Analysis on fractals is part of a program to develop "rough analysis", where the underlying space is far from smooth. Fractals possess a lot of structure that can be used to advantage in this task. The P.I. will continue his research in this area, with the general goals 1) to extend the depth and scope of the theory for basic examples, and 2) to extend the breadth of the class of fractal Laplacians. In particular, he will investigate problems in the following general categories: distribution theory, differential equations, quantum mechanics, the energy Laplacian, spectra of Laplacians, energy and Laplacians on the Hilbert gasket, and the method of outer approximation (a new method of constructing fractal Laplacians recently introduced by the P.I.). Some of the research will involve "experimental mathematics" to be carried out in collaboration with undergraduate students (mainly REU students).
Mathematical analysis provides scientists with the tools to model real world phenomena. However, classical analysis makes the tacit assumption that the underlying space is smooth. The real world is filled with rough objects. In recent years, mathematical analysts have attempted to construct theories of differential equations on rough spaces. Fractals give examples of spaces that are both extremely rough and yet have a great deal of structure that allows the development of an analytic theory. One approach was pioneered by Jun Kigami in Japan and intensely developed by the P.I. and his colleagues. This theory has produced a deep understanding of certain idealized examples, such as the Sierpinski gasket and related spaces. Although these spaces are far too symmetric to occur in objects in the natural word, they have already appeared in manmade objects (antennas, and nanomolecules). This project will continue the mathematical development of the theory of these key examples, and also broaden the theory to encompass wider classes of fractals, with the hope of developing tools that can be used in modeling naturally occurring objects. Part of the project will involve the emerging methodology of "experimental mathematics", in which computer simulations are used to explore mathematical questions in the hope of formulating conjectures that may eventually lead to conventional mathematical proofs. The P.I. has been very successful in using this approach in the past, and will continue to develop it in collaboration with undergraduate students.
Mathematics builds the infrastructure that scientists use to study the world. Subjects such as geometry, algebra, probability, and calculus have been studied by mathematicians for centuries for their own intrinsic interest, while at the same time they have provided invaluable tools to all the sciences. Calculus was developed in the 17th century, and has since broadened to an area of mathematics called analysis, which includes a vast interconnected set of basic problem areas, including the theory of differential equations. Differential equations allow you to answer basic questions about objects, such as 1) if you hit it with a hammer, how will it vibrate?, and 2) if you heat it with a blowtorch, how will the temperature change? A fundamental assumption needed in order to apply the classical theory of differential equations is that the object be smooth. Nevertheless, the world is full of nonsmooth objects, and mathematicians have been aware of the challenge of extending the basic concepts and techniques of analysis to the nonsmooth setting. In the 1970’s, Benoit Mandelbrot introduced the notion of fractal, bringing together ideas that had been developed over the previous century. A fractal is an object whose geometry exhibits a characteristic complexity at all scales. In other words, no matter how much you zoom in on it, you see more or less the same pattern. A self?similar fractal is one where you see exactly the same pattern. Mandelbrot gave a far reaching vision of many real world objects that might best be modeled by fractals, such as clouds, and ferns and coastlines, and stimulated many scientists to look a the world from the fractal perspective. At the same time, he inspired mathematicians to continue to develop the mathematics of fractals, especially fractal geometry. Starting in the 1980’s mathematicians began to study probability and analysis on fractals. The research supported by this grant has been mainly in the area of analysis on fractals. The mathematical tools developed may be expected to lead to progress on scientific questions, such as: when sunlight hits the top of a cloud, how much of the heat is reflected back into space, and how does the rest of the heat distribute through the cloud? But the mathematics has its own intrinsic interest, and has led to feedback in more classical areas of analysis. Mathematicians need to build; this means proving theorems, inventing definitions, and formulating precise conjectures. But they also need to dream; this means developing heuristic ideas that are too speculative to be made precise, but point to new developments and make connections among disparate areas. More recently, mathematicians have found a need to experiment; this means using computers to carry out numerical computations that uncover new and important patterns. The research supported by this grant has made contributions to all three endeavors. Experimentation has played a vital role in the research supported by this grant. The experiments have been carried out by undergraduate research assistants, many of whom have been supported through Research Experiences for Undergraduates grants. Not only have these students contributed vital insights into the research problems, but they have learned a methodology that promises to play an important role in the future development of mathematics.
