Spectra of three-peg Hanoi towers graphs
We consider the relationship between the Laplacians on two sequences of planar graphs, one from the theory of self-similar groups and one from analysis on fractals. By establishing a spectral decimation map between these sequences, we give an elementary calculation of the spectrum of the former, which was first computed by Grigorchuk and Šunić. Our method also gives a full description of the eigenfunctions. https://arxiv.org/abs/2107.02697
Spectral properties of graphs associated to the Basilica group
Antoni Brzoska, Courtney George, Samantha Jarvis, Luke G. Rogers, Alexander Teplyaev https://arxiv.org/abs/1908.10505
Convergence, optimization and stability of singular eigenmaps
Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter \epsilon. If \epsilon is too small or too large, then the approximation is inaccurate or completely breaks down. However, an analytic expression for the optimal \epsilon is out of reach. In our work, we use some explicitly solvable models and Monte Carlo simulations to find the approximately optimal range of \epsilon that gives on average, relatively accurate approximation of the eigenmaps. Numerically we can consider several model situations where eigen-coordinates can be computed analytically, including intervals with uniform and weighted measures, squares, tori, spheres, and the Sierpinski gasket. In broader terms, we intend to study eigen-coordinates on weighted Riemannian manifolds, possibly with boundary, and on some metric measure spaces, such as fractals. https://arxiv.org/abs/2406.19510
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A student asks the professor,
“Why is this theorem so easy to prove?”
The professor grins,
“Because I spent three weeks crafting the definition.”