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Fractal Zeta Functions and Fractal Drums [<a href="http://www.math.caltech.edu/~mathphysics/Slides/lapidus.pdf" target="_blank">PDF</a>]

<a href="http://www.math.caltech.edu/~mathphysics/ws15prog.html" target="_blank">1 hour lecture of Professor Michel L. Lapidus at Caltech</a>: We will give some sample results from the new higher-dimensional theory of complex fractal dimensions developed jointly with Goran Radunovic and Darko Zubrinic in the forthcoming 450-page research monograph (joint with these same co-authors), Fractal Zeta Functions and Fractal Drums: Higher Dimensional Theory of Complex Dimensions, to be published by Springer. We will also explain its connections with the earlier one-dimensional theory of complex dimensions developed, in particular, in the research mono- graph (by the speaker and M. van Frankenhuijsen) entitled "Fractal Geom- etry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings" (Springer Research Monographs, Springer, New York, 2013 ; 2nd rev. and enl.edn. of the 2006 edn.) In particular, to an arbitrary compact subset A of the N-dimensional Euclidean space (or, more generally, to any relative fractal drum), we will associate new distance and tube zeta functions, as well as discuss their basic properties, including their holomorphic and meromorphic extensions, and the nature and distribution of their poles (or 'complex dimensions'). We will also show that the abscissa of convergence of each of these fractal zeta functions coincides with the upper box (or Minkowski) dimension of the underlying compact set A, and that the associated residues are intimately related to the (possibly suitably averaged) Minkowski content of A. Example of classical fractals and their complex dimensions will be provided. Finally, if time permits, we will discuss and extend to any dimension the general definition of fractality proposed by the authors (and M-vF) in their earlier work, as the presence of nonreal complex dimensions. We will also provide examples of hyperfractals, for which the critical line {; ; Re(s) =D}; ; , where D is the Minkowski dimension, is not only a natural boundary for the associated fractal zeta functions, but also consist entirely of singularities of those zeta functions. These results are used, in particular, to show the sharpness of an estimate obtained for the abscissa of meromorphic convergence of the spectral zeta functions of fractal drums. Furthermore, we will also briefly discuss recent joint results in which we obtain general fractal tube formulas in this context (that is, for compact subsets of Euclidean space or forrelative fractal drums), expressed in terms of the underlying complex dimensions. We may close with a brief discussion of a few of the many open problems stated at the end of the aforementioned forthcoming book. [<a href="http://www.math.caltech.edu/~mathphysics/Slides/lapidus.pdf" target="_blank">PDF</a>]

fractal zeta functions; relative fractal drums; complex dimensions; meromorphic extensions; fractal strings; box dimension; Minkowski content

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