Uniformization and Rigidity in Metric Surfaces and in the Complex Plane

公制曲面和复平面中的均匀化和刚度

• 批准号: 2246485
• 负责人: Dimitrios Ntalampekos
• 金额: $ 23.98万
• 依托单位: SUNY at Stony Brook
• 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: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246485&HistoricalAwards=false
• 关键词: Uniformization; Rigidity; Metric; Surfaces; Complex

In this project, the PI aims to develop techniques for the deeper understanding of fractals; that is, objects whose shape is not smooth and potentially have cusps and wrinkles, or objects with possibly self-similar repeating patterns. Such objects appear in nature as coastlines, mountainous landscapes, river networks, lightning bolts, snowflakes, growth models of plants and crystals, and soap films. The questions the PI plans to study have applications whenever storage of three-dimensional information (landscapes, faces, human brain surface) in a two-dimensional image is desired without loss of information. While in the case of smooth objects (the opposite of fractals) the corresponding mathematical theory is well understood, this is not the case for fractal objects, which require the development of new techniques. Another focus of this project is on rigidity problems, asking whether it is possible to deform a fractal object that is made out of a flexible material into another fractal object, with controlled distortion. Also, fractal sets appear sometimes as boundaries of otherwise smooth objects; another rigidity problem concerns whether these fractals are removable, in the sense that their presence can be ignored for transformation purposes. Rigidity problems on fractal sets have applications in mathematical problems that require "gluing" together two functions, or two dynamical systems, or two surfaces, and could result in the better understanding of dynamical systems in physics. This project will also incorporate the training and professional development of graduate students. The main focus of the project is on two interrelated types of problems on fractals: uniformization and rigidity problems. The uniformization problem asks for geometric conditions on a fractal metric space so that it can be transformed to a smooth space with a well-behaved transformation that preserves the geometry, such as quasiconformal or quasisymmetric maps. Major progress has been made recently towards the quasiconformal uniformization problem with the involvement of the PI. The current project expects to develop an analytic theory for two-dimensional surfaces of locally finite area under no other assumption; the classical approaches in the field of analysis on metric spaces require instead several additional and restrictive geometric assumptions. Specifically, the PI will study the quasiconformal classification of non-smooth surfaces, the embedding of fractal surfaces in Euclidean space, the uniformization of 2-dimensional spheres of infinite area, and potential theory on fractal surfaces. Regarding rigidity problems, the PI will work on the problem of conformal removability, which asks whether a given compact subset of Euclidean space is negligible from the domain of a conformal map. The PI in recent works has displayed several new examples of removable and non-removable planar sets and has found a striking connection between the problems of uniformization and removability. Moreover, the PI has identified a new general class of sets that he conjectures to provide a characterization of removable sets. The PI will study this conjecture, as well as several related removability and rigidity problems in complex dynamics, geometric group theory, and circle domains.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.

在这个项目中，PI旨在开发对分形更深入了解的技术。也就是说，其形状并不光滑且可能具有尖端和皱纹的对象，或具有可能自相似重复模式的对象。这些物体出现在自然界中，例如海岸线，山区景观，河网，闪电，雪花，植物和晶体的生长模型以及肥皂膜。每当需要在二维图像中存储三维信息（景观，面部，人脑表面）时，PI计划研究的问题就会有应用。在平滑对象（与分形的对面）的情况下，相应的数学理论是充分理解的，但对于需要开发新技术的分形对象并非如此。该项目的另一个重点是僵化问题，询问是否有可能将柔性材料制成的分形对象变形为另一个分形对象，并具有受控的失真。同样，分形集有时会作为平滑对象的边界出现。另一个僵化的问题涉及这些分形是否可以去除，从某种意义上说，它们的存在被忽略是出于转化目的。分形集中的僵化问题在数学问题中具有应用，这些问题需要“粘合”两个功能，或两个动态系统或两个表面，并且可能会更好地理解物理学中的动态系统。该项目还将纳入研究生的培训和专业发展。该项目的主要重点是关于分形的两种相互关联的问题：均匀化和刚性问题。统一的问题要求在分形度空间上使用几何条件，以便可以将其转化为平稳的空间，并具有表现良好的变换，以保留几何形状，例如准形式或准对称地图。最近，与PI参与的准形式均匀化问题最近取得了重大进展。当前的项目预计将在没有其他假设的情况下为局部有限区域的二维表面开发一种分析理论。公制空间分析领域中的经典方法需要几个其他限制性的几何假设。具体而言，PI将研究非平滑表面的准表面分类，欧几里得空间中分形表面的嵌入，无限面积的二维球的均匀化以及分形表面上的潜在理论。关于刚性问题，PI将在保形可移动性问题上工作，该问题询问从保形图的域中可以忽略一个给定的紧凑型欧几里得空间子集。最近的工作中的PI显示了几个新的可移动和不可易换平面集的示例，并发现了均匀化和可移动性问题之间存在惊人的联系。此外，PI已经确定了他猜想的新的一般集合，以提供可移动集的特征。 PI将研究该猜想，以及复杂动态，几何组理论和圆形领域中的几种相关的可移动性和刚性问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估来通过评估来支持的。