Uniformization of Metric Spaces and Quasiconformal Removability

这个项目的目标是开发理解分形空间几何学的方法和几何工具。分形空间出现在许多自然现象的描述中，如闪电、植物和晶体的生长模型、雪花、海岸线和河流网络。该项目计划研究的问题适用于在不丢失信息的情况下将三维信息(风景、人脸、人脑表面)存储在二维图像中。虽然在“平滑”对象(不是使用分形学建模的对象)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。这个项目的目的是发展关于这种分形空间的数学理论。本项目的另一个重点是对可移动分形集的研究。有时，分形集看起来像是其他“平滑”对象的边界，对于许多问题来说，知道这些分形图是可移除的是很有用的，因为出于某些目的，可以忽略它们的存在。分形集的可去性在需要将两个函数、两个动力系统或两个曲面“粘合”在一起的数学问题中有着广泛的应用，并且可以更好地理解物理中的动力学系统。本项目由三部分组成，涉及Sierpinski地毯的均匀化，二维度量曲面的均匀化，以及保形映射的分形集的可去性问题。在前面工作的基础上，PI将研究与正方形Sierpinski地毯的齐化有关的问题，PI将研究已知的拟对称或离散拟共形的均匀化映射的正则性。PI还将研究均匀映射下Hausdorff维度扭曲的相关问题，以及抽象Sierpinski地毯的平面均匀理论的推广。另一个焦点是二维度量曲面的均匀化问题。在这个方向上，PI将研究欧几里德空间对二维曲面均匀化定理的可能推广，并集中精力削弱现有的几何假设。最后，PI将致力于扩展早期关于分形集可去除性的结果，找到分形集(类似于Sierpinski垫片或地毯)不可去除性的拓扑准则，研究Sobolev可去除性和共形可去除性的等价性，并探索可去除性与圆域刚性问题的联系。该奖项反映了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。

The goal of this project is to develop methods and geometric tools for understanding the geometry of fractal spaces. Fractal spaces appear in description of many natural phenomena such as in lightning bolts, growth models of plants and crystals, snowflakes, coastlines, and river networks. The questions that the project 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 (objects that are not modeled using fractals) the corresponding mathematical theory is well understood, this is not the case for fractal objects, which require the development of new techniques. This project aims to develop mathematical theory for such fractal spaces. Another focus of this project is the study of removable fractal sets. Fractal sets appear sometimes as boundaries of otherwise "smooth" objects, and for many problems it is useful to know that these fractals are removable, in the sense that their presence can be ignored for some purposes. Removability of fractal sets has 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 consists of three parts, concerning the uniformization of Sierpinski carpets, the uniformization of two-dimensional metric surfaces, and the problem of removability of fractal sets for conformal maps. Continuing earlier work, the PI will study problems related to the uniformization of Sierpinski carpets by square Sierpinski carpets and the PI will study the regularity of the uniformizing map, which is already known to be quasisymmetric or discrete quasiconformal. The PI will also work in questions related to Hausdorff dimension distortion under the uniformizing map and in generalizations of this planar uniformization theory to abstract Sierpinski carpets. Another focus is the problem of uniformization of two-dimensional metric surfaces. In this direction, the PI will investigate possible generalizations of uniformization theorems for two-dimensional surfaces by Euclidean space and concentrate efforts on weakening the existing geometric assumptions. Finally, the PI will work on extending earlier results on the removability of fractal sets, by finding topological criteria for fractal sets (resembling the Sierpinski gasket or carpet) to be non-removable, studying the equivalence of Sobolev removability and conformal removability, and exploring the connections of removability to the problem of rigidity of 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.