Mathematical Sciences: Chord-arc Domains and Quasiconformal Maps

该奖项支持专注于与复平面中定义的函数和区域相关的解析和几何问题的数学研究。该工作结合了调和分析、拟共形映射和几何函数理论的方法。已被确定为特别适合于将解析映射的扩张问题扩展到拟共形映射的某些区域，即弦弧域，形成了一类其类属性质将被分析的区域。这将包括班级是否连接的问题。此外，还将研究双Lipschitz平面映射的因式分解问题，利用柯西积分的有界性来获得区域的几何信息，并得到高维拟共形映射的偏差估计。这项研究将把毕晓普-琼斯关于准圆的最新结果推广到确定哪些准共形映射将直线发送到弦弧曲线上的问题。几何函数论和拟共形映射理论对理解平面区域和高维区域的几何变换有重要贡献，其中变换受允许的扭曲或收缩的量(有界膨胀)的约束。这个项目延续了这一传统，对过去几年出现的几个新概念进行了研究。

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.