Group Members

Madison Phelps, Brett Hungar, Johnathan Wheeler.

Supervisors

Gamal Mograby, Luke Rogers, Sasha Teplyaev.

Overview

Physicists and mathematicians have used the self-similar nature of certain fractals to develop and study analytical structures on fractal spaces. We examine the analytical structure of a class of fractals that arise as limit sets of the Schreier graphs of the action of self-similar groups on infinite n-ary trees. In particular, we consider how the spectrum of a Laplacian operator on one level of a Schreier graph relates to the spectrum on the next level, a technique known as spectral decimation.

Grigorchuk and collaborators have developed a method to spectrally decimate Schreier graphs of several
important self-similar groups, and have derived significant consequences about the structure of amenable groups. Their method is related to a notion of spectral similarity arising from the work of Fukushima-Shima and Malozemov-Teplyaev. In the latter, a sufficient condition for spectral decimation for fractal graphs is obtained. We consider the analogous question for Schreier graphs of self-similar groups with the goal of understanding the class to which Grigorchuk’s approach is applicable.

Publication

Presentation

Poster

Group Members

Elizabeth Melville, Nikhil Nagabandi.

 Supervisors

Gamal Mograby, Luke Rogers, Sasha Teplyaev.

Overview

Analytic structures on fractals have been analyzed extensively in the past 50 years both because of their interesting mathematical properties and their potential applications in physics. One important question in this area is how the spectrum of a Laplacian on a fractal reflects its geometry; one version of the corresponding problem for domains in Euclidean space was famously described in Kac’s question ”Can you hear the shape of a drum?”.
Some features of the spectra of self-similar sets, such as the asymptotic behavior of the eigenvalue counting function, can be obtained using renewal theory (as in the work of Kigami-Lapidus), but our interest is in more precise results that give the locations and multiplicities of eigenvalues explicitly. These are connected to a long strand of research in mathematical physics about the structure of spectra of Schrodinger operators ¨ and their relation to topological invariants of the underlying space (prominent results in this area are due to Landau, Peierls, Harper, Moser, Bellissard, and, recently, Avila and Jitomirskaya). One name for these results is gap-labeling theorems. For certain highly-symmetric self-similar sets, the computation of the gap structure of the Laplacian spectrum is possible using spectral decimation. We use this method to explicitly compute the gap structure for the Laplacian on a particular two-point self-similar graph and its fractal limit, and for Sierpinski graphs and the Sierpinski gasket.

Publication

Presentation

Poster

Group Members

Claire Canner, William Huang, Michael Orwin.

 Supervisors

Christopher Hayes,  Luke Rogers.

Overview

The study of analytic structures on self-similar fractal sets was initiated by physicists who discovered that heat flow on such sets had sub-Gaussian rather than Gaussian scaling, indicating that the fundamental physics of these sets was very different than on manifolds. These results were first made rigorous for sets with a finite ramification property, but in the late 1980s, Barlow and Bass developed a corresponding theory on a class of generalized Sierpinski carpets. Their approach depends on taking a (weak) limit of Brownian motions on a suitable sequence of closed sets that intersect to the carpet. A key step in proving that the limiting object has sub-Gaussian scaling is showing that the resistance of the approximating domain of scale n is bounded above and below by ρ^n
for a factor ρ that depends on the carpet. Computing the exact value of ρ remains an open problem.

We consider the resistance scaling problem for the octacarpet, and more generally for 4N-carpets, with
the goal of showing analogous bounds for the resistance and obtaining numerical estimates for the resistance scaling factors.

Publication

Presentation

Poster

Group Members

Carlos Lima,  Malcolm Gabbard.

Supervisors

Gamal Mograby,  Luke Rogers,  Sasha Teplyaev.

Overview

At the end of the 20th century studies had been conducted on the Koch Snowflake which had been motivated through work done by physicists on “fractal drum” experiments. These investigations focused on the eigenfunctions of the negative Dirichlet Lapcian generated on a planar domain with a fractal boundary, particularly with the condition that the boundary be set to zero. Here we study the eigenfunctions on the Koch Snowflake with a non-zero boundary condition and we consider a Laplacian defined on the boundary. To generate an n-level fractal and apply the Laplacian matrix, Python programming was implemented for both the n-level fractal and Laplacian matrix. This then gives us insight into the eigenvalues and visualization of the corresponding eigenfunctions on the Koch Snowflake. Initial observations indicate a kind of localization of the eigenfunctions on the fractal boundary. 

Publication: arXiv:2002.04680 (accepted to ICIAM2019 SEMA SIMAI Springer Series)

Presentation

Poster