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Removable Sets and Questions in Geometric Function Theory

几何函数论中的可移集和问题

基本信息

批准号：
1758295
负责人：
Malik Younsi
金额：
$ 10.45万
依托单位：
University of Hawaii
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-31 至 2020-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1758295&HistoricalAwards=false
关键词：
Removable Sets Questions Geometric Function

项目摘要

Conformal maps are planar changes of coordinates that locally preserve angles. The study of the properties of such maps led to the development of the subject known as geometric function theory. A cornerstone of this discipline is the remarkable fact that every planar region without holes can be conformally transformed into the round disk in an essentially unique fashion. Such coordinate changes have proven over the years to be of great value in a wide variety of applications in physics and engineering. In many of these applications, one has to deal with coordinate changes that are conformal in a given planar region except possibly some "exceptional set" of points inside the region. The question whether the exceptional set is negligible and small enough to be ignored leads to the notion of conformal removability, central to this research project and closely related to fundamental questions in complex dynamics and in random surfaces. The project will also consider several other questions in geometric function theory, such as conformal welding (a correspondence between curves in the plane and functions on the circle, recently observed to have important applications in the field of numerical vision), shapes of Julia sets (fractal sets arising from the iteration of polynomials), and the subadditivity of analytic capacity. Each of these investigations has a numerical aspect, and successful completion of the research project will enhance computational structure and build interdisciplinary connections with applied sciences such as finance and pattern recognition.The first part of the research project deals with the study of the geometric properties of conformally removable sets. More precisely, the investigator will further study various settings where removability appears naturally, including the rigidity of circle domains. This first component of the research project also includes problems related to conformal welding and fingerprints of lemniscates. The second part is devoted to the study of the possible shapes of Julia sets. More specifically, the investigator plans to explore various questions revolving around the constructive approximation of planar sets by polynomial Julia sets in the Hausdorff distance, such as sharp rates of approximation, dynamical properties of the approximating polynomials, and an efficient numerical implementation of the approximation scheme. Finally, the last part of the research project concerns the subadditivity problem for analytic capacity. The investigator will further explore a conjecture that, if true, would imply analytic capacity is indeed subadditive. This involves an improvement of a numerical method for the computation of the analytic capacity of finite unions of disks.

共形地图是局部保留角度的平面坐标变化。对此类映射性质的研究导致了几何函数理论这一学科的发展。该学科的基石是一个显着的事实，即每个没有孔的平面区域都可以以本质上独特的方式共形地转变为圆盘。多年来，这种坐标变化已被证明在物理和工程领域的各种应用中具有巨大价值。在许多这样的应用中，人们必须处理在给定平面区域中共形的坐标变化，除了该区域内可能存在的一些“特殊的”点组之外。异常集是否可以忽略不计并且小到足以被忽略的问题引出了共形可去除性的概念，它是该研究项目的核心，与复杂动力学和随机表面的基本问题密切相关。该项目还将考虑几何函数理论中的其他几个问题，例如保形焊接（平面中的曲线与圆上的函数之间的对应关系，最近观察到在数值视觉领域具有重要应用）、Julia 集的形状（由多项式迭代产生的分形集）以及分析能力的次可加性。这些研究中的每一项都涉及数值方面，研究项目的成功完成将增强计算结构并与金融和模式识别等应用科学建立跨学科联系。研究项目的第一部分涉及共形可移除集的几何性质的研究。更准确地说，研究人员将进一步研究可自然出现可移除性的各种设置，包括圆域的刚性。该研究项目的第一个组成部分还包括与保形焊接和双纽带指纹相关的问题。第二部分致力于研究 Julia 集的可能形状。更具体地说，研究人员计划探索围绕 Hausdorff 距离中的多项式 Julia 集对平面集的建设性逼近的各种问题，例如逼近率、逼近多项式的动态性质以及逼近方案的有效数值实现。最后，研究项目的最后一部分涉及分析能力的次可加性问题。研究者将进一步探索一个猜想，如果该猜想为真，则意味着分析能力确实是次加性的。这涉及对计算磁盘有限并集分析容量的数值方法的改进。