Circle Packing: Discrete Conformal Geometry

PI: Stephenson Stephenson提出研究圆填料的基本性质及其所产生的几何结构。圆填料是具有规定的切线模式的圆的结构。直到最近，Stephenson一直专注于发展与经典解析函数的联系，为现在非常全面和几何上忠实的“离散”解析函数理论做出了贡献。他的注意力现在转向了其他方向——一部分转向几何基础，一部分转向应用。目前的建议集中在三个主要问题上：(1)球上分支填料的存在性和唯一性（相当于球上的分支双曲多面体）；(2)圆填料上定制随机漫步的性质和用途；(3)组合环境中的圆诱导几何，如组合黎曼映射、格罗滕迪克设计、结投影、双曲群和图嵌入。特别值得注意的是Stephenson和他的合作者将“逆距离”引入圆填充的努力。此外，他通过使用一个复杂的软件包“测试台”，继续使用数值和实验方法来研究圆形包装中的现象。圆填充是数学中一个相对较新的课题，它涉及研究具有特定相切模式的圆的构型。乍一看，这似乎是一个奇怪而做作的话题，但结果却出人意料地丰富起来。在不提供技术细节的情况下，我们可以说，在一个圆的包装中，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，这些封装在“离散”包中携带了通常通过“连续”方程和公式描述的许多关键几何信息。当我们走向计算机所要求的更加离散的世界模型时，这一点尤为重要。其中一个重要的研究领域是解析函数和调和函数理论——例如，在流体流动、电路、扩散过程等的建模中至关重要。结果现在正在被证明使用圆形包装不能建立或甚至在经典理论中没有预料到。此外，圆形包装提供了新的结果和见解的机会，因为它允许计算机实验。应用已经开始出现在数学和计算机科学的不同领域；例如，圆封装对于图形嵌入和各种计算机可视化任务（如3-D建模）非常有用。作为这些进步基础的经典和离散数学理论是非常重要的。

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.