Projects: Fractals

Geodesic Interpolation on the Sierpinski Gasket

July 9, 2018

Group Members

Cory McCartanLaura LeGareCaitlin Davis.

Supervisors

Luke RogersSweta Pandey.

Overview

Geodesics (shortest paths) on manifolds such as planes and spheres are well understood.  Geodesics on fractal sets such as the Sierpinski Triangle are much more complicated.  We begin by constructing algorithms for building shortest paths and provide explicit formulas for computing their lengths.  We then turn to the question of interpolation along geodesics—given two subsets of the Sierpinski Triangle, we “slide” points in one set along geodesics to the other set.  We construct a measure along the interpolated sets which formalizes a notion of the interpolation of a distribution of mass, and we prove interesting self-similarity relations about this measure.

Publication

Presentation

Poster

 

Computations on the Koch Snowflake with Boundary and Interior Energies

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

Presentation

Poster

 

Lipschitz Continuity of Laplacian Eigenfunctions on a class of postcritically finite (PCF) self-similar sets

Group Members

Benjamin YorkAnchala Krishnan

Supervisors

Gamal MograbyChristopher Hayes, Luke Rogers.

Overview

We prove a general result for Lipschitz and Hölder continuity of functions defined on a class of postcritically finite (PCF) self-similar sets. Intuition for this theorem comes from formulating arguments on the unit interval, which is well understood in a classical setting. We generalize these results for many kinds of self-similar sets endowed with different measures and metrics. As a corollary to this general result, we prove that eigenfunctions of the Laplacian on the harmonic Sierpinski Gasket are Lipschitz continuous.

Publication

Presentation

Poster

 

 

Probability, Analysis and Mathematical Physics on Fractals 2018

February 23, 2018

Each year we are looking for a group of undergraduate students to work on Probability, Analysis and Mathematical Physics on Fractals. The aim of the projects will be exploration of differential equations and various operators on fractal domains. Students in the project are supposed to have the usual background in linear algebra and differential equations. Knowledge of Matlab, Mathematica, other computer algebra systems, or programming, as well as proof writing, mathematical analysis, and probability may be helpful but is not required. Previous undergraduate work includes published papers on the eigenmodes (vibration modes) of the Laplacian (2nd derivative) of functions that live on Sierpinski gasket type fractals, and the electrical resistance of fractal networks, as well as work on Laplacians on projective limit spaces. The exact choice of the topics to study will depend on the students’ background and interests. Besides being interesting, taking part in a research project like this may be very useful in the future (for instance, when applying to graduate schools).

Luke Rogers, Gamal Mograby, Sasha Teplyaev, Patricia Alonso-Ruiz

Gradients on Higher Dimensional Sierpinski Gaskets

July 28, 2017

Group Members

Luke Brown,  Giovanni E Ferrer SuarezKaruna Sangam.

Supervisors

Gamal MograbyDan KelleherLuke RogersSasha Teplyaev.

Overview

Laplacians have been well studied on post-critically finite (PCF) fractals. However, less is known about gradients on such fractals. Building on work by Teplyaev, we generalize results regarding the existence and continuity of the gradient on the standard Sierpinski Gasket to higher dimensional Sierpinski Gaskets. In particular, we find that, for functions with a continuous Laplacian, the gradient must be defined almost everywhere, and specify a set of points for which it is defined. Furthermore, we provide a counterexample on higher-dimensional Sierpinski gaskets where the Laplacian is continuous but the gradient is not defined everywhere. We conjecture that Hölder continuity of the Laplacian is a condition strong enough to guarantee that the gradient exists at each point.

Publication

Presentation

Poster

Spectral Analysis on Graphs Related to the Basilica Julia Set

Group Members

Courtney GeorgeSamantha Jarvis.

Supervisors

Dan KelleherLuke RogersSasha Teplyaev.

Overview

We analyze the spectra of a sequence of graphs constructed from the Schreier graphs of the Basilica group.  Our analysis differs from earlier work of Grigorchuk and Zuk in that it is based on a macroscopic decomposition of the graphs. This method gives precise information about the multiplicities of eigenvalues and, consequently, good information about the spectral measures of large graphs. It also permits a proof of the existence of gaps in the spectrum of limiting graphs.

