Computational and Conformal Geometry

PI Christopher Bishop将研究二维和三维几何中的问题，这些问题来自经典复分析，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已经表明，双曲几何和计算几何的思想可以帮助解决经典分析中的一个问题：如何有效地计算平面域上的保角映射。这项工作将扩展到新的设置，并将用于研究计算几何中出现的问题，例如，具有理想属性的网格的有效生成。PI将研究计算几何中木工规则问题在弦弧曲线变形分析问题中的可能应用。他还将继续研究拟共形映射的畸变性质和共形焊接问题。PI Christopher Bishop将继续研究保角和拟保角映射的理论和计算。保角映射将一个区域传送到另一个区域，这样角度（但不一定是距离）就会保持不变。拟共形映射可能会扭曲角度，但只是有限的程度。最古老和最著名的例子之一是墨卡托投影，它将球形的地球映射到一张平坦的纸上。如果没有某种形式的扭曲，这是不可能做到的，对于导航来说，保留角度比保留距离更方便（所以看着图表，你知道正确的方向，如果不是要覆盖的确切距离）。保角映射也出现在许多应用程序中，当人们想要将问题从具有复杂几何形状的域转移到更容易解决的简单域（例如磁盘）时。例子来自空气动力学、流体流动、振动膜、热流、静电学和许多其他问题。保角映射在动力学、复杂分析、概率论和几何等纯数学领域也发挥着重要作用。由于它们的重要性，有许多计算共形映射的数值技术，但不同的方法在不同的情况下效果最好，并且通常一种方法在足够复杂的区域失效。PI开发了一种算法，该算法保证适用于大量区域，并且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的非欧几里得几何之间的新联系。根据目前的提案，PI将寻求将这种理论算法转化为实用方法，研究更复杂区域和更高维度的泛化，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.