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旨在开发更深入地理解分形的技术；即形状不光滑并可能具有尖端和皱纹的对象，或具有可能自相似重复图案的对象。这类物体在自然界中表现为海岸线、山地景观、河流网络、闪电、雪花、植物和晶体的生长模型，以及肥皂片。PI计划研究的问题适用于任何需要在二维图像中存储三维信息(风景、人脸、人脑表面)而不丢失信息的情况。虽然在光滑对象(与分形学相反)的情况下，相应的数学理论被很好地理解，但对于需要开发新技术的分形物来说，情况并非如此。这个项目的另一个焦点是刚性问题，询问是否有可能将一个由柔性材质制成的分形对象变形为另一个具有受控扭曲的分形对象。此外，分形集有时显示为其他平滑对象的边界；另一个刚性问题涉及这些分形图是否可移除，因为在变换目的中可以忽略它们的存在。分形集上的刚性问题在需要将两个函数或两个动力系统或两个曲面粘合在一起的数学问题中有应用，并可能导致更好地理解物理中的动力系统。该项目还将包括研究生的培训和专业发展。该项目的主要焦点是关于分形学的两个相互关联的问题：统一化问题和刚性问题。一致化问题要求在一个分形度量空间上满足几何条件，这样它就可以变换到一个光滑空间，具有保持几何性质的良好变换，如拟共形或拟对称映射。最近在PI参与下的拟共形均匀化问题方面取得了重大进展。目前的项目期望在没有其他假设的情况下发展局部有限区域的二维曲面的解析理论；相反，度量空间上的分析领域的经典方法需要几个附加的和限制性的几何假设。具体地说，PI将研究非光滑曲面的拟共形分类，分形曲面在欧几里德空间中的嵌入，无限区域的二维球体的均匀化，以及分形曲面上的位势理论。关于刚性问题，PI将致力于共形可去除性问题，该问题询问给定的欧几里德空间的紧子集是否从共形映射域中忽略。在最近的著作中，PI展示了几个新的可移除和不可移除平面集的例子，并发现了一致性问题和可移除问题之间的显著联系。此外，PI已经确定了一类新的一般集合，他猜想这类集合提供了可去集合的特征。PI将研究这一猜想，以及复杂动力学、几何群论和圆域中的几个相关的可移动性和刚性问题。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。