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

9305852 Gehring这个项目是关于二维和高维拟共形映射和莫比乌斯群的经典复分析及其现代衍生几何问题的数学研究。群上的工作集中在与椭圆发生器离散的两个发生器群的几何相关的问题上。有可能通过从产生器的痕迹中得到的复数三元组来研究群。但是并不是所有的三元组都来自两个生成元群，我们将努力确定确定离散群或由大于三阶的椭圆生成元产生的集合的描述。将本文的工作应用于三维完全双曲轨道的研究。特别是在已知的这些轨道的体积的下界已经发现了实质性的改进。这个项目的主要目标是证明每个双曲轨道的体积与对应的群包含一个大于3阶的椭圆元素的体积相同。这个值大约是0.03905…拟共形映射的工作继续在确定一个维数大于2的域何时拟共形等价于相同维数的单位球的基本问题上进行。充分条件通过schoenfly型定理已知。如果找到必要的条件，它们可能必须在特定类型的域的k-连通性中公式化。最后，本奖项支持的学生将考虑Hardy和Littlewood发现的经典函数理论关系如何延续到有界准盘。几何函数理论的研究主要集中在如何通过光滑函数将域映射到另一个域上，并在光滑函数上设置各种畸变限制。哪些领域可以相互转化的问题自然出现，这种观点自动导致具有基本后果的群论问题。＊＊＊

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. ***