Noncommutative Geometry Seminar
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 developing analysis on fractals and is a good model for the elusive notion of a 'fractal manifold'), as well as suitable quantum graphs, Cayley graphs and other infinite graphs. We recover from the spectral triple the geodesic metric intrinsic to the fractal, as that metric is shown to coincide with the noncommutative metric naturally associated with the spectral triple. This main result is especially interesting in the case of the harmonic gasket, which will be the key example used to illustrate our theory. This work is joint with Jonathan Sarhad ("Journal of Noncommutative Geometry", vol. 8, 2014, pp. 947-985). It builds on and significantly extends earlier work of the author, 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 Sierpinski gasket (as well of other fractals built on curves). It also builds on earlier work of the author (carried out in the 1990s) in which, in particular, a broad research program was proposed for developing "noncommutative fractal geometry". The new advance highlighted here 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.