Towards a Noncommutative Fractal Geometry?
Laplacians and Volume Measures on Fractals
We review some recent work of the author (and his collaborators)
regarding spectral and fractal geometry; more specifically, the
vibrations of "fractal drums" and the spectral distribution of
Laplacians on (suitable) self-similar fractals. [In this talk, we will
focus on "drums with fractal membrane" (Laplacians on fractals) rather
than on "drums with fractal boundary."] We also discuss how this work
was combined by the author with techniques from Connes' noncommutative
geometry to construct "volume measures" on such fractals, including an
analogue in this context of the Riemannian volume measure. Further, we
present new results (joint with J. Kigami) regarding the nature of these
volume measures and, in special cases, their relationship with a
suitable notion of Hausdorff measure. If time permits, we will briefly
discuss some very recent results and propose a number of conjectures and
open problems in this area, aimed at further developing geometric
analysis on fractals and--in the long term--at laying out some of the
foundations for what may be coined "noncommutative fractal geometry."
Fall 1997 meeting of the PNGS