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

在这个项目中要做的工作集中在欧几里德空间中几何和数学分析的界面上产生的问题。该研究的基础在于复函数理论的几何发展，因为它强调解析函数理论的群结构和这些函数的无穷小行为。正是它们的无穷小规律性导致了20世纪30年代拟共形映射的概念。几十年后，这些映射的内在定义才被发现，它可以应用于更高维度的映射。这项工作是关于这些映射及其基础领域的。拟共形映射是两个域之间的一价映射，它将无穷小圆映射到其长、短轴具有有界（膨胀）比的无穷小椭圆。在理解它们的结构时，一个重要的问题是将映射分解为具有相同扩展的映射的迭代。那些没有这种分解的将构成不可约映射的基础。在几何方面，人们想知道如何确定两个域在拟共形映射下何时是等价的。即使两个域中的一个是球，这个问题也很难。舍恩菲斯型定理给出的充分条件与由局部连通性引起的必要条件相距甚远。在三维空间中，当一个域是狭缝域，另一个域是球时，已经取得了一些进展。其他的工作，集中在平面问题，将涉及克莱因群。这包括继续测量几个自然度量中元素与恒等的距离。当应用于群的生成器时，可以得到关于奇异集的信息。随着这项工作的进一步发展，高维类似物将被研究。我们还将继续研究离散收敛群，这是几年前引入的一类变换群，它反映了莫比乌斯群的许多性质（它们通过拟共形映射与莫比乌斯群共轭）。将对收敛群的精细结构进行研究。

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.