The Effros-Shen algebra corresponding to an irrational number θ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of θ encodes the dimensions of the summands, and how the matrix algebras at the nth level fit into the summands at the (n+1)th level. In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the C*-algebra of a category of paths -- a generalization of a directed graph -- determined by the continued fraction expansion of θ. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. In this thesis, the author defines a spectral triple in terms of the category of paths presentation of an Effros-Shen algebra, drawing on a construction by Christensen and Ivan. This thesis describes categories of paths, the example of Mitscher and Spielberg, and the spectral triple construction.

The author employs bundle theory to investigate dynamics on C*- algebras. Using methods old and new to define dynamics on topological spaces (often with additional structure), implications of the dynamics are investigated in the non-commutative setting, and in suitable situations the dynamics are classified. As a result, new Morita equivalence results are derived and new settings introduced in the study of crossed products, whether by group coactions or by actions of groups and groupoids.

This thesis presents a categorical approach to recover and extend some well-known results from the theory of $C^*$-correspondences. First, a detailed study on \emph{the enchilada category} is given. In this category, the objects are $C^*$-algebras, and morphisms are isomorphism classes of $C^*$-correspondences. In the \emph{equivariant enchilada category} $\mathcal{A}(C)$, all objects and morphisms are equipped with a locally compact group action satisfying certain conditions. These categories were used by Echterhoff, Kaliszweski, Quigg, and Raeburn for a study regarding a very fundamental tool to the representation theory: imprimitivity theorems. This work contains a construction of exact sequences in enchilada categories. One of the main results is that the reduced-crossed-product functor defined from $\mathcal{A}(C)$ to the enchilada category is not exact. The motivation was to determine whether one can have a better understanding of the Baum-Connes conjecture. Along the way numerous results are proven, showing that the enchilada category is rather strange. The next main study regards the functoriality of Cuntz-Pimsner algebras. A construction of a functor that maps $C^*$-correspondences to their Cuntz-Pimsner algebras is presented. The objects in the domain category are $C^*$-correspondences, and the morphisms are the isomorphism classes of $C^*$-correspondences satisfying certain conditions. Applications include a generalization of the well-known result of Muhly and Solel: Morita equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras; as well as a generalization of the result of Kakariadis and Katsoulis: Regular shift equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras.

This thesis is a supplement textbook designed with ASU’s MAT 370, or more generally, a course in introductory real analysis (IRA). With research in the realms of mathematics textbook creation and IRA pedagogy, this supplement aims to provide students or interested readers an additional presentation of the materials. Topics discussed include the real number system, some topology of the real line, sequences of real numbers, continuity, differentiation, integration, and the Fundamental Theorem of Calculus. Special emphasis was placed on worked examples of proven results and exercises with hints at the end of every chapter. In this respect, this supplement aims to be both versatile and self-contained for the different mathematics skill levels of readers.

C*-algebras of categories of paths were introduced by Spielberg in 2014 and generalize C*-algebras of higher rank graphs. An approximately finite dimensional (AF) C*-algebra is one which is isomorphic to an inductive limit of finite dimensional C*-algebras. In 2012, D.G. Evans and A. Sims proposed an analogue of a cycle for higher rank graphs and show that the lack of such an object is necessary for the associated C*-algebra to be AF. Here, I give a class of examples of categories of paths whose associated C*-algebras are Morita equivalent to a large number of periodic continued fraction AF algebras, first described by Effros and Shen in 1980. I then provide two examples which show that the analogue of cycles proposed by Evans and Sims is neither a necessary nor a sufficient condition for the C*-algebra of a category of paths to be AF.

Diophantine arithmetic is one of the oldest branches of mathematics, the search

for integer or rational solutions of algebraic equations. Pythagorean triangles are

an early instance. Diophantus of Alexandria wrote the first related treatise in the

fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat.

The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals.

The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.

Almost commuting matrices, i.e. matrices with a sufficiently small commutator, may be nearly commuting, i.e. there may exist matrices close by which do commute. By referencing current literature, this condition is studied for fixed dimension, unitary, self-adjoint, and orthogonal matrices. These proofs are made more accessible and compared to each other, providing insight to possible future progress in the field.

Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs, and as with ordinary directed graphs, there are various C*-algebraic objects that can be associated with them. This thesis adopts a functorial approach to study the relationship between k-graphs and their associated C*-algebras. In particular, two functors are given between appropriate categories of higher-rank graphs and the category of C*-algebras, one for Toeplitz algebras and one for Cuntz-Krieger algebras. Additionally, the Cayley graphs of finitely generated groups are used to define a class of k-graphs, and a functor is then given from a category of finitely generated groups to the category of C*-algebras. Finally, functoriality is investigated for product systems of C*-correspondences associated to k-graphs. Additional results concerning the structural consequences of functoriality, properties of the functors, and combinatorial aspects of k-graphs are also included throughout.