Analysis on Fractals

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Analysis on fractals is part of a program to develop ``rough analysis", where the underlying space is far from smooth. Fractals have a lot of structure that can be used to advantage in this task. Strichartz has contributed to the development of this area and its connections to classical ``smooth" analysis, and will continue his research in this area. In particular, he will investigate problems in the following general categories: Laplacians on Julia sets, Spectral theory, boundary value problems and Sobolev spaces, Hodge De Rham theory of k-forms, mean value properties of harmonic functions, the infinity-Laplacian, integral geometry, physically realistic models, Littlewood-Paley theory, sampling based on mean values, Peano curves, removable singularities , and thick carpets. Some of the research will involve ``experimental mathematics" to be carried out in collaboration with undergraduate students (mainly REU students). Strichartz has been a leader in the field for the past 15 years. He has been able to bring to this area a broad expertise in harmonic analysis, partial differential equations and analysis on manifolds, and has succeeded in broadening the scope of research in this area. This mathematics research project will continue this development.This mathematics research project deals the study of fractals. Scientists use fractals to model many real world phenomena. Electrical engineers build antennae, and chemists build molecules with the geometry of specific fractals studied in this project. The development of the mathematical theory of analysis on fractals may provide scientists with useful tools for their work. For an example of an important scientific problem that demands the technique being developed in this project, consider the question of what happens when sunlight hits the top of a cloud. Some of the heat is reflected back into space, and some is absorbed into the cloud. The best model of a cloud is as a fractal mixture of water vapor and air. A scientific experiment will be required to determine the most suitable specific fractal model to use, and this most likely will depend on the type of cloud. After the fractal geometry is chosen, it will still require the analysis of the solution of the heat equation on such fractals to resolve the problem. The heat equation is an example of the type of differential equations studied in this project. Strichartz is currently supervising the research of a graduate student in physics, with the goal of studying analysis on physically realistic fractal models. Strichartz has been a leader in the development of experimental techniques in mathematics. The proposed research will continue this development. Strichartz has successfully mentored nearly 100 undergraduate students in research work (including many women and several underrepresented minorities), largely through the REU program. Strichartz has written a book, suitable for undergraduates and graduate students to introduce them to the field and has organized conferences with a strong educational component.