Analysis on Graphs and Fractals

Complex Fractal Dimensions and Zeta Functions

M Lapidus, University of California, Riverside

Abstract

We discuss some elements of the theory of fractal complex fractal dimensions developed by the author and his collaborators over the last few years, as presented in the recent research monograph with Machiel van Frankenhuijsen, 'Fractal Geometry, Complex Dimensions and Zeta Functions: geometry and spectra of fractal strings' (Springer Monographs in Mathematics, Springer-Verlag, 2006). If time permits, we will talk about the higher-dimensional theory (joint with Erin Pearse) and the extension to multifractals (joint with John Rock and Jacques Levy Vehel) and to the p-adic realm (joint with Hung (Tim) Lu). Relevant to this lecture and general themes of this lecture--but most likely, outside its scope, due to natural time constraints-- are a recent work on "Spectral triples and Dirac operator on fractals built on curves" (joint with Cristina Ivan and Erik Christensen) and a series of three papers (joint with Daniele Guido and Tommaso Isola) on Ihara Zeta functions on infinite graphs, including periodic and self-similar graphs.

Finally, we mention that some aspects of the theory of complex fractal dimensions are used or pursued in a forthcoming book by the author, entitled 'In Search of the Riemann Zeros: strings, fractal membranes and noncommutative spacetimes' (Amer. Math. Soc., Sept. 2007, approx. 500 pp.).