Mathematical Sciences: Discrete Analytic Function Theory Via Circle Packing

数学科学：通过圆堆积的离散解析函数理论

• 批准号: 9303135
• 负责人: Kenneth Stephenson
• 金额: $ 9.75万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303135&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Discrete; Analytic; Function

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.

该项目继续研究循环堆积与复变量解析函数理论之间的联系。 圆形堆积长期以来一直是数学研究中一个有趣的概念。 1985 年，William Thurston 提出圆堆积——用不同大小的圆对平面域进行堆积——可能会带来共角映射的全新观点。 事实证明确实如此，这个项目是那场现在著名的讲座的几个副产品之一。 这个想法已经成长和繁荣。 圆堆积表现出与几何函数理论的经典结果（例如 Schwarz-Pick 引理、狄利克雷问题、均匀化定理、布朗运动等）忠实的离散类似物。当前的项目集中于长度面积方法的离散版本，这是一种强大但笨拙的方法，由少数专家使用，并且不容易被函数理论的消费者理解。 第二部分涉及无限打包，可能是该项目中最具推测性的部分。 迄今为止，大多数研究仅考虑有限堆积。 研究的最后一部分涉及使用重叠的圆圈进行包装。 有一些基本结果允许重叠，但在这种概括方面几乎没有做任何工作。 过去的研究导致了用于操作、显示圆形包装和制作动画的软件的开发。 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。 该软件是免费提供的。 复函数理论是计算复变量的复函数行为并将其可视化的数学基础。 该学科有着悠久的历史，通常与分析和几何同时研究。 该项目增加了一个新的维度：计算。 圆形堆积自然而然地导致了计算机模拟。 这反过来又导致了经典理论的新见解和证明。为了证明其价值，新观点必须带来新的、意想不到的发现。