Analysis on Fractals

Cornell University, Ithaca NY

Investigators

Abstract

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.