Mathematical Sciences: Discrete Analytic Function Theory Via Circle Packing

这个项目继续研究圆填充和复变量解析函数理论之间的联系。圆填充在数学研究中一直是一个有趣的概念。1985年，威廉·瑟斯顿提出了圆填充--用不同大小的圆对平面域进行填充--可以在保角映射中产生一种全新的观点。事实证明是这样的，这个项目是那个现在著名的讲座的几个副产品之一。这个想法得到了发展和繁荣。圆填充与几何函数论的经典结果如Schwarz-Pick引理、Dirichlet问题、均匀化定理、布朗运动等显示出忠实的离散相似.目前的项目集中在长度-面积方法的离散版本，一个强大的，但由少数专家使用的方法笨拙，不容易被函数论的消费者理解。第二部分涉及无限包装，这可能是该项目最具投机性的部分。到目前为止，大多数研究只考虑了有限填充。研究的最后一部分涉及使用带重叠的圆的填充。已经有了允许重叠的基本结果，但在这一推广方面所做的工作很少。过去的研究导致了用于操纵、显示和制作圆包装动画的软件的开发。这既是一种进行实验的研究工具，也是一种创建演示所需图片的实用方法。该软件是免费提供的。复变函数论是计算和可视化复变数复函数行为的数学基础。这门学科有着广泛的历史，通常与分析和几何同时学习。这个项目增加了一个新的维度：计算。循环填充自然地导致了计算机模拟。这反过来又带来了经典理论中的新见解和新证据。为了证明它的价值，新的观点将不得不导致新的、意想不到的发现。

This project continues investigations into the connections between circle packing and the theory of analytic functions of a complex variable. Circle packings have long been an intriguing concept in mathematical research. In 1985 William Thurston proposed that circle packings - packings of plane domains by circles of various sizes - could lead to a completely new point of view in conformal mapping. This turned out to be the case and this project is one of several by-products of that now famous lecture. The idea has grown and prospered. Circle packings are exhibiting faithful discrete analogues to the classical results of geometric function theory such as the Schwarz-Pick lemma, Dirichlet problem, uniformization theorem, Brownian motion, etc. The current project concentrates on the discrete version of the length-area method, a powerful, but unwieldy method exercised by a few experts and not easily understood by consumers of function theory. The second part concerns infinite packings, possibly the most speculative part of the project. Most research to date considers only finite packings. The final component of the research involves packings using circle with overlap. There are fundamental results which allow for overlap, but little work has been done on this generalization. Past research led to the development of software for manipulating, displaying and animating circle packings. This is both a research tool for performing experiments and a practical method for creating the pictures needed for presentation. The software is freely available. Complex function theory is the mathematical basis for computing with and visualizing the behavior of complex functions of a complex variable. The subject has an extensive history and is normally studied concurrently as analysis and geometry. This project adds a new dimension: computation. Circle packing leads naturally to computer simulation. This in turn has led to new insights and proofs in the classical theory. To prove its worth, the new point of view will have to lead to new , unexpected discoveries.