Removable Sets and Questions in Geometric Function Theory

保角映射是平面坐标的变化，局部保持角度。对这种映射性质的研究导致了被称为几何函数理论的学科的发展。这门学科的基石是一个显著的事实，即每一个没有孔的平面区域都可以以一种本质上独特的方式共形转化为圆盘。多年来，这种坐标变化已被证明在物理和工程的各种应用中具有很大的价值。在许多这样的应用中，人们必须处理在给定平面区域内的共形坐标变化，除非区域内可能有一些“特殊集”的点。异常集是否可以忽略，是否小到可以忽略的问题导致了保形可移性的概念，这是本研究项目的核心，与复杂动力学和随机曲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，如保形焊接（平面上的曲线和圆上的函数之间的对应关系，最近被观察到在数值视觉领域有重要应用），Julia集的形状（多项式迭代产生的分形集），以及解析能力的次可加性。这些研究中的每一个都有一个数值方面，研究项目的成功完成将增强计算结构，并与应用科学（如金融和模式识别）建立跨学科的联系。本课题的第一部分研究共形可移动集的几何性质。更确切地说，研究者将进一步研究各种可移除性自然出现的设置，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金属指纹有关的问题。第二部分研究了朱丽亚集合的可能形状。更具体地说，研究者计划探索围绕在Hausdorff距离上的多项式Julia集的平面集的构造近似的各种问题，如近似的急剧速率，近似多项式的动态特性，以及近似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子可加性问题。研究者将进一步探索一个猜想，如果是真的，将意味着分析能力确实是次加性的。这涉及到一种计算圆盘有限联合解析能力的数值方法的改进。

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.