Analysis and geometry on non-smooth spaces

非光滑空间的分析和几何

• 批准号: 1506099
• 负责人: Mario Bonk
• 金额: $ 37.5万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1506099&HistoricalAwards=false
• 关键词: Analysis; geometry; non; smooth; spaces

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.

在许多自然现象中，出现了无法用经典数学工具描述的几何图案。通常这些图案具有自相似或分形性质。例子包括雪花和晶体，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，有助于调查这些几何特征及其更深入的理解。PI将邀请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门知识，并将有助于维持一个为科学和工程进步提供必要数学知识的科学界。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集从动力系统中产生。在这个项目中，这些空间的分析和几何将被探索，并为他们的调查创建新的数学工具。具体来说，PI打算研究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。在该项目的第二部分中，这将应用于出现此类非光滑空间的相关模型案例，即瑟斯顿地图的动力学。