Uniformization of Metric Spaces and Quasiconformal Removability

度量空间的均匀化和拟共形可去除性

基本信息

批准号：
2000096
负责人：
Dimitrios Ntalampekos
金额：
$ 9.55万
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：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000096&HistoricalAwards=false
关键词：
Uniformization Metric Spaces Quasiconformal Removability

项目摘要

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.

该项目的目的是开发理解分形空间几何形状的方法和几何工具。分形空间出现在许多自然现象的描述中，例如在闪电，植物和晶体的生长模型，雪花，海岸线和河流网络中。每当需要在二维图像中存储三维信息（风景，面部，人脑表面）时，该项目计划研究的问题就会有应用。在“平滑”对象（未使用分形建模的对象）的情况下，相应的数学理论是充分理解的，但对于需要开发新技术的分形对象并非如此。该项目旨在为这种分形空间开发数学理论。该项目的另一个重点是研究可移动分形集。分形集有时会作为其他“光滑”对象的边界出现，并且对于许多问题而言，知道这些分形是可移动的，从某种意义上说，出于某些目的可以忽略它们的存在。分形集合的可移动性在数学问题中有应用，这些问题需要“粘合”两个功能，或两个动态系统或两个表面，并且可能会更好地理解物理学中的动态系统。该项目由三个部分组成，使得Sierpinski地毯的均匀化统一的统一范围均事范围，并将其均置入了两大方面的问题。继续进行较早的工作，PI将研究与Square Sierpinski地毯对Sierpinski地毯统一化的问题，PI将研究均匀图的规律性，该地图的规律性已知已知是准对称或离散的准圆顶形式。 PI还将在与Hausdorff尺寸扭曲的问题有关的问题中起作用，在均匀的地图和该平面统一理论对抽象的Sierpinski地毯上的概括中。另一个重点是二维度量表面均匀化的问题。在这个方向上，PI将通过欧几里得空间研究统一定理的统一定理的可能概括，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（类似于Sierpinski垫圈或地毯）的拓扑标准来扩展分形集合的可移动性的早期结果，以研究sobolev sobolev的可移动性和保质可移动性的范围，并探索cression croment of cirdive croment of Circle的连接的等效性并被认为是通过基金会的知识分子优点和更广泛的影响审查标准来评估值得支持的。