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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，有许多现象表现出分形特征，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力系统的研究中。本专题将探讨某些分形空间的几何学。首席研究员将开发更好的分析和几何工具，以更好地理解从非常特定的动力系统中产生的分形，即所谓的扩展瑟斯顿映射。早期职业研究人员参与这项活动将有助于增加该领域的专业知识，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。该项目包括培训研究生。扩展瑟斯顿地图提供了一个令人惊讶的丰富景观与分形，Teichmüller理论，几何群论和双曲几何。虽然这一领域有许多有趣的问题，但在本项目中，将挑出一些具体问题，这些问题的解决将导致对这一主题的深入了解。这些问题涉及到瑟斯顿障碍，诱导拉回地图上的Teichmüller空间，和几何形状的视觉领域与扩大瑟斯顿地图。一个重要的数值不变量，这些领域将进行研究，即他们的Ahlfors经常保形尺寸。一个基本的技术工具，这项调查是组合模的概念的道路家庭。瑟斯顿映射的障碍物似乎与退化模的路径族有关。该项目的目标之一是更好地把握这种联系，这是目前还不太了解的。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。