New approaches to potential theory and conformal mapping

黎曼映射定理是一个古老的、美丽的、非常有用的工具，它可以将单位圆盘上经典势论对象的显式公式拉回平面上任意单连通区域。单位圆盘是双正交域的典型例子，即。即面积和边界弧长的正交域。贝尔教授和他的合作者一直致力于改进黎曼映射定理，用双正交域取代单位圆盘，而不是像单位圆盘那样遥远，而是接近原始域。贝尔证明了许多与双正交域有关的势理论的经典对象都是初等函数，它们是坐标变量和代数Schwarz函数的简单组合，因此很容易计算。因此，有可能在一个区域进行微小但微妙的改变，使其具有单位圆盘的许多理想特性。这种方法具有潜在的适用性，即使在多重连通和黎曼曲面的情况下，黎曼映射定理是不可用的。势理论和保角映射的对象在数学、科学和工程中无处不在。虽然这些对象已经得到了很好的研究，但它们仍然是令人兴奋和潜在应用的新数学的来源。Bell教授将用简单的、可计算的解析对象来表达与有孔的二维表面相关的势理论的经典对象。这些结果可能会产生新的和实用的方法来表达和压缩微分方程、保角映射和势理论中的经典问题的解。因为人类最好地感知高维物体是通过拍摄一系列二维切片，所以开发的工具可以找到重要的应用。这个项目背后的一些关键思想源于贝尔在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从事数学职业的本科生和研究生进行指导和培训的重要承诺。

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.