Uniformization and Rigidity in Metric Surfaces and in the Complex Plane

在这个项目中，PI旨在开发更深层次理解分形的技术；也就是说，物体的形状不光滑，可能有尖头和皱纹，或者物体可能具有自相似的重复图案。这些物体在自然界中出现，如海岸线、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膜。PI计划研究的问题在二维图像中存储三维信息（风景，面孔，人脑表面）而不丢失信息的情况下具有应用价值。在光滑物体（与分形相反）的情况下，相应的数学理论被很好地理解，而分形物体的情况并非如此，这需要开发新技术。这个项目的另一个重点是刚性问题，询问是否有可能将由柔性材料制成的分形物体变形为另一个分形物体，并控制变形。此外，分形集有时作为光滑物体的边界出现；另一个刚性问题涉及这些分形是否可移动，在某种意义上，它们的存在可以被忽略以进行转换。分形集上的刚性问题在需要“粘合”两个函数、两个动力系统或两个表面的数学问题中有应用，并且可以更好地理解物理学中的动力系统。该项目还将包括研究生的培训和专业发展。该项目的主要重点是分形的两种相互关联的问题：均匀化和刚性问题。均匀化问题要求分形度量空间上的几何条件，使其可以通过保持几何形状的良好变换（如拟共形映射或拟对称映射）转换为光滑空间。近年来，在PI的参与下，准共形均匀化问题取得了重大进展。目前的项目期望在没有其他假设的情况下发展局部有限面积的二维曲面的解析理论；度量空间分析领域的经典方法需要一些附加的和限制性的几何假设。具体而言，PI将研究非光滑表面的拟共形分类，分形表面在欧几里得空间中的嵌入，无限面积的二维球体的均匀化以及分形表面的势理论。关于刚性问题，PI将研究保形可移性问题，它询问欧几里得空间的给定紧子集在保形映射的域上是否可以忽略。PI在最近的作品中展示了几个可移动和不可移动平面集的新例子，并发现了均匀化和可移动问题之间的惊人联系。此外，PI已经确定了一个新的一般集合类，他推测它提供了可移动集合的表征。PI将研究这一猜想，以及复杂动力学、几何群论和圆域中几个相关的可移性和刚性问题。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。

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.