Lipschitz映射。这延续了主要研究者早期的工作，使用一种新的多面体近似方案作为主要方法。这种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地图的不同几何类相关的各种附加问题。其中包括双Lipschitz映射的因子分解以及Lipschitz映射和拟对称映射的扩展。这个奖项反映了NSF的法定使命，并被认为值得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来支持。