Computational Discrete Conformal Geometry and Applications

调查人员进一步开发了软件工具CirclePack，并使用它来探索在几个应用领域出现的数学问题。圈填充是由瑟斯顿引入的，它首先是在奥比福尔德结构的背景下引入的，然后是在共形映射的近似下引入的。这个主题已经发展到了这样一个点，它提供了一个令人惊讶的全面的离散理论，既平行又近似于经典的解析函数理论，即二维共形映射。发展在很大程度上是由经典的联系推动的-离散解析性、极值长度、调和度量、黎曼映射等-反过来，这些相同的联系解释了圆填充在各种其他主题中的用途。在函数论中，主要研究Smer猜想中的圆填充技术、保角平铺和Grothendieck‘s dessins d’fants的构造。在应用中，圆填充为图嵌入提供了一种独特的新工具，特别强调了在大脑平面化和二维猝灭方面的合作。圆填充的可计算性是一个核心优势，但不断增长的需求给“CirclePack”软件包带来了相当大的压力。该项目解决了移植和模块化问题，并研究了实现并行布局算法的策略。科学、技术和数学的许多领域在根本上依赖于组合信息，即具体和抽象的模式-DNA缠绕链中的交叉模式，晶体中的原子位置，视觉皮质中的激活位置。在二维环境中，这样的图案通常被表示为平面图，然后需要在视觉上呈现。从抽象拓扑学中出现了一种令人愉快的新方法。“圆填充”是具有特定切线图案的圆的配置。它具有两面性：在规定的图案中是“组合的”，但在实际圆的半径中是“几何的”。人们应该认为，圆圈将一种“自发”的几何图形带入了一种纯粹抽象的组合情况。幸运的是，这种几何继承了一个重要的经典数学主题-共形几何的关键特征，因此它带来了超过150年的理论发展的应用。然而，与经典理论不同的是，圆填充既是具体的，又是可计算的。该项目涉及圆填充算法和显示软件的开发和应用，特别强调与美国国立卫生研究院的一个项目合作，该项目涉及“脑图谱”、二维淬火的物理学以及数学分析中的离散解析函数理论。

9972769The investigator further develops the software tool CirclePack and uses it to explore mathematical questions that arise in several areas of application. Circle packing was introduced by Thurston, first in the context of orbifold constructions and then in the approximation of conformal mappings. The topic has been developed to the point that it provides a surprisingly comprehensive discrete theory that both parallels and approximates the classical theory of analytic functions, i.e., conformal mappings in two dimensions. Developments have been driven largely by classical connections --- discrete analyticity, extremal length, harmonic measure, Riemann mappings, and so forth --- and in turn those same connections account for circle packing's utility in a variety of other topics. In function theory, the research focuses on circle packing techniques in studies of a conjecture of Smale, conformal tilings, and construction of Grothendieck's dessins d'enfants. In applications, circle packing provides a unique new tool for graph embedding, and particular emphasis is placed on collaborations in brain-flattening and 2-dim quenching. Computability of circle packings is a central strength, but growing demands have placed considerable stress on the software package "CirclePack." The project addresses porting and modularization issues, and investigates strategies for implementing parallel packing algorithms.Many areas of science, technology, and mathematics rely in fundamental ways on combinatorial information, that is, patterns, both concrete and abstract --- crossing patterns in a twisted strand of DNA, atomic locations in a crystal, activation sites in the visual cortex. In 2-dimensional settings such patterns are often represented as planar graphs, which then need to be presented visually. A pleasing new approach has emerged from abstract topology. A "circle packing" is a configuration of circles having a specified pattern of tangencies. It has dual natures: "combinatoric" in the prescribed pattern, but "geometric" in the radii of the actual circles. One should think of the circles as bringing a "spontaneous" geometry to what began as a purely abstract combinatorial situation. Fortuitously, this geometry inherits key features of an important classical mathematical topic, conformal geometry, so it brings to applications over 150 years of theoretical development. However, unlike the classical theory, circle packings are both concrete and computable. This project involves the development and application of circle packing algorithms and display software, with particular emphasis on collaboration with an NIH project on "brain mapping," on the physics of 2-dimensional quenching, and on discrete analytic function theory in mathematical analysis.