Mathematical Sciences: Problems in Complex Analysis

这项数学研究的中心焦点是 复变函数论和拟共形映射中的问题。 这项工作的主要目标之一是将 欧几里得空间的某些同胚群。 中 最好理解的群是所谓的莫比乌斯群， 元素由最简线性表示的群 （分数）变换。 分类问题是一个 确定哪些群拓扑共轭 一个莫比乌斯群，并了解障碍是什么， 可能会阻止这种群体同构。在目标群体中 是一个称为收敛群的群，它有许多性质， 拟共形映射的群。 许多趋同群体 已知与莫比乌斯群共轭，但到目前为止，没有 已经找到了完整的分类。工作将在一个 努力确定这些群体在多大程度上享有 与Mobius群有相同的性质 第二条调查路线涉及注入性标准 局部单叶函数 这项工作寻求条件 保证了局部单叶亚纯函数的扩张 域的邻域到拟共形映射的映射 整个复杂的平面。 延长期将以 复杂的膨胀仍然以一个固定的量为界。 关于这类问题有大量文献。 这项工作的一个具体目标是消除条件 的邻域内全纯 一个域，并且仅在闭域中假设此属性。 如果放弃初始函数为 亚纯的，那么扩张问题就变得太一般了。 然而，一个自然条件对交比的复杂性 在磁盘内的函数导致更严格的类， 拟共形扩展（内射性问题）已经被 显示存在。 将继续努力确定 保证注入性的交比。 //

The central focus of this mathematical research concerns problems in complex function theory and quasiconformal mappings. One of the primary objectives of the work is that of classifying certain groups of homeomorphisms of Euclidean space. Among the best understood groups are the so-called Mobius groups, those groups whose elements are represented by simplest linear (fractional) transformations. The classification problem is one of determining which other groups are topologically conjugate to a Mobius group and to understand what the obstruction is that might prevent such a group isomorphism. Among the target groups is one known as the convergence group which has many properties of groups of quasiconformal mappings. Many convergence groups are known to be conjugate to Mobius groups, but to date, no complete classification has been found. Work will be done in an effort to determine the extent to which these groups all enjoy the same properties relative to Mobius groups. A second line of investigation concerns injectivity criteria for locally univalent functions. This work seeks conditions which guarantee the extension of locally univalent meromorphic mappings from neighborhoods of a domain to a quasiconformal map of the entire complex plane. The extension is to be carried so that the complex dilatation remains bounded by a fixed amount. There is a sizeable literature concerning questions of this type. One specific objective of this work is to remove the condition that the meromorphic function be holomorphic in a neighborhood of a domain and only assume this property within a closed domain. If one drops the condition that the initial function be meromorphic, then the extension problem becomes too general. However, a natural condition on the cross-ratio of a complex function within a disc leads to a more restrictive class where quasiconformal extensions (the injectivity problem) have been shown to exist. Work will continue in determining precise conditions on cross-ratios which will guarantee injectivity. //