Mathematical Sciences: Chord-arc Domains and Quasiconformal Maps

该奖项支持专注于解析和几何问题的数学研究，这些问题与复平面上定义的函数和域有关。本文结合了调和分析、拟共形映射和几何函数理论等方法。某些领域已经被确定为特别适合于分析映射到拟共形映射的扩展问题，弦弧域，形成了一类，其一般性质将被分析。这将包括类是否连接的问题。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域的几何信息，并获得高维拟共形映射的畸变估计。这项研究包括将Bishop-Jones关于准圆的最新成果扩展到确定哪些准共形映射将直线投射到弦弧曲线上的问题。几何函数理论和拟共形映射理论对理解平面域和高维域的几何变换做出了重大贡献，高维域的变换受到允许的扭曲或收缩量（有界膨胀）的限制。该项目通过研究过去几年出现的几个新概念来延续这一传统。

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.