Complexity in Complex Analysis

摘要 复分析对象的复杂性 （DMS-0305958） Steven R.贝尔教授Bell指出，与某些有限Riemann曲面相关的复分析中最广泛使用的核函数和度量是与该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初等组合。 贝尔将研究更深层次的问题复杂性复杂的分析和潜在的理论所提出的他最近的发现，他将延长他的成果在平面正交域和一般有限黎曼曲面。 Bell为非齐次Cauchy-Riemann方程建立了一个独特的延拓原理，他证明了这个延拓原理可以提供关于复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试图获得对多复变数中的伯格曼投影的更多的几何理解。位势论和保角映射的数学对象在科学、数学和工程学中是普遍存在的。 它们内部编码了大量关于平面区域几何特性的信息。 虽然这些对象是熟悉的和良好的研究，他们仍然是一个有趣的和适用的新数学的来源。 贝尔教授将用简单得多的分析对象来表达与带孔的二维表面相关的势能理论的经典对象。 这些结果将产生新的和实用的方法来理解和压缩微分方程，保角映射和势理论中的许多问题的解决方案，这不仅对数学家，而且对科学家和工程师都应该感兴趣。因为人类最好通过采取一系列二维切片来感知高维物体，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.