Mathematical Sciences: Deformations of Complex Structures

该项目涉及复分析和函数理论的几个不同方面：通过Kleinian群的Eichler上同调的Riemann曲面的模，Schottky问题，二维量子引力的几何公式，以及Hausdorff维数和共形映射的边界行为是提出的主题之一。函数论是研究单自变量复变函数的学科，有一个经典的起源。函数理论的一个值得注意的方面是黎曼曲面的使用；给定一个局部定义的复函数，我们可以通过解析延拓来关联一个抽象（多片）曲面，即该函数的黎曼曲面。这个黎曼曲面包含了原始函数及其可能的正则扩展的所有基本信息。所提议的项目与分类和理解这些黎曼曲面的整体有关。

9500557 Maskit This project is concerned with several different aspects of complex analysis and function theory: moduli of Riemann surfaces via Eichler cohomology of Kleinian groups, the Schottky problem, geometric formulation of 2-dimensional quantum gravity, and Hausdorff dimension and the boundary behavior of conformal mappings are among the topics proposed. Function theory is the study of functions of one independent complex variable, and has a classical origin. A notable aspect of function theory is the use of Riemann surfaces; given a locally defined complex function one associates an abstract (multi-sheeted) surface, called the Riemann surface of the function, via analytic continuation. This Riemann surface carries all the essential information about the original function and its possible regular extensions. The proposed project has to do with classifying and understanding the totality of such Riemann surfaces.