Spectral and vector analysis on fractafolds

Alexander Teplyaev
University of Connecticut

A fractafold, a space that is locally modeled on a fractal, is the fractal equivalent of a manifold. This notion was introduced by Strichartz, who showed how to compute the discrete spectrum of the Laplacian on compact Sierpinski fractafolds in terms of the spectrum of ?nite graph Laplacians, in particular producing isospectral fractafolds. In a joint work with Strichartz it was furthermore shown how to extend these results for unbounded fractafolds which have continuous spectrum. In parallel to the spectral analysis, a recent progress was made in understanding differential forms and vector analysis on fractafolds and more general spaces. For instance, self-adjointness of the magnetic Laplacian, the Hodge theorem, and the existence and uniqueness for the Navier-Stokes equations have been proved (jointly with Michael Hinz) for topologically one-dimensional spaces with strong local Dirichlet forms that can have arbitrary large Hausdorff and spectral dimensions. These and related joint results with Marius Ionescu, Dan Kelleher, Luke Rogers, Michael Roeckner will be discussed.

Presentation (PDF File)

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