Date of Completion

5-7-2014

Embargo Period

5-6-2014

Keywords

Fractals, Dirichlet Forms, Self-Similar groups, Graphs, Metric Geometry, Probability, Neural Networks

Major Advisor

Alexander Teplyaev

Associate Advisor

Maria Gordina

Associate Advisor

Luke Rogers

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

Our study of the analysis on fractals is broken into three parts: Analysis of post- critically finite (p.c.f.) fractals, non-p.c.f. fractals and applications to Neurobiology. In the first chapter, we focus on how to construct and compute the harmonic structure on p.c.f. fractals. In the second chapter we calculate the spectrum of the Laplacian on an infinite class of fractals. In chapter 3 we discuss the relationship between p.c.f. self-similar structures and self-similar groups. In chapter 4 we discuss a new example of a non-p.c.f. fractal, namely the hexacarpet. Finally in chapter 5 we discuss applications of this analysis to analyzing neural networks.

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