Analysis on Fractals

Cornell University, Ithaca NY

Investigators

Abstract

Proposal Number: DMS-00140194 PI: Robert Strichartz ABSTRACT Research will be conducted in analysis on fractals. Analysis on fractals studies the properties of functions defined on fractals, centering on certain analogs of differential equations. Over the past few years, the P.I. has been actively involved in developing this area, in particular establishing connections with more traditional areas of mathematical analysis, including harmonic analysis, analysis on manifolds, partial differential equations, and numerical analysis. On a limited class of self-similar fractals it is possible to construct operators that play the role of the Laplacian, one of the central objects in traditional analysis on Euclidean spaces and manifolds. The work of the P.I. has led to a deeper understanding of these Laplacians. At the same time he has studied certain related nonlinear problems on these fractals, and has worked on extending the scope of the theory to include a wider class of fractals. This project will build on and continue this work. A second area of research is the study of harmonic analysis of fractal measures and functions with fractal spectrum. Previous research has shown that for a special class of fractal measures there is a generalization of the theory of Fourier series. By duality this yields a sampling theorem for functions whose spectrum lies in certain special fractals. The proposed research will attempt to find out what happens outside this narrow class of examples. Analysis on fractals is an emerging area with tremendous potential in both pure and applied mathematics. Scientists in diverse areas have realized that many objects in the real world can be modeled by fractals, and mathematicians have been exploring the properties of fractal sets and measures. This project will enhance the mathematical theory, both by making connections with more traditional areas of mathematics, and by developing numerical tools that could be used by applied mathematicians and scientists in the future. This project includes collaborative work with undergraduate students (REU) on computer experiments to explore examples and generate conjectures. It is expected that this experimental work will lead to deeper theoretical understanding of the subject.