Analysis on Fractals

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AbstractStrichartz Analysis on fractals studies the properties of functions defined on fractals, centering on certain analogs of differential equations. 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. This project will attempt to extend the construction of Laplacians to a broader class of fractals, and to explore the fundamental properties of these Laplacians and other differential-type operators. In addition, this project will attempt to broaden the connections between this new area of research and more traditional areas, including global analysis, harmonic analysis, splines and the finite element method, pseudodifferential operators, and special functions. 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. Scientists in diverse areas have realized that many objects in the real world can be modeled by fractals. In order to study physical processes on such fractal objects, it is necessary to develop a mathematical theory that will give the analog of differential equations on fractals. Such a theory is in the early stages of development, and this project will contribute to that development, bringing to the problem ideas and techniques from other areas of mathematics. It is expected that this project will develop tools that will be of use to applied mathematicians and scientists, since part of the project focuses on numerical methods and algorithms for solving concrete problems. The contribution of undergraduate students to the project will be significant, and in turn will be a positive educational experience for the students.