Removable Sets and Questions in Geometric Function Theory

保角映射是局部保持角度的坐标的平面变化。研究这些地图的性质导致了被称为几何函数论的学科的发展。这门学科的基石是一个了不起的事实，即每个平面区域没有孔可以共形地转化为圆盘在一个本质上独特的方式。多年来，这种坐标变化已被证明在物理学和工程学的各种应用中具有很大的价值。在许多这些应用中，一个必须处理的坐标变化是保形在一个给定的平面区域内，除了可能的一些“例外”的区域内的点。问题是否例外集是可以忽略的，小到可以忽略不计，导致共形可移动性的概念，中央本研究项目和密切相关的基本问题，在复杂的动力学和随机表面。该项目还将考虑几何函数理论中的其他几个问题，例如保角焊接（平面曲线与圆上函数之间的对应关系，最近观察到在数值视觉领域有重要应用）、朱莉娅集的形状（产生于多项式迭代的分形集），以及分析能力的次可加性。这些研究都有一个数值方面，研究项目的成功完成将增强计算结构，并与金融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共形可去集的几何性质的研究。更确切地说，研究人员将进一步研究自然出现可移除性的各种设置，包括圆域的刚性。这个研究项目的第一个组成部分还包括与保形焊接和双纽动物指纹有关的问题。第二部分研究Julia集的可能形状。更具体地说，研究人员计划探讨各种问题围绕建设性的近似平面集多项式朱莉娅集的Hausdorff距离，如尖锐的近似率，近似多项式的动力学性质，以及有效的数值实现的近似方案。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，如果这个猜想成立，将意味着分析能力确实是次加性的。这涉及到一个改进的数值方法计算的分析能力的有限工会的磁盘。

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.