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。在数学研究中，圆填料一直是一个有趣的概念。1985年，威廉·瑟斯顿提出圆填充——用不同大小的圆来填充平面域——可能会导致保角映射的一个全新观点。事实证明确实如此，这个项目是那个著名讲座的副产品之一。这个想法已经发展壮大。圆填充表现出忠实的离散类似于几何函数理论的经典结果，如Schwarz-Pick引理、Dirichlet问题、均匀化定理、布朗运动等。目前的项目集中在长度-面积法的离散版本上，这是一种强大但笨拙的方法，只有少数专家使用，而且不容易被函数理论的消费者理解。第二部分涉及无限包装，可能是项目中最具投机性的部分。迄今为止，大多数研究只考虑有限的填料。研究的最后一个组成部分涉及使用重叠的圆包装。有一些基本的结果允许重叠，但在这种概括上做的工作很少。过去的研究导致了用于操纵、显示和动画化圆形包装的软件的开发。这既是进行实验的研究工具，也是创建演示所需图片的实用方法。该软件是免费的。复变函数理论是计算和可视化复变函数行为的数学基础。这门学科有着悠久的历史，通常与分析和几何同时研究。这个项目增加了一个新的维度：计算。圆形填充自然导致了计算机模拟。这反过来又导致了经典理论的新见解和新证明。为了证明它的价值，新的观点必须带来新的、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.