Analysis on Fractals

提案号：DMS-00140194PI: Robert strichartz1摘要：研究将进行分形分析。分形分析研究在分形上定义的函数的性质，集中于微分方程的某些类似物。在过去的几年中，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联系，包括谐波分析、流形分析、偏微分方程和数值分析。在有限类的自相似分形上，可以构造起拉普拉斯算子的作用，拉普拉斯算子是传统欧几里得空间和流形分析的中心对象之一。私家侦探的工作让我们对这些拉普拉斯人有了更深入的了解。同时，他还研究了这些分形的一些相关的非线性问题，并致力于将理论的范围扩展到包括更广泛的分形类。这个项目将建立并继续这项工作。第二个研究领域是分形测度和分形谱函数的调和分析。以往的研究表明，对于一类特殊的分形测度，存在傅里叶级数理论的推广。通过对偶性，给出了谱不确定的特殊分形函数的抽样定理。拟议的研究将试图找出在这类狭窄的例子之外发生了什么。分形分析在纯数学和应用数学中都是一个具有巨大潜力的新兴领域。不同领域的科学家已经意识到，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和测度的性质。该项目将通过与更传统的数学领域建立联系，并通过开发未来应用数学家和科学家可以使用的数值工具，加强数学理论。该项目包括与本科生（REU）合作进行计算机实验，以探索示例并产生猜想。期望这项实验工作将导致对该主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.