Circle Packing: Discrete Conformal Geometry

摘要建议：DMS-962280 PI：斯蒂芬森·斯蒂芬森建议研究圆填充的基本性质及其诱导的几何结构。圆填充是具有规定相切图案的圆的配置。直到最近，斯蒂芬森一直专注于与经典解析函数的联系的发展，贡献了现在非常全面和在几何上忠实的“离散”解析函数理论。他的注意力现在已经转向其他方向-部分转向几何基础，部分转向应用。目前的研究集中在三个主要问题上：(1)球面上分支填充的存在唯一性(等价于球面上分支双曲多面体)；(2)圆填充上定制随机游动的性质和用途；(3)组合环境中的圆诱导几何，如组合Riemann映射、Grothendieck dessins、纽结投影、双曲群和图嵌入。特别值得注意的是，斯蒂芬森和他的合作者努力将“倒置距离”引入圆圈包装。此外，他通过使用一个复杂的软件包“测试工作台”，继续用数值和实验的方法来研究圆包装中的现象。圆填充在数学中是一个相对较新的课题，它涉及到研究具有特定相切模式的圆的构形。乍一看，这似乎是一个奇怪而虚假的话题，但事实证明，这个话题出人意料地丰富。在不给出技术细节的情况下，人们可以说，在一个圆中，包装关于圆之间接触的局部组合信息变成了关于整体构型的刚性全局几何信息。这种方法的独特价值在于，事实证明，这些填料以“谨慎”的方式携带了许多通常通过“连续”方程和公式描述的关键几何信息。随着我们走向计算机所需的更离散的世界模型，这一点尤为重要。其中重要的研究领域是解析函数理论和调和函数理论--例如，在流体流动、电路、扩散过程等的建模中至关重要。结果现在正被证明使用圆填充，这在经典理论中是不能建立的，甚至是预料不到的。此外，循环填充提供了新结果和新见解的机会，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，圆填充对于图形嵌入和各种计算机可视化任务都很好，例如3D建模。而支撑这些进步的经典和离散数学理论是重要的。

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.