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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.

该项目的目标是开发用于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，例如闪电、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需要在二维图像中存储三维信息（风景、面孔、人脑表面）且不丢失信息的情况。虽然在“平滑”对象（不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，这需要开发新技术。该项目旨在开发此类分形空间的数学理论。该项目的另一个重点是可移动分形集的研究。分形集有时表现为“平滑”对象的边界，对于许多问题来说，知道这些分形是可移除的是有用的，因为出于某些目的可以忽略它们的存在。分形集的可移除性在需要将两个函数、两个动力系统或两个表面“粘合”在一起的数学问题中得到应用，并且可以更好地理解物理学中的动力系统。该项目由三个部分组成，涉及谢尔宾斯基地毯的均匀化、二维度量表面的均匀化以及共形地图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 地毯均匀化 Sierpinski 地毯相关的问题，并且 PI 将研究均匀化图的规律性，已知该图是拟对称或离散拟共形的。 PI 还将研究与均匀化图下的豪斯多夫维度畸变相关的问题，以及将平面均匀化理论推广到抽象谢尔宾斯基地毯。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将研究欧几里德空间对二维表面均匀化定理的可能推广，并集中精力弱化现有的几何假设。最后，PI 将致力于扩展分形集可移除性的早期成果，通过寻找不可移除的分形集（类似于谢尔宾斯基垫片或地毯）的拓扑标准，研究 Sobolev 可移除性和共形可移除性的等效性，并探索可移除性与圆域刚性问题的联系。该奖项反映了 NSF 的法定使命通过使用基金会的智力优点和更广泛的影响审查标准进行评估，并被认为值得支持。