Analysis and Geometry of Conformal and Quasiconformal Mappings

共形和拟共形映射的分析和几何

• 批准号: 2350530
• 负责人: Malik Younsi
• 金额: $ 21.13万
• 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: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350530&HistoricalAwards=false
• 关键词: Analysis; Geometry; Conformal; Quasiconformal; Mappings

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.

该项目旨在更好地理解共形和拟共形映射的解析和几何特性。共形映射是局部保留角度的平面变换。一个重要的例子是制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，人们对准共形映射的研究投入了很多注意力，准共形映射是允许控制角度畸变量的共形映射的推广。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各种领域中具有根本重要性。这些应用中的许多应用涉及给定区域内拟共形的平面变换，除了该区域内的一些特殊点集之外。对这一特殊集合的研究引出了可移除性的概念，它是该研究项目的核心，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。该项目的另一个重点是研究某些称为全纯运动的拟共形映射族。首席研究员将研究尺寸和面积等量在全纯运动下如何变化，从而更好地理解拟共形映射的几何特性。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针对当地高中生的科学和数学推广计划。计划的工作包括两部分研究。第一个部分涉及保形可去除性的研究。受长期存在的 Koebe 均匀化猜想的启发，首席研究员将研究圆域的可移除性和刚性之间的关系。该项目的这一部分还涉及保形焊接的研究，即平面乔丹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重新燃起，并出现了新的概括和变体，特别是在随机表面理论以及计算机视觉和数字模式识别应用方面。该项目的第二个组成部分涉及全纯运动。首席研究员将研究全纯运动下几种尺寸概念的变化。首席研究员和他的合作者使用中频调和函数对这个主题提出了一种新方法，已经对关于拟共形映射的几个著名定理进行了统一的处理，并且随着在更好地理解全纯运动。这部分项目还涉及全局拟共形维数和共形维数之间的关系。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。