Mercoledì 6 Luglio 2005
	Aula 1101, Dip. Matematica, U. Roma "Tor Vergata"
12.00 - 13.00	Prof. Michel L. Lapidus, UC Riverside COMPLEX DIMENSIONS OF SELF-SIMILAR STRINGS AND SYSTEMS: FROM FRACTAL GEOMETRY TO GEOMETRIC MEASURE THEORY, AND BACK. (1st part)
16.00 - 17.00	Prof. Michel L. Lapidus, UC Riverside COMPLEX DIMENSIONS OF SELF-SIMILAR STRINGS AND SYSTEMS: FROM FRACTAL GEOMETRY TO GEOMETRIC MEASURE THEORY, AND BACK. (2nd part)
17.30 - 18.30	Prof. Erik Guentner, Univ. of Hawaii at Manoa PERMANENCE PROPERTIES OF EXACT AND UNIFORMLY EMBEDDABLE GROUPS
ABSTRACT OF THE TALK OF M.L. LAPIDUS: In the first part of this talk, we will briefly summarize some aspects of the theory of fractal strings and of the associated complex fractal dimensions, with emphasis on the case of self-similar strings (corresponding to self-similar subsets of the real line). These complex dimensions-defined initially as the poles of a suitable geometric zeta function associated with the fractal string (or harp), capture some of the essential features --and in particular, the intrinsic oscillations in the geometry, the spectrum or the dynamics--of the underlying fractal geometry, via suitable 'explicit formulas'. In turn, such formulas yield precise asymptotic expansions for the geometric or the spectral counting functions of fractal strings, in terms of the underlying complex dimensions. An important example of explicit formula is provided by the 'tube formula' (for the volume of the inner epsilon-neighborhoods of the fractal boundary of the string). For self-similar strings, the error term in all of these formulas can be estimated in terms of the Diophantine properties of the associated scaling ratios, while the quasiperiodic structure of the patterns of complex dimensions can be understood via Diophantine approximation (and numerical experimentation). Building on earlier work of the author on the vibrations of fractal drums---including joint articles with Carl Pomerance, Helmut Maier, and Christina He, on the connections between direct and inverse spectral problems for fractal strings and the Riemann zeta function or the Riemann hypothesis--- this theory of complex dimensions of fractal strings was developed over the last few years in a series of papers and in the research monograph (joint with Machiel van Frankenhuysen) "Fractal Geometry and Number Theory: complex dimensions of fractal strings and zeros of zeta functions" (Birkhauser, Boston, 2000, 280 pp.; second revised and enlarged edition in press, 2005, 440 pp.). We note that recently, an extension of some of these results to random fractal stings (such as random self-similar strings or more general recursively constructed strings, and the zero set of Brownian motion) was provided by Ben Hambly and the author (Trans. Amer. Math. Soc., 2005, in press). In the second part of the talk, we will present some elements of a higher-dimensional theory currently being developed in a series of papers with Erin Pearse, one of the speaker's Ph.D. students. We will begin by obtaining a tube formula for the Koch snowflake curve and deduce from it the associated (possible) complex dimensions. In that case, the coefficients of the tube formula are expressed in terms of the Fourier coefficients of a suitable nonlinear and periodic analog of the Cantor-Lebesgue function (or "Devil's staircase"). We will then move towards setting-up a more general theory of complex dimensions for self-similar systems (or more general 'scaling fractals'), based on suitable tilings of the complement (of a natural compactification) of the self-similar attractor. A key result in this context is a general tube formula (now for the inner epsilons neighborhoods of the cells of the tilings, or equivalently, for the associated self-similar fractal spray, with a finite or countably infinite number of generators). It makes use both of an extended explicit formula (obtained in the abovementioned monograph joint with MvF) and of recent results (by D. Hug, G. Last and W. Weil) from geometric measure theory extending the classical work of Federer on the Steiner and Weyl tube formulas for compact convex sets and for the epsilon neighborhoods of smooth submanifolds of Euclidean space. The latter Steiner-type formula is applied to each of the generators of the self-similar system or tiling, which typically is of a much simpler nature than the self-similar set (or more general 'scaling fractal') under consideration. (For the Koch snowflake curve, for example, there is a single generator, an equilateral triangle, For the pentagasket, there are two generators, a triangle and a pentagon. For a fractal string, the single generator is the unit interval.) The resulting tube formula involves not only the complex dimensions of the self-similar string by which each generator of the self-similar system is scaled, but also the (typically integer) dimensions of the generators. Extending the work of Federer (and his followers) to the present fractal setting, one can define suitable 'complex curvature measures' (now indexed by the 'complex dimensions of the self-similar fractal') and interpret accordingly the coefficients of the tube formula. This answers positively in this context a long-standing open problem in this theory, which was open even for self-similar fractal strings (cf. the end of chapter 10 (first ed.) or 12 (2nd ed.) of the above monograph with MvF); namely, it is now possible to interpret geometrically in terms of (integrals of suitably defined curvatures) the coefficients of our tube formulas and hence, the residues of the corresponding geometric zeta functions. Naturally, much work remains to fully develop the higher-dimensional theory of the complex dimensions of fractals, but the present work gives hope that we may now expect it to reach maturity in the not so distant future, thereby realizing a new synthesis of aspects of geometric and fractal geometry. Finally, we mention that in the speaker's forthcoming book/essay, entitled "In Search of the Riemann Zeros: strings, fractal membranes, and noncommutative spacetimes" (approx. 510pp., May 2005), motivated in part by some aspects of the above theory of fractal strings (discussed in the first two paragraphs) and of modern theoretical physics (including standard string theory), a theory of quantized fractal strings (or 'fractal membranes') is provided. Moreover, the parallels uncovered between fractal, self-similar geometries and artihmetic geometries are further developed to provide a general framework within which to try to understand the Riemann hypothesis and its ramifications, in terms of a suitable (noncommutative) flow on the moduli space of fractal membranes. A rigorous and noncommutative geometric version of part of this research program on 'fractal membranes' is currently being provided in joint work with Ryszard Nest, with whom very recently (i.e., over the last few months), we have begun developing a mathematical version of 'complex homology/cohomology' or 'fractal cohomology theory', as suggested in the abovementioned two research monographs. Most (if not all) of the talk (and of the accompanying additional 'elaboration') will be dedicated to a discussion of some of the work described in the first three paragraphs of this abstract and should be understandable to non-experts and graduate students.