Fractal Geometry and Dynamical Systems, with Applications

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In this project, the principal investigator and his collaborators (including his graduate students and postdocs) plan to pursue the investigation of the geometry, dynamics and spectra of fractal drums, both drums with fractal boundary and drums with fractal membrane (analysis and PDEs off or on fractals). This entails the further development of the theory of complex dimensions, along with the investigation of the underlying oscillatory phenomena, of (approximately) self-similar fractals, multifractals, and associated nonlinear dynamical systems. This also entails the development and the use of new tools in noncommutative geometry and global analysis in order to study the geodesic flow and the geodesic metric of 'fractal manifolds'. A connecting thread throughout this project is provided by the interplay between fractal geometry and dynamical systems, from two main perspectives: (i) the use of (multivariable) complex dynamics in order to obtain a detailed understanding of the spectra of certain fractal drums, via a generalization of the classic decimation method. (ii) The development of a theory of fractal billiards and (in the longer term) of corresponding trace formulas connecting the geometry (e.g., the geodesic flow) and spectra of the associated fractal drums. Computer-aided experiments as well as the interpretation of physical experiments play a significant role in forming appropriate intuition in this context.This projects aims at addressing the following question: "Can one hear the shape of a fractal drum" That is, how much information can one recover about the geometric contours of a rough or complex shape by making it vibrate and just listening to the resulting sounds? This question, even in the classical setting of smooth, Euclidean (or Riemannian) geometry, plays a central role in contemporary mathematics. Fractals are mathematical idealizations of many complex shapes occurring in nature, such as coastlines, river beds, trees, computer networks, networks of blood vessels, lungs, ore and oil distribution, etc. Understanding how waves propagate through these fractal shapes (or 'manifolds') or how light (and electromagnetic radiation) reflect on or off them, is a key scientific and mathematical problem. Potential applications of this work involve a variety of domains, including high technology (e.g, computer microchips, computer networks, and fractal antenna for use in cell phone technology), mathematical biology and medicine (cancer research, blood circulation), geology, applied and theoretical physics (microwaves cavities, random surfaces of use in models of quantum gravity), astronomy (large-scale structure of the universe), and engineering (very efficient sound and heat insulators, catalysts in chemical reactions). The principal investigator, whose previous NSF supported research has already had a significant impact on this area (both in mathematics, physics and other sciences), plans to continue his broad integration of research and educational activities, via the continued mentoring of his many graduate students and postdocs, as well as of promising and creative undergraduate students. lecturing in many scientific venues and summer schools, the creation, teaching and supervision of new courses and seminars connected with this research area, as well as the writing of research and educational books and scientific papers connected with this field.