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

该项目的工作重点是解决 在几何和数学分析的界面上， 欧几里得空间 研究的基础在于 复变函数理论的几何发展，因为它 强调分析函数背后的群体结构 理论和无穷小的行为，这样的功能。 这是 他们的无穷小规律性导致了 在1930年的拟共形映射。 几十年过去了， 找到了这些映射的内在定义， 应用于更高维度的映射。 这项工作是关于 这些映射及其底层域。 拟共形映射是两个 将无穷小圆映射为无穷小的域 椭圆的长轴和短轴有界（ 膨胀）比率。 理解中的一个重要问题 它们的结构涉及到地图分解成 具有相同伸缩的映射的迭代。 那些没有这样的 分解将形成不可约映射的基础。 从 几何方面，一想知道如何确定时，两个 域在拟共形映射下是等价的。 即使 两个域中的一个是球，这个问题非常 难 Schoenflies型给出的充分条件 定理和由局部连通性产生的必要定理 相距甚远。 在三个方面取得了一些进展 当一个域是狭缝域而另一个域是狭缝域时， 是一个球。 其他工作，集中在平面问题，将 关于Kleinian群 这包括继续开展工作， 测量元素与几个单位的距离， 自然度量当应用于群的生成元时， 可以获得关于奇异集的信息。 随着这项工作 进一步发展，将研究更高维的类似物。 工作也将继续离散收敛群，一类 几年前引入的转换组， 反映了Mobius群的许多性质（它们与 这类群的拟共形映射）。 的调查 进行衔接小组的精细化建设。

Work to be done on this project focuses on questions arising at the interface of geometry and mathematical analysis in Euclidean spaces. The basis for the research lies within the geometric development of complex function theory, in that it emphasizes the group structure underlying analytic function theory and the infinitesimal behavior of such functions. It was their infinitesimal regularity that led to the notion of quasiconformal mappings in the 1930's. Decades passed before an intrinsic definition of these mappings was found which could be applied to mappings in higher dimensions. This work is about those mappings and their underlying domains. A quasiconformal mapping is a univalent mapping between two domains which maps infinitesimal circles into infinitesimal ellipses whose major and minor axes have bounded (the dilatation) ratios. An important question in the understanding of their structure concerns the decomposition of maps into iterates of maps with the same dilatation. Those with no such decomposition would form a basis of irreducible maps. From the geometric side, one would like to know how to determine when two domains are equivalent under quasiconformal mappings. Even when one of the two domains is a ball, this question is very difficult. The sufficient conditions given by Schoenflies type theorems and the necessary ones arising from local connectivity are very far apart. Some progress has been made in three dimensions when one of the domains is a slit domain and the other is a ball. Other work, concentrating on planar problems, will be concerned with Kleinian groups. This includes continuing work on measuring the distance of elements from the identity in several natural metrics. When applied to generators of groups, information about the singular set can be obtained. As this work develops further, higher dimensional analogues will be studied. Work will also continue on discrete convergence groups, a class of transformation groups introduced several years ago which reflect many properties of Mobius groups (they are conjugate to such groups by quasiconformal maps). Investigations into the fine structure of convergence groups will be conducted.