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Abstract

Classical analysis is not able to treat functions whose domain is fractal. We present an introduction to analysis on a particular class of fractals known as post-critically finite (PCF) self-similar sets that is suitable for the undergraduate reader. We develop discrete approximations of PCF self-similar sets, and construct discrete Dirichlet forms and corresponding discrete Laplacians that both preserve self-similarity and are compatible with a notion of harmonic functions that is analogous to a classical setting. By taking the limit of these discrete Laplacians, we construct continuous Laplacians on PCF self-similar sets. With respect to this continuous Laplacian, we also construct a Green's function that can be used to find solutions to the Dirichlet problem for Poisson's equation.

Author Bio

Lucas Yong is a final-year undergraduate at Reed College in Portland, OR, who will receive an undergraduate degree in Mathematics-Computer Science in May 2021. After college, he plans to get a PhD in Mathematics, and pursue a career in research and teaching.

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