Analysis on Fractals

分形分析

• 批准号: 1162045
• 负责人: Robert Strichartz
• 金额: $ 15.3万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1162045&HistoricalAwards=false
• 关键词: Analysis; Fractals

Analysis on fractals is part of a program to develop ``rough analysis", where the underlying space is far from smooth. Fractals have a lot of structure that can be used to advantage in this task. Strichartz has contributed to the development of this area and its connections to classical ``smooth" analysis, and will continue his research in this area. In particular, he will investigate problems in the following general categories: Laplacians on Julia sets, Spectral theory, boundary value problems and Sobolev spaces, Hodge De Rham theory of k-forms, mean value properties of harmonic functions, the infinity-Laplacian, integral geometry, physically realistic models, Littlewood-Paley theory, sampling based on mean values, Peano curves, removable singularities , and thick carpets. Some of the research will involve ``experimental mathematics" to be carried out in collaboration with undergraduate students (mainly REU students). Strichartz has been a leader in the field for the past 15 years. He has been able to bring to this area a broad expertise in harmonic analysis, partial differential equations and analysis on manifolds, and has succeeded in broadening the scope of research in this area. This mathematics research project will continue this development.This mathematics research project deals the study of fractals. Scientists use fractals to model many real world phenomena. Electrical engineers build antennae, and chemists build molecules with the geometry of specific fractals studied in this project. The development of the mathematical theory of analysis on fractals may provide scientists with useful tools for their work. For an example of an important scientific problem that demands the technique being developed in this project, consider the question of what happens when sunlight hits the top of a cloud. Some of the heat is reflected back into space, and some is absorbed into the cloud. The best model of a cloud is as a fractal mixture of water vapor and air. A scientific experiment will be required to determine the most suitable specific fractal model to use, and this most likely will depend on the type of cloud. After the fractal geometry is chosen, it will still require the analysis of the solution of the heat equation on such fractals to resolve the problem. The heat equation is an example of the type of differential equations studied in this project. Strichartz is currently supervising the research of a graduate student in physics, with the goal of studying analysis on physically realistic fractal models. Strichartz has been a leader in the development of experimental techniques in mathematics. The proposed research will continue this development. Strichartz has successfully mentored nearly 100 undergraduate students in research work (including many women and several underrepresented minorities), largely through the REU program. Strichartz has written a book, suitable for undergraduates and graduate students to introduce them to the field and has organized conferences with a strong educational component.

对分形的分析是开发“粗略分析”计划的一部分，其中底层空间远非平滑。 分形有很多结构，可以在这项任务中发挥优势。 Eschenhartz为这一领域的发展及其与经典“平滑”分析的联系做出了贡献，并将继续他在这一领域的研究。 特别是，他将调查问题在以下一般类别：拉普拉斯对朱莉娅集，谱理论，边值问题和Sobolev空间，霍奇德拉姆理论的k-形式，平均值性质的调和函数，无穷大，拉普拉斯，积分几何，物理现实模型，利特尔伍德-佩利理论，采样的基础上平均值，皮亚诺曲线，可移动的奇异性，厚地毯。 一些研究将涉及与本科生（主要是REU学生）合作进行的“实验数学”。 在过去的15年里，科尔哈茨一直是该领域的领导者。 他已经能够把这一领域的广泛专业知识，调和分析，偏微分方程和分析流形，并成功地扩大了范围的研究在这一领域。 本数学研究计画将延续此一发展，本数学研究计画将探讨分形之研究。科学家们用分形来模拟许多真实的世界现象。 电子工程师制造天线，化学家制造具有本项目研究的特定分形几何形状的分子。 分形分析的数学理论的发展可以为科学家的工作提供有用的工具。 举一个重要的科学问题的例子，需要在这个项目中开发的技术，考虑当阳光照射到云层顶部时会发生什么。 一部分热量被反射回太空，一部分被吸收到云层中。 云的最佳模型是水蒸气和空气的分形混合物。 需要进行科学实验来确定最合适的特定分形模型，这很可能取决于云的类型。 在选择分形几何之后，仍然需要分析热方程在这种分形上的解来解决这个问题。 热方程是本项目中研究的一类微分方程的一个例子。 Eschenhartz目前正在指导一名物理学研究生的研究，其目标是研究物理现实分形模型的分析。 Eschenhartz一直是领导者在发展中国家的实验技术在数学。 拟议的研究将继续这一发展。 Eschenhartz已经成功地指导了近100名本科生的研究工作（包括许多妇女和几个代表性不足的少数民族），主要是通过REU计划。Eschenhartz写了一本书，适合本科生和研究生介绍他们的领域，并组织了具有很强的教育成分的会议。