Removable Sets and Questions in Geometric Function Theory

几何函数论中的可移集和问题

• 批准号: 1664807
• 负责人: Malik Younsi
• 金额: $ 12.69万
• 依托单位: University of Washington
• 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: 2017-05-01 至 2017-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664807&HistoricalAwards=false
• 关键词: Removable; Sets; Questions; Geometric; Function

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.

共形图是局部保留角度的坐标的平面变化。对此类地图的性质的研究导致了被称为几何函数理论的主题的发展。这个学科的基石是一个了不起的事实，即每个没有孔的平面区域都可以以本质上独特的方式将其共同转变为圆盘。多年来，这种协调变化在物理和工程领域的各种应用中都具有巨大价值。在这些应用中的许多应用中，必须处理在给定平面区域中符合条件的坐标更改，除了该区域内的某些“特殊集合”。这个问题是否可以忽略不计，足够小，以至于被忽略了，导致了保形可移动性的概念，这是该研究项目的核心，并且与复杂动态和随机表面中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（飞机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），朱利娅集的形状，朱利娅集的形状（分形设置）（由iTmials的迭代产生），以及下属的分析能力。这些研究中的每一个都有一个数字方面，并且成功完成研究项目将增强计算结构，并通过诸如财务和模式识别等应用科学建立跨学科的联系。研究项目的第一部分涉及对共同解动集的几何特性的研究。更确切地说，研究人员将进一步研究可移动性自然出现的各种环境，包括圆形域的刚度。研究项目的第一个组成部分还包括与结构化焊接和指纹有关的问题。第二部分专门研究朱莉娅套装的可能形状。更具体地说，研究人员计划探索各种问题，这些问题围绕多项式Julia集在Hausdorff距离中的建设性近似，例如近似近似速率，近似多项式的动力学特性，以及近似方案的有效数值实施。最后，研究项目的最后一部分涉及分析能力的亚功能问题。研究人员将进一步探讨一个猜想，即如果是真的，则意味着分析能力确实是亚基。这涉及改进数值方法，用于计算有限磁盘的分析能力。