Circle Packing: Discrete Conformal Geometry

圆堆积：离散共形几何

• 批准号: 9622803
• 负责人: Kenneth Stephenson
• 金额: $ 4.52万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622803&HistoricalAwards=false
• 关键词: Circle; Packing; Discrete; Conformal; Geometry

ABSTRACT Proposal: DMS-962280 PI: Stephenson Stephenson proposes to study fundamental properties of circle packings and the geometric structures they induce. Circle packings are configurations of circles with prescribed patterns of tangency. Until recently Stephenson has concentrated on the development of connections with classical analytic functions, contributing to what is now a very comprehensive and geometrically faithful "discrete" analytic function theory. His attention has now turned in other directions --- partly in towards the geometric foundations and partly out towards applications. The current proposal concentrates on three main issues: (1) existence and uniqueness of branched packings on the sphere (equivalently, of branched hyperbolic polyhedra in the ball); (2) properties of and uses for tailored random walks on circle packings; and (3) circle induced geometries in combinatoric settings, such as combinatorial Riemann mapping, Grothendieck dessins, knot projections, hyperbolic groups, and graph embedding. Of particular note is the effort by the Stephenson and his collaborators to introduce "inversive distances" into circle packing. Also, he continues numerical and experimental approaches to investigating phenomena in circle packing through use of a sophisticated software package, "test bench". Circle packing is a relatively new topic in mathematics which involves the study of configurations of circles with specified patterns of tangency. It seems an odd and artificial topic at first, but turns out to be surprisingly rich. Without giving technical details, one can say that in a circle packing the local combinatoric information about contacts between circles turns into rigid global geometric information on the overall configuration. The unique value in this approach lies in the fact that these packings turn out to carry in "discreet" packets much of the key geometric information normally described via "continuous" equations and formulas. This is especially important as we move towards the more discrete model of the world required by computers. Among the important areas of study are analytic and harmonic function theory -- crucial, for instance, in modeling of fluid flow, electrical circuits, diffusion processes, and so forth. Results are now being proven using circle packing which could not be established or were even unanticipated in the classical theories. Moreover, circle packing gives opportunities for new results and insights because it permits computer experimentation. Applications have begun to emerge in diverse areas of mathematics and in computer science; for instance, circle packings are good for graph embedding and for various computer visualization tasks, such as 3-D modeling. And the classical and discrete mathematical theory which underlies these advances is of importance.

摘要提案：DMS-962280 PI：斯蒂芬森 斯蒂芬森提出研究圆填充的基本性质及其几何结构， 诱导。圆填充是具有规定的相切模式的圆的配置。直到最近斯蒂芬森一直集中在发展的联系与经典的解析函数，有助于什么是现在一个非常全面和几何忠实的“离散”解析函数理论。他的注意力现在已经转向其他方向-部分在几何基础上，部分在应用上。目前的建议集中在三个主要问题：（1）存在性和唯一性的分支包装上的球（相当于，分支双曲多面体在球）;（2）性质和用途的裁剪随机游动的圆包装;和（3）圆诱导几何在组合设置，如组合黎曼映射，Grothendieck dessins，结投影，双曲群，和图嵌入。特别值得注意的是努力由斯蒂芬森和他的合作者介绍“反演距离”到圆包装。此外，他继续通过使用一个复杂的软件包，“测试台”的数值和实验方法来调查圆包装现象。 圆填充是数学中一个相对较新的课题，它涉及对具有特定相切模式的圆的构型的研究。这似乎是一个奇怪的和人为的话题在第一，但结果是令人惊讶的丰富。在不给出技术细节的情况下，可以说，在圆包装中，关于圆之间接触的局部组合信息变成关于整体配置的刚性全局几何信息。这种方法的独特价值在于这样一个事实，即这些包装原来携带在“离散”包通常通过“连续”方程和公式描述的关键几何信息。当我们转向计算机所需的更加离散的世界模型时，这一点尤其重要。其中重要的研究领域是分析和调和函数理论-至关重要的，例如，在流体流动，电路，扩散过程等建模。结果现在正在证明使用循环包装不能建立或甚至是在经典理论中没有预料到的。此外，圆包装为新的结果和见解提供了机会，因为它允许计算机实验。在数学和计算机科学的不同领域已经开始出现应用;例如，圆填充适用于图形嵌入和各种计算机可视化任务，如3D建模。 这些进展背后的经典和离散数学理论是重要的。