Analysis of conformal and quasiconformal maps

摘要奖：DMS-1006309主要研究员：Christopher Bishop主要研究共形和拟共形映射的几何性质，重点研究它与其他领域的联系，如数值分析、计算几何、复动力学和几何测度论。他将继续他在快速保角映射算法方面的工作，包括Schwarz-Christoffel迭代类。这些方法包括作为特例的Davis方法和Driscoll和Vavasis的CRDT算法，以及涉及中轴和拟共形映射的其他方法。其中一个基本主题是从某些拟共形映射开始，这些拟共形映射近似于所需的共形映射，但更容易计算并具有更好的性质。这些地图与布伦南猜想等几个众所周知的问题有着天然的联系，PI将调查这些联系。PI还将继续他在各种相关问题上的工作，包括最优网格划分算法、扩散限制聚集、布朗运动的几何、拟共形雅可比问题和保角焊接。数学和工程的一个基本问题是快速准确地求解与复杂区域上的流体流动、热传导和波传播有关的各种微分方程。对于二维曲面，一种标准的方法是使用保角映射(即，在小尺度上保持角度)，将区域替换为更简单的区域，如圆盘或矩形。这种方法已经研究了150多年，但直到1980年代的S，计算机才使保角映射在高度复杂的地区变得实用。PI将研究如何改进现有方法，并开发更快、更可靠的新方法。他还将研究极复杂区域上的共形映射的理论行为，如分形图，以及共形映射与概率论之间的联系。PI已经将从这些问题中获得的见解应用到啮合中。这是将复杂区域划分为简单的部分(如三角形)的过程。网格化是大多数数值方法的基本部分，网格在应用中的有用性取决于网格的数量(越少越好)及其形状(最好避免小角度和大角度)。很难构建一个在这两个方面都好的网格，但PI使用非欧几里德几何为简单的多边形构造了具有最佳大小和形状的二维四边形网格，并正在努力将其扩展到更一般的区域。在涉及裂纹形成、材料之间的界面和计算机学习的各种问题上，需要更大的普遍性。高效网格化在高性能计算中也有大量的应用，如工程曲面建模和计算机图形学。该奖项由分析和几何分析项目共同资助。

AbstractAward: DMS-1006309Principal Investigator: Christopher BishopThe principal investigator will study the geometric properties of conformal and quasiconformal maps, with an emphasis on the connections with other areas such as numerical analysis, computational geometry, complex dynamics and geometric measure theory. He will continue his work on fast conformal mapping algorithms, including the class of Schwarz-Christoffel iterations. These include Davis' method and the CRDT algorithm of Driscoll and Vavasis as special cases, and other methods involving the medial axis and quasiconformal mappings. One of the basic themes is to start with certain quasiconformal maps that approximate the desired conformal map, but are easier to compute and have better properties. These maps have natural connections to several well known problems such as Brennan's conjecture, and the PI will investigate these connections. The PI will also continue his work on various related problems involving optimal meshing algorithms, diffusion limited aggregation, the geometry of Brownian motion, the quasiconformal Jacobian problem and conformal welding.A fundamental problem of mathematics and engineering is to quickly and accurately solve various differential equations related to fluid flow, heat conduction and wave propagation on complex regions. For 2-dimensional surfaces, a standard method is to use a conformal mapping (i.e., angle preserving on small scales) to replace the region with a simpler one, such as a disk or rectangle. This approach has been studied for over 150 years, but not until the 1980's did computers made conformal mapping practical for highly complex regions. The PI will investigate how to improve existing methods and develop new ones that are faster and more reliable. He will also investigate the theoretical behavior of conformal maps on extremely complex domains such as fractals and the connections between conformal maps and probability theory. The PI has already applied the insights gained from these problems to meshing. This is the process of dividing a complex region into simple pieces such as triangles. Meshing is a basic part of most numerical methods and the usefulness of a mesh in applications depends on the number of pieces (fewer is better) and their shapes (better to avoid small and large angles). It is difficult to construct a mesh which is good in both respects, but PI has used non-Euclidean geometry to construct 2-dimensional quadrilateral meshes with optimal size and shapes for simple polygons and is working to extend this to more general domains. Greater generality is needed in various problems involving crack formation, interfaces between materials and computer learning. Efficient meshing also has numerous applications in high-performance computing such as modeling surfaces for engineering and computer graphics. This award is jointly funded by the programs in Analysis and Geometric Analysis.