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

这个项目结合了经典几何函数理论的数学和新发现的圆填充的应用。这种联系的根源在于W.瑟斯顿在1985年提出的一个猜想，即人们应该能够通过对简单连通平面区域进行圆填充，然后用规定的算法对圆进行映射来获得对该区域的Riemann映射的良好逼近。事实证明就是这样，现在被称为有限黎曼映射定理。这一证明在复分析领域内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，通过这种新技术可以打开什么新的发现？随着可能性的扩大，这项工作现在还在继续。要研究的主题包括六角填充和最大与最小半径的渐近性和比率，多连通区域的正则共形映射的经典问题--任意区域是否与具有圆或线边界分量的区域共形等价。我们还将推广关于圆填充的密切Mobius变换的结果，已知这些变换是收敛的，并给出黎曼映射的一阶和二阶导数。高阶导数是否也可以近似的问题将被研究。复变函数论研究复变数的可微函数和相关的函数类，如调和函数和拟共形映射。这个主题是高度几何的；许多问题涉及上述类中的一个函数变换下的各种集合的性质。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.