Spectral Triples and Fractal Geometry
Fractal sets are sets that show self-similarity meaning that if one zooms in on some part of the fractal, the close up view exhibits the same patterns as the larger whole. Fractals are difficult to study using the usual tools of geometry and analysis; often classical notions from calculus cannot be meaningfully defined on fractals. The study of analysis on fractals seeks to develop analytic tools analogous to those used on ``nice" spaces but that can be used on fractal sets. One can then ask if these fractal tools give results analogous to the results in the classical setting. This text contributes to a new way of thinking about fractals by developing operator algebraic tools that can provide an alternative way of studying geometry and analysis on fractals.
Work in noncommutative fractal geometry involves an operator algebraic tool kit known as a spectral triple which is constructed based on the fractal being studied. Building upon previous works, we give the construction of a spectral triple for the fractal sets known as the stretched Sierpinski gasket and the Harmonic Sierpinski gasket. We show how these spectral triples can be used to describe fractal geometric properties: Hausdorff dimension, geodesic distance, and certain "fractal" measures. We then describe a spectral triple which can be used to describe the standard energy on the Sierpinski gasket.