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具有应用程序，只要在二维图像中存储三维信息（景观，面孔，人脑表面），而不会丢失信息。虽然在“平滑”对象（未使用分形建模的对象）的情况下，相应的数学理论是很好理解的，但分形对象的情况并非如此，需要开发新技术。这个项目旨在发展这种分形空间的数学理论。本项目的另一个重点是可去分形集的研究。分形集有时会作为其他“光滑”物体的边界出现，对于许多问题来说，知道这些分形是可移动的是有用的，在某种意义上，它们的存在可以被忽略。分形集的可移除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，并且可以使人们更好地理解物理学中的动力系统。本项目包括三个部分，分别涉及Sierpinski地毯的一致化、二维度量曲面的一致化和共形映射的分形集的可移除性问题。继续早期的工作，PI将研究与方形Sierpinski地毯的Sierpinski地毯的均匀化有关的问题，PI将研究均匀化映射的规律性，这是已知的准对称或离散准共形。PI还将在均匀化映射下处理与Hausdorff维数失真相关的问题，并将此平面均匀化理论推广到抽象的谢尔宾斯基地毯。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将研究欧氏空间二维表面的一致化定理的可能推广，并集中精力削弱现有的几何假设。最后，PI将通过寻找分形集的拓扑准则来扩展早期关于分形集可移除性的结果（类似于Sierpinski垫圈或地毯）不可移除，研究Sobolev可移除性和适形可移除性的等效性，以及探索可移动性与圆域刚性问题之间的联系。该奖项反映了NSF的法定使命，并被认为值得支持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。

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.