Analysis, Geometry, and Spectral Theory On or Off Fractals

University Of California-Riverside, Riverside CA

Investigators

Abstract

ABSTRACT The PI intends to pursue and amplify his investigations of the relationships between spectral and fractal geometry. 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). The proposed problems are closely connected to Kac's question "Can one hear the shape of a drum?" and to its beautiful extensions from the "smooth" to the "fractal" domain by the physicist Michael Berry. Although the proposed theory is mathematically rigorous, it is also naturally physically motivated (with, for example, applications to the scattering of waves by fractal surfaces and the study of porous media), and has recently drawn some of its impetus from the use of computer graphics. Moreover, we propose to use and extend the theory of "complex dimensions" of "fractal strings" (one-dimensional drums with fractal boundary)-recently developed extensively by the PI and Machiel van Frankenhuysen in the research monograph [La-vF2] on "Fractal Geometry and Number Theory" and motivated in part by the PI's earlier joint work with Carl Pomerance [LaPo] and Helmut Maier [LaMa] on (direct and) inverse spectral problems for fractal strings and the Riemann hypothesis-in order to study the fascinating oscillatory phenomena occurring in the geometry and in the spectrum of "drums with fractal boundary" and of "drums with fractal membrane". ("Complex dimensions" are defined as the poles of a suitable geometric zeta function. Further, in [La-vF], a detailed study of their structure is given, for example, in the case of self-similar fractal strings.) We plan to further develop analysis and spectral theory on fractals and on regions with fractal boundary, as well as to investigate problems of a 'dynamical nature', 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. We also intend to pursue our mathematical and computer graphics-aided study ([LaPa], [LaNRG]) of partial differential equations (PDEs)-such as the Laplace, heat and (linear or nonlinear) wave equations-on regions with fractal boundary or on fractals themselves. According to appealing experiments and interpretations by the physicist Bernard Sapoval, this work may help understand the formation of fractal structures (for example, coastlines, trees and blood vessels) in nature. In the long term, it is hoped that the tools and results developed in this project (and in the PI's earlier investigations) will help us to probe more deeply than has been so far possible the fine geometric structure of fractals and of related 'objects' occurring in mathematics and in physics.