Mathematical Sciences: The Geometry of Image Surfaces of Complex Functions

这个项目将涉及使用势理论和黎曼曲面方法的复函数理论。这两个调查领域不同但又相关。首先，工作将在解析函数的像面几何上完成-特别是在圆盘中定义的内函数和整个函数。该研究源自Gross关于小Bloch空间中函数奇异性结构的旧工作。这些方法包括利用布朗运动和势理论构造黎曼曲面。由瑟斯顿最近的一个猜想所引起的，第二个重点将放在飞机上的圆形填料问题上。现在已经确定可以用圆填充来近似经典的共形映射。对于这个主题的另一种方法将基于随机漫步进行探讨，它有望提供图与布朗运动近似之间的联系。这项工作有望应用于偏微分方程的数值近似以及势理论和概率论。

This project will involve work on complex function theory using potential theory and Riemann surface methods. The two areas of investigation are distinct but related. First, work will be done on the geometry of image surfaces of analytic functions - particularly inner functions defined in a disc, and entire functions. The research derives from older work of Gross on the structure of singularities of functions in the little Bloch space. The methods involve the construction of Riemann surfaces using Brownian motion and potential theory. A second thrust will be made into problems of circle packings in the plane, motivated by a recent conjecture of Thurston. It is now established that one may use circle packings to approximate classical conformal mappings. An alternate approach to this theme will be pursued based on random walks which are expected to provide a link between the graph and approximations to Brownian motion. This work is expected to have application to numerical approximations to partial differential equations as well as to potential theory and probability.