Fractal Geometry and Applications

University Of California-Riverside, Riverside CA

Investigators

Abstract

This project pursues and amplifies the PI's earlier investigations of the relationships between fractal geometry, spectral geometry, and dynamical systems. We plan to study the vibrations of "fractal drums," both "drums with fractal boundary" (Laplacians on open sets with very irregular boundary) and "drums with fractal membrane" (Laplacians on fractals themselves), "fractal billiards," as well as the associated "complex fractal dimensions" (a suitable measure of the oscillations intrinsic to nonsmooth geometries and their spectra) and "fractal curvatures" (a suitable measure of the way space warps in a fractal space). Although the proposed theory is mathematically rigorous, it is also naturally physically motivated (with, e.g., applications to the scattering of waves by fractal surfaces and to the study of porous media), and has recently drawn (and will continue to draw) its impetus from the use of computer graphics and computer experiments. We also intend to pursue our mathematical and computer graphics aided study of partial differential equations such as the Laplace, heat, and wave equations on regions with fractal boundary or on fractals themselves. The proposed problems are closely connected to Mark Kac's question "Can one hear the shape of a drum?" and to its beautiful extensions from the "smooth" to the "fractal" domain by Michael Berry. The expected results should be of physical significance in condensed matter and solid state physics; for example, in the study of mechanical or electrical transport in porous or in random media, as well as of heat diffusions on fractals and in disordered systems. They should also be relevant to subterranean imaging and the study of the way information and electromagnetic signals propagate across rough terrain. (Recent engineering and physical applications include new types of cell phones, fractal antennas, loudspeakers, heat insulators, nearly optimal soundproof walls, radar detection, catalytic chemical reactions, and computer microchips.) This work may help understand the formation of fractal structures (e.g. coastlines, trees and blood vessels) in nature as well as the reason why certain biological structures, such as lungs, are both fractal and nearly optimal to fulfill their biological functions.