Analysis and geometry on non-smooth spaces

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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.