Analysis on Fractals

建议编号：DMS-00140194PI：Robert Strichartz研究将在分数分析中进行。分数分析研究定义在分数上的函数的性质，以某些类似的微分方程为中心。在过去的几年里，P.I.一直积极参与这一领域的发展，特别是与更传统的数学分析领域建立了联系，包括调和分析、流形分析、偏微分方程组和数值分析。在有限类自相似分形图上，可以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传统分析的中心对象之一。P.I.的工作使人们对这些拉普拉斯算子有了更深的理解。同时，他还研究了关于这些分形图的某些相关的非线性问题，并致力于将该理论的范围扩大到包括更广泛的分形图。该项目将在此基础上继续这项工作。第二个研究领域是研究具有分形谱的分形度和函数的调和分析。前人的研究表明，对于一类特殊的分形测度，傅里叶级数理论具有一定的推广意义。通过对偶性，这给出了谱位于某些特殊分形图中的函数的采样定理。这项拟议的研究将试图找出在这一狭窄的例子类别之外发生了什么。分形学分析是一个新兴领域，在纯数学和应用数学中都具有巨大的应用价值。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图来建模，数学家们也一直在探索分形集和度量的性质。该项目将通过与更传统的数学领域建立联系，以及通过开发可供未来应用数学家和科学家使用的数值工具，来加强数学理论。这个项目包括与本科生(REU)在计算机实验方面的合作，以探索例子和产生猜想。预计这项实验工作将导致对这一问题的更深层次的理论理解。

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.