Mathematical Sciences: The Geometry of Image Surfaces of Complex Functions

这个项目将涉及使用位势理论和黎曼曲面方法的复函数理论的工作。这两个调查领域既不同，又相互关联。首先，我们将研究解析函数的象面几何，特别是定义在圆盘上的内函数和整函数。这项研究源于Gross关于小Bloch空间中函数奇点结构的较早工作。这些方法涉及到利用布朗运动和位势理论来构造黎曼曲面。在最近瑟斯顿猜想的激励下，第二个推力将被用来研究平面中的圆填充问题。现在已经证实，人们可以使用圆填充来逼近经典的共形映射。这一主题的另一种方法将基于随机行走，该随机行走有望在图形和布朗运动的近似之间提供联系。这项工作有望应用于偏微分方程组的数值逼近以及位势理论和概率论。

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.