Complexity in Complex Analysis

复杂分析对象的复杂性（DMS-0305958）Bell已经证明，与某些有限黎曼曲面相关的复分析的最广泛使用的核函数和度量是只有三个，有时甚至是两个的初等组合，一个复变量的解析函数与该域的特殊保角映射有关。贝尔将研究有关复杂分析和他最近的发现所提出的势理论的复杂性的更深层次的问题，他将把他在平面上的结果扩展到正交域和一般有限黎曼曲面。Bell为非齐次Cauchy-Riemann方程制定了一个独特的延拓原理，他展示了复空间中域间全纯映射行为的信息。他将把这个原理应用于上面提到的一些问题，他也将使用这个性质来尝试获得对几个复杂变量的伯格曼投影的更多几何理解。势理论和共形映射的数学对象在科学、数学和工程中无处不在。它们携带着大量关于平面区域几何特性的编码信息。虽然这些对象很熟悉，研究得很透彻，但它们仍然是有趣和适用的新数学的来源。贝尔教授将用更简单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。这些结果将产生新的和实用的方法来理解和简化微分方程、保角映射和势理论中许多问题的解决方案，这些问题不仅是数学家，而且是科学家和工程师都感兴趣的。由于人类最好地感知高维物体是通过拍摄一系列二维切片，因此开发的工具可以找到许多应用。

ABSTRACT Complexity of the objects of complex analysis (DMS-0305958) Steven R. BellProf. Bell has shown that the most widely used kernelfunctions and metrics of complex analysis associated tocertain finite Riemann surface are elementary combinationsof only three, and sometimes even two, analytic functions ofone complex variable related to special conformal mappingsof the domain. Bell will study deeper questions aboutcomplexity in complex analysis and potential theory posed byhis recent findings and he will extend his results in theplane to quadrature domains and general finite Riemannsurfaces. Bell has formulated a unique continuationprinciple for the inhomogeneous Cauchy-Riemann equationsthat he has shown yields information about the behavior ofholomorphic mappings between domains in complex space. Hewill apply this principle to some of the questions mentionedabove and he will also use the property to try to gain amore geometric understanding of the Bergman projection inseveral complex variables.The mathematical objects of potential theory and conformalmapping are ubiquitous in Science, Mathematics, andEngineering. They carry encoded within them a vast amountof information about geometric properties of regions in theplane. Although these objects are familiar and wellstudied, they continue to be a source of interesting andapplicable new mathematics. Professor Bell will express theclassical objects of potential theory associated to a twodimensional surface with holes in terms of much simpleranalytic objects. These results will give rise to new andpractical methods for understanding and zipping the solutionsto many problems in differential equations, conformal mapping,and potential theory that should be of interest, not only tomathematicians, but to scientists and engineers as well.Because humans best perceive higher dimensional objects bytaking a series of two dimensional slices, the toolsdeveloped could find many applications.