Mathematical Sciences: Teichmuller Spaces & Analysis

本项目扩展了以往在复变函数方面的研究 理论及其对拟共形映射的推广， 拟正则映射和Teichmuller理论。 所有人都被一个 关于平滑变换的几何观点， 限制变形 这一特殊的主题 活动涉及新开发的保形焊接方法。 主要目标之一是研究 保形焊接域与零面积的补充。 在理论上也得到了初步的结果。 迭代 本次调查的目的是确定 类型的朱莉娅集，人们可以期望通过焊接得到， 内部功能。 还将在以下问题上开展工作： 我们是否可以深入了解希尔伯特变换， 极大化Fredholm行列式的连通域 的变换，以获得一个等价的域，所有的 边界组件是圆盘。 复变函数理论包括研究 复变量的可微函数及相关函数 函数类，如调和函数和 拟共形映射 这个问题是高度几何;许多 的问题有关的性质，各种集下 通过上述类之一的函数进行转换。 该理论在势理论和流体力学中的应用 现在是工程界的标准。

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.