Dynamics and Quasiconformal Geometry

动力学和拟共形几何

基本信息

批准号：
1808856
负责人：
Mario Bonk
金额：
$ 27万
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：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1808856&HistoricalAwards=false
关键词：
Dynamics Quasiconformal Geometry

项目摘要

This research project explores the geometry of fractal spaces. Many natural phenomena exhibit fractal features, including lightning bolts, growth patterns of plants and crystals, snowflakes, coastlines, and river networks. In mathematics, fractal objects arise in the study of dynamical systems such as Julia sets of rational maps, limit sets of Kleinian groups, and attractors in iterated function systems, as well as in probabilistic models such as continuum random trees. The goal of this project is to develop better analytic and geometric tools for an improved understanding of fractals. The involvement of young researchers in this activity will contribute to increasing the expertise in the field and will help to maintain a scientific community with the mathematical knowledge necessary for progress. The project consists of three parts, concerning trees in dynamics and probability, expanding Thurston maps, and solenoids in dynamics. In each of these subprojects, geometric aspects of fractal spaces are studied that are related to their quasiconformal geometry. One is interested in geometric features that only depend on relative shape sizes and are scale invariant. In more technical terms, this means that one wants to study the geometry of the relevant spaces up to quasisymmetric equivalence. In the past, this conceptual framework has been quite successful for the study of geometric properties of self-similar fractals; the project focuses on some open questions that seem particularly amenable to further investigation.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.

该研究项目探讨了分形空间的几何形状。许多自然现象表现出分形特征，包括闪电螺栓，植物和晶体的生长模式，雪花，海岸线和河流网络。在数学中，分形对象在动态系统（例如朱利娅的理性图，限制克莱琳组的限制集以及迭代功能系统中的吸引子以及概率模型（例如延性随机树）中出现。该项目的目的是开发更好的分析和几何工具，以提高对分形的了解。年轻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的进步所需的数学知识。该项目由三个部分组成，涉及动态和概率中的树木，扩展瑟斯顿地图和动力学中的电磁阀。在这些子运动中的每一个中，都研究了分形空间的几何方面，这些方面与它们的准形式几何形状有关。一个人对仅取决于相对形状大小且比例不变的几何特征感兴趣。用更技术的术语来说，这意味着要研究相关空间的几何形状，直到准对称等效性。过去，这个概念框架在研究自相似分形的几何特性方面非常成功。该项目着重于一些似乎特别适合进一步调查的开放问题。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准被认为值得通过评估来支持。