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.

在当前项目中，PI打算探讨有关自相似分形和相关几何结构的一些开放问题。所考虑的分形通常是由群体的动力学或迭代下地图的动力学产生的，并且通常对它们的准文形几何形状有更好的了解至关重要。调查的具体主题包括Sierpinski地毯的几何形状，扩展瑟斯顿地图的动力学以及粗糙和度量的几何形状中的熵刚度实例。 在许多自然现象中可以看到分形和相似的特征，例如沿海和断层线，雪花和晶体生长，电气排放或植物生长模式。 该项目打算为开发基本方法和工具的开发做出贡献，这些方法和工具是对这种结构的更深入理解所必需的。这项活动的重要部分是年轻研究人员的参与。目的是为他们提供对分形现象的独立研究所需的数学专业知识。