Quasiconformal Mappings in Geometry and Analysis

拟共形映射和相关映射是目前可以用解析方法研究的最大的一类映射。因此，它们的理论和应用处于几何和分析的交叉点，并与数学和物理的许多其他领域有联系。虽然准共形映射理论的基本问题仍未解决，但度量空间分析的最新进展扩大了这些映射的适用范围。他们的理论现在可能导致以前无法解决的问题在其他领域，如几何群论。这个项目的目的是探索该地区的当前趋势。具体的研究课题包括拟对称均匀化，分形球上的动力学，拟共形雅可比问题，拟正则映射和椭圆流形。这门学科的根源可以追溯到高斯在19世纪上半叶对地图学和表面几何的研究。他创造了“共形映射”一词，并推导出了平面准共形映射理论的方程。这个理论的全部重要性直到大约一个世纪后才被认识到。到目前为止，拟保角映射已经发展成为当代几何函数理论的主要工具之一。

ABSTRACTQuasi-conformal and related mappings form the largest class of maps that can be studied by analytic methods. Accordingly, their theory and their applications lie at the intersection of geometry and analysis and have connections to many other areas of mathematics and physics. While fundamental questions at the foundation of the theory of quasi-conformal maps remain open, recent advances in analysis on metric spaces have enlarged the range of applicability of these maps. Their theory may now lead to solutions of previously inaccessible problems in other fields such as geometric group theory. The purpose of this project is to explore current trends in the area. Specific topics of research include quasi-symmetric uniformization, dynamics on fractal spheres, the quasi-conformal Jacobian problem, quasi-regular maps and elliptic manifolds.The roots of this subject can be traced back to Gauss's work on cartography and surface geometry in the first half of the 19th century. He coined the phrase ``conformal map" and derived equations that govern the theory of planar quasi-conformal mappings. The full importance of this theory was only realized about a century later. By now quasi- conformal mappings have developed into one of the major tools in contemporary Geometric Function Theory.