Expanding Thurston Maps and Fractal Geometry

扩展瑟斯顿图和分形几何

基本信息

批准号：
2054987
负责人：
Mario Bonk
金额：
$ 34.05万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054987&HistoricalAwards=false
关键词：
Expanding Thurston Maps Fractal Geometry

项目摘要

In nature, there are many phenomena that exhibit fractal features, such lightning bolts, growth patterns of plants and crystals, snowflakes, or coastlines and river networks. In mathematics, fractal objects often appear in the study of dynamical systems. This project will explore the geometry of certain fractal spaces. The principal investigator will develop better analytic and geometric tools for an improved understanding of fractals that arise from very specific dynamical systems, namely so-called expanding Thurston maps. The involvement of early-career researchers in this activity will contribute to increasing the expertise in the field, and will help to maintain a scientific community that provides the necessary mathematical knowledge for progress in science and engineering. The project includes the training of graduate students. Expanding Thurston maps provide a surprisingly rich landscape with ties to fractals, Teichmüller theory, geometric group theory, and hyperbolic geometry. While there are many interesting questions in this field, in this project specific problems will be singled out whose resolution will lead to advanced insights into the subject. These problems are related to Thurston obstructions, the induced pull-back map on Teichmüller space, and the geometry of the visual sphere associated with an expanding Thurston map. An important numerical invariant of these spheres will be studied, namely their Ahlfors regular conformal dimension. A fundamental technical tool for this investigation is the notion of combinatorial modulus of path families. It seems that obstructions for Thurston maps are tied to path families of degenerating modulus. One of the goals of this project is to get a better grasp of this connection, which is only poorly understood at present.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在自然界中，有许多现象暴露了分形特征，例如闪电，植物和晶体的生长模式，雪花或海岸线和河流网络。在数学中，分形对象经常出现在动态系统的研究中。该项目将探索某些分形空间的几何形状。主要研究者将开发更好的分析和几何工具，以改善对非常特定的动态系统（即所谓的扩展瑟斯顿地图）产生的分形的理解。早期研究人员参与这项活动的参与将有助于提高该领域的专业知识，并有助于维持一个科学界，该社区为科学和工程的进步提供必要的数学知识。该项目包括对研究生的培训。扩展的瑟斯顿地图提供了与分形，蒂奇穆勒理论，几何群体理论和双曲线几何形状相关的令人惊讶的丰富景观。尽管该领域有许多有趣的问题，但在该项目中，将要挑出特定问题的解决方案，其决议将导致对该主题的高级见解。这些问题与Thurston障碍物，Teichmüller空间上的诱发背面图以及与膨胀瑟斯顿图相关的视觉球的几何形状有关。这些球体的重要数值不变将是研究的，即它们的ahlfors常规保形维度。这项投资的基本技术工具是路径家庭组合模量的概念。看来瑟斯顿地图的对象与脱落模量的路径家族有关。该项目的目标之一是更好地掌握这种联系，目前只能理解这一联系。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估，被认为是珍贵的支持。