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

这个项目结合了古典几何学的数学 函数论与圆的新应用 打包 这种联系的来源在于一种猜想， W. Thurston在1985年认为，一个人应该能够获得良好的 单连通平面Riemann映射的逼近 域通过域的圆形包装，然后是 通过规定的算法对圆进行映射。 这竟然 现在被称为有限黎曼映射 定理 这一证据引发了一系列新的调查， 复分析领域：什么类型的定理可以 证明了这一点，更重要的是， 新的发现是通过这种新技术来实现的吗？ 随着可能性的扩大，工作仍在继续。 研究的主题包括六方堆积和 渐近性和最大和最小半径之比， 乘正则共形映射的经典问题 连通区域-任意域是否共形 等价于具有圆的边界分量的域或 线 工作也将在密切扩展结果 已知的圆填充的Mobius变换， 收敛并给出黎曼的一阶和二阶导数 地图 高阶导数是否也可以 将进行近似检验。 复变函数理论包括研究 复变量的可微函数及相关函数 函数类，如调和函数和 拟共形映射 这个问题是高度几何;许多 的问题有关的性质，各种集下 通过上述类之一的函数进行转换。 应用于势理论和流体动力学， 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.