Publication

Presentation

Poster

Spectrum of the Magnetic Laplacian on the Diamond Fractal

May 22, 2016

Group MembersIMG_5606

Stephen Loew, Madeline Hansalik, Aubrey Coffey

Supervisors

Luke Rogers, Antoni Brzoska

Overview

The diamond fractal is a fractal that is obtained in the following manner.  Start with a graph with two vertices and an edge and replace the edge with two new vertices connected to our original vertices to obtain a diamond shaped graph.   The diamond fractal is defined to be the limiting object after continuing with the edge replacement indefinitely.  In the project, the spectrum of magnetic Laplacian operators on graph approximations to the diamond fractal was computed.

Given a level n approximation to the fractal with known magnetic field strengths through cells and holes, it is possible to determine the net magnetic field through the cells and holes of the preceding graph approximations.  The spectral similarity relation between the operators on successive graph approximations was worked out, with the corresponding spectral decimation polynomial depending on the magnetic field strengths.  A poster and talk on this work was presented at the REU Mini-Symposium at UConn.

Publication

arXiv:1704.01609

Presentation

Magnetic Spectral Decimation

Poster

Power Dissipation in Fractal AC Circuits

May 21, 2016

Group Members

Loren Anderson, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen LoewIMG_5534

Supervisors

Alexander TeplyaevLuke Rogers, Antoni Brzoska, Ulysses Andrews

Overview

In this project, Feynman’s analysis of an infinite ladder circuit is extended to fractal circuits, in particular, a Sierpinski Ladder circuit and two variants of a Hanoi circuit.   These circuits are constructed by taking the limit of graph approximations with inductors and capacitors placed along edges in a well-defined manner.  Inductors, capacitors and resistors all contribute to the impedance within a circuit; but whereas a resistor imparts a real impedance, inductors and capacitors impart a purely imaginary impedance.

For each circuit, the following was accomplished.  First, the net impedance between certain boundary points of the circuit was computed.  Second, the filter conditions for each circuit were found.  A circuit becomes a filter when the net impedance between boundary vertices has a positive real part!  Third, it was proved that these impedances can be obtained by placing a small positive resistance epsilon on each edge of the graph approximations, finding the limiting impedance between the boundary vertices, and then taking epsilon to zero.  Finally, the construction of harmonic functions on these circuits was outlined.

Publication

arXiv:1605.03890

Presentation

Fractal AC Circuits

Poster

Wave Propagation through a Fractal Medium

May 2, 2016

Group Members

Edith Aromando, Lee FisherPoster - Wave Propagation through a Fractal Medium

Supervisors

Alexander TeplyaevLuke RogersUlysses Andrews

Overview

We consider the wave equation on the unit interval with fractal measure, and use two numerical models to study wave speed and propagation distance.  The first approach uses a Fourier series of eigenfunctions of the fractal Laplacian, while the second uses a Markov chain to model the transmission and reflection of classical waves on an approximation of the fractal. These models have complementary advantages and limitations, and we conjecture that they approximate the same fractal wave.

 

Presentation

Poster

Spectrum of the Magnetic Laplacian on the Sierpinski Gasket

April 21, 2016

Jessica Hyde, Jesse Moeller

Group Members

Jessica Hyde, Jesse Moeller, and Luis Seda

Supervisors

Luke Rogers, Dan Kelleher

Overview

One project investigated magnetic gauge fields on the Sierpinski Gasket. After numerical experimentation, using Mathematica and MatLab, this team determined that specific portions of the spectrum of the Laplacian are unaffected by a perturbating magnetic field given by a harmonic 1-form of finite topological type and continued on to prove that this. In a specific case they also gave a description of the spectrum via a covering space and symmetry argument. They presented a poster and a talk on this work at the REU mini-conference at the University of Massachusetts, Amherst and gave a talk at the Young Mathematician’s Conference at Ohio State University.

Presentation

Spectrum of the Magnetic Laplacian

Publication

arXiv.org