Quasiconformal methods in analysis, geometry and dynamics

PI，克里斯托弗·毕晓普，将研究共形和拟共形映射的几何性质，重点是与其他领域的联系，如动力学，计算几何，数值分析和几何测度理论。 该提案描述了PI将使用拟共形映射作为主要工具进行调查的几个领域。第一个是平面上复解析函数的迭代理论。 PI以前曾使用拟共形方法来解决该领域的一些问题，并计划改进这些技术来解决几个剩余的开放问题。第二个领域是在计算几何中使用拟共形映射和双曲几何，主要是在与最佳复杂性和几何形状的平面区域网格化算法相关的问题中。在三个维度上获得类似的结果是该领域最重要的目标之一，该提案也描述了解决这个问题的一些想法。建议的最后一部分描述了拟共形映射的维数失真性质。这是一个最有趣的问题内在的研究拟共形映射，但正如该提案所述，也可以链接到著名的开放问题在计算geometrical.The拟议的工作将探讨如何组合和离散的想法产生新的结果，在分析和思想如何从共形分析和双曲几何可以证明新的定理离散几何和网格。 保角保角映射这些地图已经被深入研究了150多年，对分析、几何、概率、物理和工程中的各种问题都具有根本的重要性。 经典上，保角映射已被用于研究与流体流动、热传导和波传播有关的各种微分方程。 最近，共形映射已基本的统计力学，渗流和随机增长模型的研究。 准共形映射允许控制角度失真量。这些映射是共形映射的一个灵活且非常有用的推广，帮助我们更好地理解共形映射的特殊情况，但也引入了许多重要的新问题和新技术。 PI以前的工作已经使用拟共形映射给出了最著名的算法，用于计算到多边形上的共形映射，以及最著名的算法，用于将平面域网格化为具有最佳几何特性的三角形或四边形。 这种网格在许多数值问题中起着重要的作用，从计算机图形到偏微分方程的有限元方法，如果底层网格具有良好的几何特性，这些方法中的许多方法会更有效地工作。 拟议的工作将扩展和锐化已经获得的结果，并将使用类似的想法来攻击其他问题中出现的动力学和几何。

The PI, Christopher Bishop, will study the geometric properties of conformal and quasiconformal maps, with an emphasis on the connections with other areas such as dynamics, computational geometry, numerical analysis and geometric measure theory. The proposal describes several areas that the PI will investigate using quasiconformal mappings as a primary tool. The first is the iteration theory of complex analytic functions on the plane. The PI has previously used quasiconformal methods to solve a number of problems in the field and plans to refine these techniques to attack several remaining open problems. The second area is to use quasiconformal maps and hyperbolic geometry in computational geometry, mostly in problems related to algorithms for meshing planar regions with optimal complexity and geometry. Obtaining similar results in three dimensions is one of the most important goals of the field, and the proposal describes some ideas for attacking this problem too. The final part of the proposal describes the dimension distortion properties of quasiconformal maps. This is one of the most interesting problems intrinsic to study of quasiconformal maps, but, as the proposal describes, can also be linked to famous open problems in computational geometry.The proposed work will investigate how combinatorial and discrete ideas yield new results in analysis and how ideas from conformal analysis and hyperbolic geometry can prove new theorems about discrete geometry and meshing. Conformal maps preserve angles; these maps have been intensively studied for over 150 years and are of fundamental importance to a wide variety of problems in analysis, geometry, probability, physics and engineering. Classically, conformal maps have been used in the study of various differential equations related to fluid flow, heat conduction and wave propagation. More recently, conformal maps have been fundamental to the study of statistical mechanics, percolation and random growth models. Quasiconformal maps allow a controlled amount of angle distortion. These maps are a flexible and extremely useful generalization of conformal maps that help us better understand the special case of conformal mappings, but also introduces many important new problems and techniques. The PI's previous work has used quasiconformal maps to give the best known algorithm for computing conformal maps onto polygons and the best known algorithms for meshing planar domains into triangles or quadrilaterals with optimal geometric properties. Such meshes play an important role in many numerical problems from computer graphs to finite element methods for PDEs and many of these methods work more effectively if the underlying mesh has good geometric properties. The proposed work will extend and sharpen the results already obtained and will use similar ideas to attack other problems that arise in dynamics and geometry.