Analysis, Geometry, and Spectral Theory On or Off Fractals

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ABSTRACTThe PI intends to pursue and amplify his investigations of therelationships between spectral and fractal geometry. We plan to study thevibrations of "fractal drums", both "drums with fractal boundary"(Laplacians on open sets with very irregular boundary) and "drums withfractal membrane" (Laplacians on fractals themselves). The proposedproblems are closely connected to Kac's question "Can one hear the shape ofa drum?" and to its beautiful extensions from the "smooth" to the "fractal"domain by the physicist Michael Berry. Although the proposed theory ismathematically rigorous, it is also naturally physically motivated (with,for example, applications to the scattering of waves by fractal surfacesand the study of porous media), and has recently drawn some of its impetusfrom the use of computer graphics. Moreover, we propose to use and extendthe theory of "complex dimensions" of "fractal strings" (one-dimensionaldrums with fractal boundary)-recently developed extensively by the PI andMachiel van Frankenhuysen in the research monograph [La-vF2] on "FractalGeometry 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-inorder to study the fascinating oscillatory phenomena occurring in thegeometry and in the spectrum of "drums with fractal boundary" and of "drumswith fractal membrane". ("Complex dimensions" are defined as the poles ofa suitable geometric zeta function. Further, in [La-vF], a detailed studyof 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 onregions with fractal boundary, as well as to investigate problems of a'dynamical nature', of physical significance in condensed matter and solidstate physics; for example, in the study of mechanical or electricaltransport in porous or in random media, as well as of heat diffusions onfractals and in disordered systems. We also intend to pursue ourmathematical and computer graphics-aided study ([LaPa], [LaNRG]) of partialdifferential equations (PDEs)-such as the Laplace, heat and (linear ornonlinear) wave equations-on regions with fractal boundary or on fractalsthemselves. According to appealing experiments and interpretations by thephysicist Bernard Sapoval, this work may help understand the formation offractal structures (for example, coastlines, trees and blood vessels) innature. In the long term, it is hoped that the tools and results developedin this project (and in the PI's earlier investigations) will help us toprobe more deeply than has been so far possible the fine geometricstructure of fractals and of related 'objects' occurring in mathematicsand in physics.