Mathematical Sciences: Conformal Mapping, Riemann Surfaces, and Circle Packings

本课题将经典几何函数理论的数学与新发现的圆填充的应用相结合。这种联系的来源是W. Thurston在1985年的一个猜想，即人们应该能够通过对一个简单连接平面区域进行圆填充，然后用规定的算法对圆进行映射，从而获得简单连接平面区域的黎曼映射的良好近似。事实证明确实如此，现在被称为有限黎曼映射定理。这一证明在复分析领域催生了一条新的研究路线：什么类型的定理可以通过圈填充来证明，更重要的是，通过这种新技术开辟了哪些新发现？随着可能性的扩大，工作仍在继续。研究的主题包括六边形填充及其渐近性和最大最小半径之比，多连通区域的正则共形映射的经典问题-任意域是否与具有圆或线的边界分量的域共形等价。我们还将扩展圆填充的密切莫比乌斯变换的结果，这些结果已知是收敛的，并给出黎曼映射的一阶和二阶导数。高阶导数是否也能被近似的问题将被研究。复变函数理论包括复变量的可微函数和相关的函数类，如调和函数和拟共形映射的研究。这个主题是高度几何化的；许多问题涉及由上述类之一的函数变换的各种集合的属性。位势理论和流体动力学的应用现在是工程界的标准。

This project combines the mathematics of classical geometric function theory with newly discovered applications of circle packing. The source of this connection lies in a conjecture of W. Thurston in 1985 that one should be able to obtain good approximations to the Riemann map of a simply connection planar domain through a circle packing of the domain following by a mapping of the circles by a prescribed algorithm. This turned out to be the case and is now known as the Finite Riemann Mapping Theorem. The proof spawned a new line of investigation within the field of complex analysis: what type of theorems can be proved by circle packing and, more importantly, what new discoveries are opened up by means of this new technique? Work now continues as the breadth of possibilities broadens. Topics to be investigated include hexagonal packing and the asymptotics and the ratios of maximum and minimum radii, the classical problem of canonical conformal maps of multiply connected regions - whether arbitrary domains are conformally equivalent to domains with boundary components of circles or line. Work will also be done in extending results on osculating Mobius transformations of a circle packing which are known to converge and give the first and second derivatives of the Riemann map. The question of whether higher order derivatives can also be approximated will be examined. Complex function theory encompasses the study of differentiable functions of a complex variable and related classes of functions such as harmonic functions and quasiconformal mappings. The subject is highly geometric; many of the problems concern the properties of various sets under transform by functions from one of the above classes. Applications to potential theory and fluid dynamics is now standard in engineering circles.