Quasiconformal geometry of fractals

在目前的项目中，PI打算探索一些关于自相似分形和相关几何结构的开放性问题。考虑的分形通常产生于群的动力学或迭代下的映射的动力学，并且通常更好地理解它们的拟共形几何是至关重要的。具体的调查课题包括几何的谢尔宾斯基地毯，动力学的扩大瑟斯顿地图，并在粗糙和度量几何的熵刚性的实例。 分形和自相似特征可以在许多自然现象中看到，如海岸线和断层线，雪花和晶体生长，放电或植物生长模式。 该项目旨在帮助开发加深对此类结构的理解所必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。目标是为他们提供独立调查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.