# Fractals

# Gaps in the Spectrum of the Laplacian on 3N-Gaskets

## Publication: D. Kelleher, N. Gupta, M. Margenot, J. Marsh, W. Oakley, A. Teplyaev, *Gaps in the spectrum of the Laplacian on 3N-Gaskets*. Communications on Pure and Applied Analysis (CPAA) Pages: 2509 – 2533, Volume 14, Issue 6, November 2015

*Gaps in the spectrum of the Laplacian on 3N-Gaskets*

## Project

# Spectrum of the Magnetic Laplacian on the Sierpinski Gasket

# Measurable Riemannian Structure on Higher Dimensional Sierpinski Gaskets

## Events

UConn Summer Research Symposium

UMass Summer Research Symposium

## Presentation

Geodesics and a Riemannian Metric on Harmonic Sierpinski Gaskets

## Project

# Analyzing Properties of the C. Elegans Neural Network: Mathematically Modeling a Biological System

# From Self-Similar Structures to Self-Similar Groups

# Spectrum of the Magnetic Laplacian on the Sierpinski Gasket

## Group Members

Jessica Hyde, Jesse Moeller, and Luis Seda

## Supervisors

## 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: 2017 CPAA 16 6 2299- 2319 doi: 10.3934/cpaa.2017113

# Measurable Riemannian Structure on Higher Dimensional Sierpinski Gaskets

## Group Members

## Supervisors

## Overview

This team worked on the existence of measurable Riemannian structures in the sense of Kigami on higher dimensional Sierpinski-type gaskets. Some time ago, Kusuoka proved existence of a measure, metric (in the Riemannian sense) and gradient operator on a class of fractals that includes these gaskets, such that these objects bear the same relation to the Dirichlet form as do the Riemannian volume, metric and gradient on Euclidean space. Kigami later completed this picture in the case of the usual 3-vertex Sierpinski gasket by constructing a geodesic length that is the analogue of that occuring in the Riemannian case and proving Gaussian heat kernel estimates, and Kajino has subsequently proved very re ned estimates for the heat kernel in this setting. It was believed that although Kigami’s approach relied on certain 2-dimensional techniques the results would also be valid on Sierpinski-type gaskets with more vertices, and that is what our research team have proved. They presented a poster on this work at the REU mini-conference at the University of Massachusetts, Amherst, and are writing the results up for publication.

## Publication

# arXiv:1703.03380

## Presentation

Geodesics and a Riemannian Metric on Harmonic Sierpinski Gaskets

# Determining the Spectrum of the Laplacian on 3N-Gaskets

## Group Members

Jason Marsh, Nikhaar Gupta, Max Margenot, and Will Oakley

## Overview

The Laplacian is the central object of analysis on fractals. While most work on the Laplacian has been focused on computing the Laplacian spectrum on specific fractals, this group instead found the spectrum on an entire class of fractals—the 3N-Gaskets. This is the class of finitely ramified fractal 3N-Gons that are the attractors of iterated function systems containing 3N contraction mappings. For example, the 3-Gasket is the Sierpinski triangle.

The Laplacian on the fractal, and thus its eigenvalues, must be studied by examining the graph Laplacian on approximating graphs. The group found a function R(z) relating the Laplacian eigenvalues on consecutive graph approximations. They used R(z) along with the eigenvalues of the Laplacian on graph approximations to derive part of the spectrum on the next level. The rest of the spectrum was found using an “exceptional set”— a set of values which the group provided for arbitrary N. By repeating this infinitely many times, the spectrum of the Laplacian on the fractal could be found.

## Presentation

Determining the Spectrum of Laplacian on 3N-Gaskets

Poster

## Publication

# Analyzing Properties of the C. Elegans Neural Network: Mathematically Modeling a Biological System

## Contributors

Tyler Reese, Dylan Yott, Antoni Brzoska, and Daniel Kelleher

## Overview

The brain is one of the most studied and highly complex systems in the biological world. This group analyzed the brain of the nematode Caenorhabditis elegans. They used eigenvalues and eigenvectors of the Laplacian matrix — a matrix representation — of the neural network of the nematode brain with an eye on indicators of self-similarity.

The following is a list of the programs this group used in their research. Most of them were written in Matlab.