Mathematical Sciences: Spectral and Fractal Geometry: Analysis on Fractals, Noncommutative Geometry, and PDEs in the Fractal Domain

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Abstract Lapidus Lapidus will study the vibrations of "fractal drums," both "drums with fractal boundary" and "drums with fractal membrane." More specifically, he will develop analysis on fractals and on regions with fractal boundary, as well as investigate and use new connections with Connes' noncommutative geometry and the associated "quantized calculus." The long term objective is gain a better understanding of the fine geometric structure of fractals by combining the traditional tools from harmonic analysis, probability theory, dynamical systems, and PDEs with methods from noncommutative geometry. This project has many potential physical and engineering applications, including transport in porous media and percolation networks, fractal resonators such as microwave cavities, phase transitions, and diffusions on fractally structured media. Further, it is directly relevant to current technology, such as oil recovery, computer microchips, radar imaging, and very efficient heat insulators called "aerogels."