Analysis on Fractals

提案编号：摘要研究将在分形分析中进行，分形分析研究定义在分形上的函数的性质，以微分方程的某些类似物为中心。在过去的几年里，P.I.一直积极参与这一领域的发展，特别是与数学分析的更多传统领域建立联系，包括调和分析，流形分析，偏微分方程和数值分析。 在有限的一类自相似分形上，有可能构造起拉普拉斯算子作用的算子，拉普拉斯算子是欧几里得空间和流形的传统分析的中心对象之一。使我们对这些拉普拉斯算子有了更深入的了解。 与此同时，他研究了这些分形的某些相关的非线性问题，并致力于扩展理论的范围，以包括更广泛的分形类。 这个项目将建立在并继续这项工作.第二个研究领域是分形测度和函数的调和分析与分形谱的研究。 已有的研究表明，对于一类特殊的分形测度，傅里叶级数理论有一个推广。 通过对偶性，这产生了一个关于其谱不确定的函数的分形定理。 分形分析是一个新兴的领域，在纯数学和应用数学中都具有巨大的潜力。 不同领域的科学家已经认识到现实世界中的许多物体都可以用分形来建模，数学家也一直在探索分形集合和测度的性质。 该项目将通过与更多传统数学领域的联系以及开发未来可供应用数学家和科学家使用的数值工具来增强数学理论。 该项目包括与本科生（REU）合作进行计算机实验，以探索示例并生成图表。 希望通过这项实验工作，能对这一课题有更深层次的理论认识。

Proposal Number: DMS-00140194PI: Robert StrichartzABSTRACTResearch will be conducted in analysis on fractals.Analysis on fractals studies the properties of functionsdefined on fractals, centering on certain analogs ofdifferential equations. Over the past few years, the P.I.has been actively involved in developing this area, inparticular establishing connections with more traditionalareas of mathematical analysis, including harmonic analysis,analysis on manifolds, partial differential equations, andnumerical analysis. On a limited class of self-similarfractals it is possible to construct operators that play therole of the Laplacian, one of the central objects intraditional analysis on Euclidean spaces and manifolds.The work of the P.I. has led to a deeper understanding ofthese Laplacians. At the same time he has studied certainrelated nonlinear problems on these fractals, and hasworked on extending the scope of the theory to include awider class of fractals. This project will build on andcontinue this work. A second area of research is the studyof harmonic analysis of fractal measures and functions withfractal spectrum. Previous research has shown that for aspecial class of fractal measures there is a generalizationof the theory of Fourier series. By duality this yields asampling theorem for functions whose spectrum lies incertain special fractals. The proposed research willattempt to find out what happens outside this narrow classof examples.Analysis on fractals is an emerging area with tremendouspotential in both pure and applied mathematics. Scientistsin diverse areas have realized that many objects in thereal world can be modeled by fractals, and mathematicianshave been exploring the properties of fractal sets andmeasures. This project will enhance the mathematicaltheory, both by making connections with more traditionalareas of mathematics, and by developing numerical toolsthat could be used by applied mathematicians and scientistsin the future. This project includes collaborative workwith undergraduate students (REU) on computer experimentsto explore examples and generate conjectures. It is expectedthat this experimental work will lead to deeper theoreticalunderstanding of the subject.