| Contents |
| In this talk, we will report on recent work connecting aspects of geometric analysis on fractals and noncommutative fractal geometry. We construct spectral triples and Dirac operators on a class of fractals built on curves, including the Sierpinski gasket, the harmonic gasket (which is ideally suited for analysis on fractals and is a good model for a 'fractal manifold'), as well as suitable quantum graphs, Cayley graphs and other infinite graphs. This work is joint with Jonathan Sarhad (Math.arXiv:1207.6681v2 [math-ph], 2014, IHES preprint, IHES/M/12/22, 2012) and will be published in the "Journal of Noncommutative Geometry". It builds on earlier work, joint with Eric Christensen and Cristina Ivan (published in "Advances in Math.", vol. 217, 2008, pp. 1497-1507) in which we constructed geometric Dirac operators allowing us to recover the natural geodesic metric and the natural Hausdorff measure of the Euclidean Sieroinski gasket (as well of other fractals built on curves). It also builds on earlier work of the author in which, in particular, a broad research program was proposed for developing "noncommutaitve fractal geometry". The new advance in the present paper is that we can now deal with a significantly broader class of fractals, including the harmonic Sierpinski gasket (which can be viewed as a kind of "measurable Riemannian manifold", according to the recent work of Jun Kigami), allowing us to get one step closer to developing aspects of geometric analysis truly connected with the study of fractal manifolds and their intrinsic families of geodesic curves. |
|