权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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.

这项研究项目探索了分形空间的几何学。许多自然现象表现出分形特征，包括闪电、植物和晶体的生长模式、雪花、海岸线和河流网络。在数学中，分形对象出现在动力系统的研究中，例如有理映射的Julia集、Klein群的极限集和迭代函数系统中的吸引子，以及连续统随机树等概率模型中的吸引子。这个项目的目标是开发更好的分析和几何工具，以提高对分形学的理解。青年研究人员参与这项活动将有助于增加该领域的专门知识，并有助于维持一个拥有取得进展所必需的数学知识的科学界。该项目由三个部分组成，涉及动力学和概率中的树，扩展的瑟斯顿地图，以及动力学中的螺线管。在每个子项目中，都研究了与其准共形几何有关的分形空间的几何方面。一种是对仅依赖于相对形状大小且尺度不变的几何特征感兴趣。用更专业的术语来说，这意味着人们想要研究相关空间的几何，直到达到准对称等价。过去，这个概念框架在研究自相似分形学的几何性质方面已经相当成功；该项目聚焦于一些似乎特别适合进一步研究的开放问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。