Mathematical Sciences: Chord-arc Domains and Quasiconformal Maps

该奖项支持数学研究，重点是 与函数有关的分析和几何问题， 定义在复平面上的域。 作品结合了 调和分析、拟共形映射和几何 功能理论 某些领域已被确定为 特别适合于分析的扩展问题， 映射到拟共形映射，弦弧域，形成一类 将分析其通用属性。 这将包括 问题是类是否是连通的。 此外，本发明还提供了一种方法， 将对因式分解问题进行研究 双Lipschitz平面映射，利用Cauchy的有界性 求域几何信息积分 和获得的失真估计的拟共形映射， 更高的维度 这项研究将包括以下工作： 将Bishop-Jones关于拟圆的最新结果推广到 确定哪些拟共形映射发送直线的问题 到弦弧曲线上 几何函数论与拟共形理论 地图绘制极大地促进了对 平面域和域的几何变换 更高的维度，其中变换受到 允许的扭曲或收缩量 （有界扩张）。 该项目延续了这一传统， 研究了几个新的概念， 过去几年

This award supports mathematical research focusing on analytic and geometric problems associated with functions and domains defined in the complex plane. The work combines methods of harmonic analysis, quasiconformal mapping and geometric function theory. Certain domains which have been identified as being particularly suited to questions of extensions of analytic maps to quasiconformal maps, chord-arc domains, form a class whose generic properties will be analyzed. This will include the question of whether the class is connected. In addition, work will be done on the problem of factorizing bi-Lipschitz planar maps, using the boundedness of a the Cauchy integral to obtain geometric information about domains and obtaining distortion estimates for quasiconformal maps in higher dimensions. Included in the research will be work to extend the recent results of Bishop-Jones on quasicircles to the problem of determining which quasiconformal maps send the line onto a chord-arc curve. The theories of geometric function theory and quasiconformal mapping have contributed significantly to the understanding of geometric transformations of plane domains and domains in higher dimensions where the transformations are constrained by the amount of twisting or shrinking that is permitted (bounded dilatation). This project continues this tradition by taking up the study of several new concepts which have arisen in the past few years.