Uniformization and Rigidity in Metric Surfaces and in the Complex Plane

在这个项目中，PI的目标是开发更深入理解分形的技术;也就是说，形状不光滑并且可能有尖点和皱纹的对象，或者可能具有自相似重复模式的对象。这些物体在自然界中出现为海岸线、山区景观、河流网络、闪电、雪花、植物和晶体的生长模型以及肥皂膜。PI计划研究的问题具有应用程序，只要在二维图像中存储三维信息（景观，人脸，人脑表面），而不会丢失信息。虽然在光滑物体（分形的反面）的情况下，相应的数学理论是很好理解的，但分形物体的情况并非如此，这需要开发新的技术。该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对象变形为另一个分形对象，并控制变形。此外，分形集有时表现为光滑物体的边界;另一个刚性问题涉及这些分形是否是可移动的，在这个意义上，它们的存在可以忽略变换的目的。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，并且可以导致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两个相互关联的问题类型的分形：均匀化和刚性问题。一致化问题要求分形度量空间上的几何条件，以便它可以通过保持几何形状的良好变换（如拟共形或拟对称映射）变换到光滑空间。最近取得了重大进展的准共形一致化问题的参与PI。目前的项目预计将开发一个分析理论的局部有限区域的二维表面下没有其他的假设;在度量空间的分析领域的经典方法，而不是需要几个额外的和限制性的几何假设。具体来说，PI将研究非光滑表面的拟共形分类，分形表面在欧氏空间中的嵌入，无限区域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将研究共形可移除性问题，该问题询问欧几里得空间的给定紧致子集是否可以从共形映射的域中忽略。PI在最近的作品中展示了几个新的例子，可移动和不可移动的平面集，并发现了一个惊人的联系之间的问题，一致化和可移动性。此外，PI已经确定了一个新的一般类的集合，他的结构提供了一个特征的可移动集。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可移动性和刚性问题。该奖项反映了NSF的法定使命，并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，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.