Quasiconformal Mappings in Geometry and Analysis

几何和分析中的拟共形映射

• 批准号: 1058772
• 负责人: Mario Bonk
• 金额: $ 9.39万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1058772&HistoricalAwards=false
• 关键词: Quasiconformal; Mappings; Geometry; Analysis

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.

抽象Quasi-grom-grom-grom-grom-norfral和相关映射构成了可以通过分析方法研究的最大类图。因此，它们的理论及其应用在于几何和分析的交集，并与许多其他数学和物理学领域建立了联系。 虽然在准形式图理论基础上的基本问题仍然开放，但对度量空间分析的最新进展扩大了这些地图的适用性范围。他们的理论现在可能导致在其他领域（例如几何群体理论）中的以前无法访问的问题的解决方案。该项目的目的是探索该地区的当前趋势。研究的具体主题包括准对称均匀化，分形球的动力学，准符合形式的雅各布问题，准规范地图和椭圆形歧管。该主题的根源可以追溯到19世纪上半叶在制图和表面几何上的工作。他创造了``共形图表''一词，并得出了控制平面准形式映射理论的方程式。该理论的全部重要性仅在一个世纪后才实现。到目前为止，准整形型映射已经发展成为当代几何学函数理论的主要工具之一。