Quasiconformal geometry of fractals

分形的拟共形几何

• 批准号: 1162471
• 负责人: Mario Bonk
• 金额: $ 40.5万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1162471&HistoricalAwards=false
• 关键词: Quasiconformal; geometry; fractals

In the current project the PI intends to explore some open problems about self-similar fractals and related geometric structures. The fractals considered often arise from the dynamics of groups or the dynamics of maps under iteration, and often a better understanding of their quasiconformal geometry is of crucial relevance. Specific topics for investigation include the geometry of Sierpinski carpets, dynamics of expanding Thurston maps, and instances of entropy rigidity in coarse and metric geometry. Fractal and self-similar features can be seen in many natural phenomena such as coast- and fault lines, snowflakes and crystal growth, electric discharge or plant growth patterns. This projects intends to contribute to the development of basic methods and tools that are necessary for a deeper understanding of such structures. An important part of this activity is the involvement of young researchers. The goal is to provide them with the mathematical expertise necessary for independent investigations of fractal phenomena.

在当前的项目中，PI 打算探索一些有关自相似分形和相关几何结构的开放问题。所考虑的分形通常源自迭代下的群动态或图动态，并且通常更好地理解它们的拟共形几何具有至关重要的相关性。研究的具体主题包括谢尔宾斯基地毯的几何形状、瑟斯顿图扩展的动力学以及粗略和公制几何中的熵刚性实例。 分形和自相似特征可以在许多自然现象中看到，例如海岸线和断层线、雪花和晶体生长、放电或植物生长模式。 该项目旨在促进开发深入了解此类结构所必需的基本方法和工具。这项活动的一个重要组成部分是年轻研究人员的参与。目标是为他们提供独立研究分形现象所需的数学专业知识。