UC Riverside

This dissertation consists of two separate parts. The first part, Chapters 1--4, concerns the construction of a Dirac operator and spectral triple on the harmonic Sierpinski gasket in order to recover aspects of Jun Kigami's measurable Riemannian geometry via methods of spectral or noncommutative geometry. In particular, we recover Kigami's geodesic distance function on the gasket. Chapters 2 and 3 cover prerequisite mathematics for understanding and potentially building upon the results in Chapter 4. One of the main theorems in Chapter 4 is an example of a fractal analog to Connes' theorem on a compact Riemannian manifold. Chapter 4 also contains a more general theorem which pertains to a class of sets built on curves. \

The Dirac operator on a compact spin Riemannian manifold, in conjunction with the $C^{*}$-algebra of complex-valued continuous functions on the manifold represented as multiplication operators on the Hilbert space of $L^{2}$-spinors forms what is called a spectral triple. Connes' theorem states, in particular, that the triple contains enough information to recover the geodesic distance function on the manifold. Connes' theory on the Riemmanian manifold serves as a model for defining `natural' geometries on spaces via suitable analogs of Dirac operators and spectral triples.\

Michel Lapidus, in collaboration with Christina Ivan and Eric Christensen, has done work using Dirac operators and spectral triples to construct geometries of some fractal sets built on curves. These sets include graphs, infinite trees, and the Sierpinski gasket in Euclidean metric (i.e. the gasket as it is usually constructed, from equilateral triangles). They were able to recover the geodesic distance, Hausdorff dimension, and the standard self-similar measure on the gasket from a Dirac operator and spectral triple on the gasket.\

Jun Kigami has constructed a prototype for a measurable Riemannian geometry using the Sierpinski gasket in harmonic metric. The Sierpinski gasket in harmonic metric, also referred to as the harmonic gasket,

can be constructed using the harmonic functions on the Sierpinski gasket as a single coordinate chart. Kigami has related measurable analogs of Riemannian energy, gradient, metric, volume, and geodesic distance in formulas which are analogous to their counterparts in Riemannian geometry.\

In Chapter 4, we construct several Dirac operators and their associated spectral triples for the harmonic gasket. There are three different constructions, all of which recover Kigami's geodesic distance on the harmonic gasket. The spectrum of each Dirac operator is also described in terms of the lengths of the edges or cells of the harmonic gasket, depending on the construction. One of these constructions can be generalized to a certain class of sets built on curves in $R^{n}$. Chapter 4 closes with a description of work in progress and future directions.\

The second part, Chapter 5, is an adaptation of an article I have written, textsc{nonlinear poisson equation via a newton-embedding procedure} (15 pages, 2010), which has been accepted for publication in the journal textsl{Complex Variables and Elliptic Equations}. Chapter 5 is a version of the article, augmented to include some background on second order elliptic equations and the Newton-embedding procedure.\