Computational and Conformal Geometry

克里斯托弗·毕晓普将研究二维和三维几何中的问题，这些问题来自经典的复分析、拟共形映射理论、双曲几何和计算几何，特别强调这些领域之间的相互作用。在最近的工作中，PI证明了来自双曲和计算几何的想法可以帮助解决经典分析的一个问题：如何有效地计算平面区域上的共形映射。这项工作将扩展到新的环境，并将用于研究计算几何中出现的问题，例如，具有期望属性的网格的有效生成。PI将研究木匠规则问题从计算几何到弦弧曲线变形的分析问题的可能应用。他还将继续研究拟共形映射的变形性质和共形焊接问题。PI，Christopher Bishop将继续他对共形和拟共形映射的理论和计算的研究。保角贴图将一个区域发送到另一个区域，以便保留角度(但不一定是距离)。拟共形贴图可能会扭曲角度，但幅度有限。最古老和最著名的例子之一是墨卡托投影，它将球形地球映射到一张扁平的纸上。这是不可能在没有某种失真的情况下完成的，对于导航来说，保留角度比保留距离更方便(因此，查看图表，您知道正确的方向，如果不是要覆盖的确切距离的话)。保角映射也出现在许多应用中，在这些应用中，人们想要将问题从具有复杂几何的区域转移到更简单的区域(如圆盘)，在那里更容易求解。例如空气动力学、流体流动、振动膜、热流、静电学和许多其他问题。保角映射在诸如动力学、复杂分析、概率和几何等领域的纯数学中也扮演着基本的角色。由于它们的重要性，有许多方法可以用来数值计算保角映射，但不同的方法在不同的情况下工作得最好，而且对于足够复杂的区域，一种方法往往失败。PI开发了一种算法，该算法保证适用于大类区域，并能够估计达到给定精度所需的时间。该算法基于计算机科学中的几何概念与纯数学中出现的非欧几里德几何之间的新联系。在目前的提议下，PI将寻求将这一理论算法转化为实用方法，研究更复杂的区域和更高维度的概括，并将该方法应用于计算机科学中出现的问题。

The PI, Christopher Bishop, will study problems in two and three dimensional geometry which come from classical complex analysis, the theory of quasiconformal mappings, hyperbolic geometry and computational geometry, with a particular emphasis on interactions between these areas. In recent work the PI has shown that ideas from hyperbolic and computational geometry could help resolve a question of classical analysis: how to efficiently compute conformal maps onto planar domains. This work will be extended to new settings, and will be used to study problems arising in computational geometry, e.g., efficient generation of meshes with desirable properties. The PI will investigate the possible application of the carpenter's rule problem from computational geometry to an analysis problem about deforming chord-arc curves. He will also continue his work on the distortion properties of quasiconformal mappings and on the conformal welding problem.The PI, Christopher Bishop, will continue his investigations into the theory and computation of conformal and quasiconformal maps. Conformal maps send one region to another so that angles (but not necessarily distances) are preserved. Quasiconformal maps may distort angles, but only by a limited amount. One of the oldest and most famous examples is the Mercator projection which maps the spherical Earth to flat piece of paper. This can't be done without some sort of distortion, and for navigation it is more convenient to have angles preserved than distances (so looking at a chart you know the correct direction to head, if not the exact distance to be covered). Conformal maps also appear in many applications where one wants to transfer a problem from a domain with complicated geometry to a simpler one (such as a disk) where it is easier to solve. Examples come from aerodynamics, fluid flow, vibrating membranes, heat flow, electrostatics and many other problems. Conformal maps also play a fundamental role within pure mathematics in areas such as dynamics, complex analysis, probability and geometry. Because of their importance, there are numerous techniques for computing conformal maps numerically, but different methods work best in different situations and often a method fails for sufficiently complicated regions. The PI has developed an algorithm that is guaranteed to work for a large class of regions, and is able to estimate the time needed to achieve a given accuracy. The algorithm is based on new connections between geometric concepts from computer science and non-Euclidean geometry arising in pure mathematics. Under the current proposal the PI will seek to turn this theoretical algorithm into a practical method, investigate generalizations more complicated regions and to higher dimensions, and apply the method to problems arising from computer science.