Complexity in Complex Analysis

摘要 复杂分析对象的复杂性 (DMS-0305958) Steven R. Bell 教授。贝尔已经表明，与某些有限黎曼曲面相关的最广泛使用的核函数和复分析度量是与域的特殊共形映射相关的一个复变量的仅三个（有时甚至两个）解析函数的基本组合。 贝尔将研究他最近的发现所提出的复分析和势理论中的复杂性的更深层次问题，并将他在平面上的结果扩展到正交域和一般有限黎曼曲面。 贝尔为非齐次柯西-黎曼方程制定了独特的连续原理，他证明该原理可以产生有关复杂空间中域之间全纯映射行为的信息。 他将把这个原理应用到上面提到的一些问题上，他还将利用这个性质来尝试对几个复变量中的伯格曼投影获得更多的几何理解。位势论和共形映射的数学对象在科学、数学和工程中无处不在。 它们内部编码了大量有关平面区域几何特性的信息。 尽管这些物体为人们所熟悉并得到充分研究，但它们仍然是有趣且适用的新数学的源泉。 贝尔教授将用更简单的分析对象来表达与带孔二维表面相关的势论经典对象。 这些结果将带来新的实用方法，用于理解和压缩微分方程、共形映射和势论中许多问题的解决方案，这些问题不仅应该引起数学家的兴趣，而且也引起科学家和工程师的兴趣。由于人类最好通过一系列二维切片来感知高维物体，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.