Laboratoire J.-V. Poncelet
Michel L. Lapidus
University of California, Riverside
Zeta Functions and Complex Fractal Dimensions: Geometry and Spectra of Fractal Strings and Membranes.
In the main part of this lecture, we will give an overview of the theory of fractal strings and of the associated complex fractal dimensions, with emphasis on the case of self-similar strings.
[The most recent account of this theory can be found in the joint book of the speaker and Machiel van Frankenhuisjen, "Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and spectra of fractal strings (Springer Monographs in Mathematics, Springer-Verlag, July 2006, 500 pp.), building on the earlier monograph by the same authors "Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions (Birkhauser, Boston, 2000, 270 pp.).]
These complex dimensions, initially defined as the poles of a suitable geometric zeta function attached to a fractal string, capture some of the essential features--and, in particular, the intrinsic oscillations in the geometry, dynamics or the spectra of fractal strings (as well as, more generally, of fractal manifolds). These oscillations are expressed in terms of suitable explicit formulas (in the sense of number theory, but more general) and yield precise asymptotic expansions for certain geometric, dynamical or spectral counting functions associated to fractal strings. An important example of explicit formula is provided by the tube formula for the volume of epsilon neighborhoods of the fractal boundary of a string, which has a very clear geometric intepretation. In the self-similar case, Diophantine approximation techniques are used to obtain appropriate dimension-free regions and deduce good error estimates for such explicit formulas.
Within this theory, one can naturally reformulate and extend the earlier work of the speaker with Carl Pomerance and with Helmut Maier, showing intimate connections between direct and inverse spectral problems for fractal strings and the Riemann zeta functions in the critical strip. In particular, a geometric reformulation of the (extended) Riemann hypothesis is provided and new results about the vertical distribution of critical zeros of arithmetic zeta functions and other Dirichlet series are obtained.
If time permits, we will briefly discuss recent work of the author and one of his Ph.D. students, Erin Pearse, in which aspects of the theory (particularly, tube formulas) are extended to higher dimensions, especially for self-similar geometries. We may also very briefly mention how a quantization of fractal strings leads to the theory of fractal membranes and how the latter is proposed to be used (in conjunction with techniques inspired by string theory and noncommutative geometry) to understand and reformulate the (extended) Riemann hypothesis in terms of a noncommutative flow on the moduli space of fractal membranes. This new theory is presented in the fortcoming book/essay by the author, "In Search of the Riemann Zeros: Strings, fractal membranes and noncommutative spacetimes" (American Mathematical Society, 2007, approx. 500pp.).
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