Mathematical Sciences: Discrete Groups, Quasiconformal Mappings, and Function Theoretic Properties

小行星9305852 该项目关注的是与经典复分析及其更现代的准共形映射和莫比乌斯群在二维和更高维的衍生物相关的几何问题的数学研究。 工作组的重点是有关问题的几何离散两个发电机群的椭圆发电机。 可以通过由迹导出的复数的三元组来研究群 的发电机。 但不是所有的三元组来自两个发电机组和努力确定描述的集合确定离散群或集合所产生的椭圆发电机的顺序大于三个将进行。 本文将这一工作应用于三维完全双曲轨道褶皱的研究. 特别是大量的改进，在已知的下限为这些orbifolds的体积已经发现。本项目的主要目标是表明，每双曲orbifold的体积是相同的，相应的组包含一个椭圆形元素的顺序大于3。 该值约为0.03905...工作在quasiconformal映射继续对基本问题的确定时，一个域的尺寸大于2 quasiconformally相当于单位球的相同尺寸。 充分条件是已知的Schoenflies型定理。 如果找到了必要的条件，它们可能必须在域的某些类型的k-连通性中公式化。 最后，这个奖项支持的学生将考虑如何经典的函数理论关系发现的哈代和Littlewood结转到有界准磁盘。 几何函数论的研究主要集中在域如何通过光滑函数相互映射，在光滑函数上施加各种变形限制。 问题的领域可以转化为另一个自然出现，这一观点自动导致群论问题的基本后果。 ***

9305852 Gehring This project is concerned with mathematical research on geometric problems associated with classical complex analysis and its more modern derivatives of quasiconformal mapping and Mobius groups in two and higher dimensions. The work on groups focuses on questions related to the geometry of discrete two generator groups with an elliptic generator. It is possible to study the groups through triples of complex numbers derived from the traces of the generators. But not all triples come from two-generator groups and efforts to determine description of the sets which determine discrete groups or the sets arising from elliptic generators of order greater than three will be undertaken. Applications of this work are made to the study of three- dimensional complete hyperbolic orbifolds. In particular substantial improvements in known lower bounds for the volume of these orbifolds has been found. The primary goal of this project is to show that the volume of every hyperbolic orbifold is the same as those for which the corresponding group contains an elliptic element of order greater than three. This value is approximately .03905... Work in quasiconformal mapping continues on the basic issue of determining when a domain in dimension greater than two is quaiconformally equivalent to the unit ball of the same dimension. Sufficient conditions are known through Schoenflies type theorems. If necessary conditions are found, they will probably have to be formulated in certain types of k- connectivity of the domains. Finally, students supported by this award will consider how classical function theory relations discovered by Hardy and Littlewood carry over to bounded quasidisks. Studies in geometric function theory focus on how domains are mapped into one another by smooth functions on which various distortion restrictions are placed. Questions of which domains can be transformed into one another arise naturally and this point of view automati cally leads to group theoretic questions of fundamental consequence. ***