Computational Discrete Conformal Geometry and Applications

9972769研究者进一步开发了软件工具CirclePack，并使用它来探索几个应用领域中出现的数学问题。 圆填充是由瑟斯顿首先在轨道构造的上下文中，然后在共形映射的近似中引入的。 这个主题已经发展到这样一个地步，它提供了一个令人惊讶的全面离散理论，既平行，又近似于经典理论的解析函数，即，二维共形映射 发展主要是由经典的连接--离散解析性，极值长度，调和测度，黎曼映射，等等--反过来，这些相同的连接占圆包装的效用在各种其他主题。 在函数论中，研究重点是Smale猜想研究中的圆填充技术，共形镶嵌和Grothendieck的dessins d 'enfants的构建。 在应用中，圆包装提供了一个独特的新工具，图嵌入，并特别强调在大脑平坦化和2-dimquenching的合作。 圆包装的可计算性是一个核心优势，但不断增长的需求已经把相当大的压力放在软件包“CirclePack。该项目解决了移植和模块化问题，并研究了实现并行打包算法的策略。科学、技术和数学的许多领域都以基本的方式依赖于组合信息，即模式，既有具体的也有抽象的--DNA扭曲链中的交叉模式，晶体中的原子位置，视觉皮层中的激活位点。 在二维环境中，这种模式通常表示为平面图，然后需要可视化地呈现。 从抽象拓扑学中出现了一种令人愉快的新方法。 “圆填充”是具有特定相切图案的圆的配置。 它具有双重性质：“组合”在规定的模式，但“几何”的半径的实际圆。 人们应该把圆看作是把一种“自发的”几何引入到最初纯粹抽象的组合情形中。 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，圆填充既具体又可计算。 该项目涉及圆包装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关于“脑映射”、二维淬火物理学和数学分析中的离散解析函数理论的项目的合作。

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.