Quasiconformal geometry of fractals

在当前的项目中，PI打算探索一些关于自相似分形及其相关几何结构的公开问题。所考虑的分形图通常来自于群的动力学或迭代下的映射的动力学，并且通常对它们的准共形几何有更好的理解是至关重要的。具体的研究主题包括Sierpinski地毯的几何，扩展瑟斯顿映射的动力学，以及粗略几何和公制几何中的熵刚性实例。在海岸线和断层线、雪花和晶体生长、放电或植物生长模式等许多自然现象中都可以看到分形和自相似特征。这些项目旨在促进制定加深了解这类结构所需的基本方法和工具。这项活动的一个重要部分是青年研究人员的参与。其目的是为他们提供独立研究分形现象所需的数学专业知识。

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.