Analysis and geometry on non-smooth spaces

在许多自然现象中，出现了经典数学工具无法描述的几何图案。通常这些模式具有自相似或分形的性质。例子包括雪花和水晶、城市和植物生长、山脉和海岸线、闪电或河流网络。该项目旨在促进分析方法的发展，这些方法有助于研究这些几何特征并加深对它们的理解。PI将让他的博士生和其他年轻研究人员参与这项活动。这将有助于增加这一领域的专业知识，并有助于维持一个为科学和工程的进步提供必要的数学知识的科学界。在数学中，分形或非光滑空间通常作为Kleinian群的极限集或有理映射的Julia集出现在动力系统中。在这个项目中，将探索这些空间的分析和几何，并为他们的研究创造新的数学工具。具体地说，PI打算研究在相当一般的度量空间上的某些sobolev型函数与其双曲填充上的相应函数之间的关系。在项目的第二部分，这将应用于一个相关的模型案例，其中这种非光滑空间出现，即瑟斯顿地图的动态。

In many natural phenomena geometric patterns arise that cannot be described by classical mathematical tools. Often these patterns have a self-similar or fractal nature. Examples include snowflakes and crystals, urban and plant growth, mountain ranges and coastlines, lightning bolts, or river networks. This project intends to contribute to the development of analytic methods that are useful for the investigation of such geometric features and their deeper understanding. The PI will involve his PhD students and other young researchers in this activity. This will contribute to increasing the expertise in this area and will help to maintain a scientific community that provides the necessary mathematical knowledge for progress in science and engineering. In mathematics fractal or non-smooth spaces often arise from dynamical systems as limit sets of Kleinian groups or as Julia sets of rational maps. In this project the analysis and geometry of such spaces will be explored and new mathematical tools for their investigation be created. Specifically, the PI intends to study the relation between certain Sobolev-type functions on fairly general metric spaces and corresponding functions on their hyperbolic fillings. In the second part of the project this will be applied to a relevant model case where such non-smooth spaces appear, namely the dynamics of Thurston maps.