Analysis and Geometry of Conformal and Quasiconformal Mappings

本项目旨在更好地理解共形和拟共形映射的解析性质和几何性质。保角映射是局部保持角度的平面变换。一个重要的例子是地图学中的墨卡托投影，用于将地球表面投影到二维地图上。最近，人们对拟共形映射的研究投入了大量的注意力，拟共形映射是对共形映射的一种推广，在这种映射中允许有一定数量的角畸变。由于这种额外的灵活性，多年来，准共形映射已被证明在数学和应用的广泛领域具有重要的基础意义。这些应用中有许多涉及平面变换，这些变换在给定区域内是拟共形的，除了可能在区域内的一些特殊点集。对这一特殊集合的研究导致了可移除性的概念，这是本研究项目的核心，与复杂分析、动力系统、概率和相关领域的基本问题密切相关。这个项目的另一个重点是研究某些被称为全纯运动的拟共形映射族。首席研究员将研究在全纯运动下尺寸和面积等量是如何变化的，从而更好地理解拟共形映射的几何性质。该项目还为包括研究生在内的早期职业研究人员提供培训和指导的机会。此外，首席研究员将继续参与当地高中生的科学和数学推广计划。计划中的工作包括两方面的研究。第一部分涉及共形可移除性的研究。在长期存在的Koebe均匀化猜想的激励下，首席研究员将研究可移除性和圆域刚性之间的关系。项目的这一部分还涉及到保形焊接的研究，即平面约当曲线与圆上函数之间的对应关系。近年来，人们对保形焊接的兴趣随着新的推广和变体而重新燃起，特别是在随机表面理论以及与计算机视觉和数字模式识别相关的应用方面。该项目的第二个组成部分涉及全纯运动。首席研究员将研究几个维度概念在全纯运动下的变化。由首席研究员和他的合作者使用次调和函数的新方法已经产生了关于拟共形映射的几个著名定理的统一处理，并且随着对全纯运动的更好理解继续取得进展，预计会有更多富有成果的联系。项目的这一部分还涉及到全局拟共形维数和共形维数之间的关系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to better understand the analytic and geometric properties of conformal and quasiconformal mappings. Conformal mappings are planar transformations which locally preserve angles. An important example is the Mercator projection in cartography, used to project the surface of the Earth to a two-dimensional map. More recently, much attention has been devoted to the study of quasiconformal mappings, a generalization of conformal mappings where a controlled amount of angle distortion is permitted. Because of this additional flexibility, quasiconformal mappings have proven over the years to be of fundamental importance in a wide variety of areas of mathematics and applications. Many of these applications involve planar transformations that are quasiconformal inside a given region except possibly for some exceptional set of points inside the region. The study of this exceptional set leads to the notion of removability, central to this research project and closely related to fundamental questions in complex analysis, dynamical systems, probability and related areas. Another focus of this project is on the study of certain families of quasiconformal mappings called holomorphic motions. The principal investigator will study how quantities such as dimension and area change under holomorphic motions, leading to a better understanding of the geometric properties of quasiconformal mappings. The project also provides opportunities for the training and mentoring of early career researchers, including graduate students. In addition, the principal investigator will continue to be involved in a science and mathematics outreach program for local high school students.Two strands of research comprise the planned work. The first component involves the study of conformal removability. Motivated by the long-standing Koebe uniformization conjecture, the principal investigator will investigate the relationship between removability and the rigidity of circle domains. This part of the project also involves the study of conformal welding, a correspondence between planar Jordan curves and functions on the circle. Recent years have witnessed a renewal of interest in conformal welding along with new generalizations and variants, notably in the theory of random surfaces and in connection with applications to computer vision and numerical pattern recognition. The second component of the project concerns holomorphic motions. The principal investigator will study the variation of several notions of dimension under holomorphic motions. A new approach to this topic by the principal investigator and his collaborators using inf-harmonic functions has already yielded a unified treatment of several celebrated theorems about quasiconformal mappings, and many more fruitful connections are anticipated as progress continues to be made towards a better understanding of holomorphic motions. This part of the project also involves the relationship between global quasiconformal dimension and conformal dimension.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.