Abstract:

Abstract: This is a report on a paper by Lapidus and Sarhad titled "Dirac Operators and Geodesic Metric on the Harmonic Sierpinski Gasket and Other Fractal Sets". First I will define the graph approximation of the Sierpinski gasket. Then I will talk about Kusuoka's measurable Riemannian geometry on Sierpinski gasket and introduce counterparts of Riemannian volume, Riemannian metric and Riemannian energy in that setting. Thereafter harmonic functions on Sierpinski gasket will be introduced as energy minimizing functions. Using those functions we can define harmonic gasket. I will also talk about Kigami's geodesic metric on harmonic gasket. Then I will construct two spectral triples on harmonic gasket and we will see that those two triples induce the same spectral metric as Kigami's geodesic metric.