New approaches to potential theory and conformal mapping

黎曼映射定理是一个古老、美丽和非常有用的工具，它可以用来将单位圆盘上经典位势理论对象的显式公式拉回到平面上任意的单连通区域。单位圆盘是双正交域的典型例子，即关于面积测量和边界弧长的正交域。贝尔教授和他的合作者一直致力于对黎曼映射定理的改进，将单位圆盘替换为双正交域，而不是像单位圆盘那样远离，而是接近原始域。Bell已经证明了与双求积域有关的许多经典势论对象都是初等函数，它们是坐标变量和代数Schwarz函数的简单组合，因此可以很容易地计算。因此，可以在区域中进行微小但微妙的改变，以使其具有单位盘的许多所需属性。这种方法具有潜在的优点，即使在多重连通的黎曼曲面上，黎曼映射定理也是不成立的。位势理论和保角映射的对象在数学、科学和工程中是普遍存在的。尽管对这些物体进行了很好的研究，但它们仍然是令人兴奋和潜在适用的新数学的来源。贝尔教授将用简单和可计算的分析对象来表示与带有洞的二维表面有关的经典势论对象。这些结果可能为表达和压缩微分方程组、保角映射和位势理论中的经典问题的解提供新的实用方法。由于人类通过获取一系列二维切片来感知更高维度的物体是最好的，因此开发的工具可以找到重要的应用。该项目背后的一些关键想法源于贝尔在2008年指导的一个夏季本科生研究项目，该项目的一个值得注意的组成部分是对参与数学职业生涯的本科生和研究生进行指导和培训。

The Riemann mapping theorem is an old, beautiful, and tremendously useful device which can be used to pull back the explicit formulas for the classical objects of potential theory on the unit disc to arbitrary simply connected regions in the plane. The unit disc is the quintessential example of a double quadrature domain,i.e., a quadrature domain both with respect to area measure and boundary arc length. Professor Bell and his collaborators have been working on an improvement to the Riemann mapping theorem which replaces the unit disc by a double quadrature domain which, rather than being far away like the unit disc, is close to the original domain. Bell has proved that many of the classical objects of potential theory associated to a double quadrature domain are elementary functions which are simple combination of the coordinate variable and the algebraic Schwarz function, and consequently may be easily computed. Thus, it becomes possible to make minor, but subtle, alterations in a region to make it have many of the desirable properties of the unit disc. This approach has the virtue of being potentially applicable even in the multiply connected and Riemann surface setting, where the Riemann mapping theorem is unavailable.The objects of potential theory and conformal mapping are pervasive in Mathematics, Science, and Engineering. Although these objects are well studied, they continue to be a source of exciting and potentially applicable new mathematics. Professor Bell will express the classical objects of potential theory associated to a two dimensional surface with holes in terms of simple and computable analytic objects. These results may give rise to new and practical methods for expressing and zipping the solutions to classical problems in differential equations, conformal mapping, and potential theory. Because humans best perceive higher dimensional objects by taking a series of two dimensional slices, the tools developed could find important applications. Some of the key ideas behind this program grew out of a summer undergraduate research project that Bell directed in 2008, and a noteworthy component of the project is a significant commitment to mentoring and training undergraduates and graduate students involved for a career in mathematics.