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.
PI: Michel L. Lapidus Contributions within the Discipline: The PI and his collaborators have contributed a broad understanding of oscillatory phenomena in mathematics and physics, and connected various aspects of fractal geometry, number theory, and noncommutative geometry in new and unexpected ways. In particular, the commonly used notion of a fractal dimension has been extended to become a much more robust, precise, and accurate tool to understand non-smooth and singular geometries, which arise in many aspects of nature, as well as previously unrelated parts of mathematics. Furthermore, new bridges have been built between the very abstract and ethereal arithmetic geometry and more concrete fractal geometries, via the theory of fractal strings and their quantization known as 'fractal membranes'. The quantization process is a precise mathematical prescription that produces linear operators whose spectra are the very complex dimensions originally defined by the PI. Contributions to Other Disciplines: One characteristic feature of the PI's research program and work is that it is not limited to a single discipline within or outside mathematics, but that it establishes new connections between a number of areas of knowledge, within and outside mathematics, including aspects of theoretical and condensed matter physics, as well as engineering and the biological sciences. In further detail, various parts of this project and its findings have established connections with several aspects of theoretical physics, in particular string theory, quantum physics, thermodynamics, statistical physics, condensed matter, critical phenomena, and phase transitions. More specifically, the construction of spectral triples and the associated Dirac operators were explored in detail, in the context of geometric theories of noncommutative space-time. The result elucidated the interplay between geometry and the spectral characteristics of typical physical linear operators such as the Dirac operator on such space times. Other topics, in theoretical physics, include applications of fractal geometry to the various types of theories of quantum gravity. Further connections to engineering such as cell phone networks, fractal antennae, models of the worldwide web, and boundary/impurity affects on the production of microchips were realized. In particular, applications of the new and emerging theory of complex dimensions of multifractals to engineering endeavors such as those mentioned thus far were exposed, as well as the engineering design of improved efficiency for catalytic systems. The broader applicability of the mathematical research done did not end there. In the biological sciences, the formation of complex structures (such as the folding of genomes), the flow of the human circulatory/pulmonary system, networks of rivers, streams, and tributaries--together with their flow of local flora and fauna, all found fruitful application of the general theory of fractal geometry. Additionally, the applications of fractal geometry have found uses in coastline and erosion models of ecological systems, multi-scale modularity in the human brain, anomalous diffusion inside of living cells, and population dynamics in advective media. Contributions Beyond Science and Engineering: Some aspects of this project have been used in applications and commercial products ranging from loudspeaker technology to insulators, antennas, cell phones, RADAR technology and microchips. These products benefit from the physical science and engineering applications discussed above. Moreover, the social implications of the biological applications could result in cleaner water, more optimal river and water supply allocation to both the human and animal populations, and increased understanding of the human body for purposes of medical science and pharmacology leading to increased health benefits. Finally, fractals and their mathematical theory produce graphs, charts, diagrams, and dynamic computer simulations that have long inspired the visual arts. During the period of the grant, the PI has continued mentoring and advising a large number of undergraduate, graduate (Masters) and Ph.D. students. At the University of California, Riverside, he was awarded the campus-wide Ph.D. Advisor and Graduate Mentoring Award (in 2012). He has lectured widely throughout the world and continued to be involved as the official of the American Mathematical Society (AMS) representing the Western US and Canada in the preparation of the scientific program of a number of regional, national and international meetings, as well as of lectures accessible to a broad cross-section of the general public (AMS Einstein Public Lectures).
