Mathematical Sciences: Teichmuller Spaces & Analysis

本课题将复变函数理论及其推广推广到拟共形映射、拟正则映射和Teichmuller理论。所有这些都是由一个几何观点联系在一起的，关于具有有限失真的平滑变换。这一特殊活动的主题涉及新开发的保形焊接方法。其中一个主要目标将是研究具有零面积补的共形焊接域的唯一性。在迭代理论方面也得到了初步的结果。本研究的目的是确定可以通过内部功能通过焊接获得的朱莉娅组的类型。我们还将讨论是否可以通过最大化变换的Fredholm行列式来获得所有边界分量都是圆盘的等效域，从而深入了解有限连通域上的希尔伯特变换。复变函数理论包括复变量的可微函数和相关的函数类，如调和函数和拟共形映射的研究。这个主题是高度几何化的；许多问题涉及由上述类之一的函数变换的各种集合的属性。该理论在位势理论和流体力学中的应用现已成为工程界的标准。

This project extends previous studies in complex function theory and its generalizations to quasiconformal mappings, quasiregular mappings and Teichmuller theory. All are tied by a geometric point of view regarding smooth transformations with restricted distortion. The main themes of this particular activity involve newly developed methods of conformal welding. One of the primary goals will be to study the uniqueness of conformally welded domains with complements of zero area. Preliminary results have also been obtained in the theory of iteration. The object of this investigation is to determine the types of Julia sets one can expect to get through welding by inner functions. Work will also be done on the question of whether one can gain insight into the Hilbert transform on finitely connected domains by maximizing the Fredholm determinant of the transform to obtain an equivalent domain all of whose boundary components are discs. Complex function theory encompasses the study of differentiable functions of a complex variable and related classes of functions such as harmonic functions and quasiconformal mappings. The subject is highly geometric; many of the problems concern the properties of various sets under transform by functions from one of the above classes. Applications of the theory to potential theory and fluid dynamics is now standard in engineering circles.