published_or_final_version / Mathematics / Master / Master of Philosophy

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg, May 2016. / This dissertation provides the main results leading to its primary aim, the proof of the Primitive Divisor Theorem, by appealing to an electric potpourri of mathematical machinery. The employment of binary recurrent sequences with related results is crucial to the approach adopted. The various forms in which the theorem manifests are attributed, among others, to K. Zsigmondy, P.D. Carmichael, and Y. Bilu, G. Hanrot and P.M. Voutier. The proof is confined to instances where the roots of the characteristic polynomial are integers, and when the roots are reals. This dissertation culminates in the resolution of a Diophantine equation which serves as an application of the Primitive Divisor Theorem that is attributed to Carmichael. / GR 2016

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M.Phil.)--University of Hong Kong, 2003. / Includes bibliographical references (leaves 52) Also available in print.

The intent of this thesis is to explore whether any patterns emerge among families or through graph operations regarding the appearance of Weierstrass vertices on graphs. Currently, patterns have been identified and proven on cycles, complete graphs, complete bipartite graphs, and the house and house-x graphs. A Python program developed as part of this thesis to perform the algorithms used in this analysis confirms these findings. This program also revealed a pattern: if v is a Weierstrass vertex, then the vertex v* added to the graph as a pendant vertex to v is also a Weierstrass vertex. The converse is also true: if v is not a Weierstrass vertex, v* will not be either.

Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this set up to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as “combinatorial shadows” of curves. The development of this tropical relationship between graphs and algebraic curves has allowed for beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs. In Chapter 1 we give an overview of the history of chip-firing and its connections to algebraic geometry. In Chapter 2 we describe a reinterpretation of chip-firing in the language of partial graph orientations and apply this setup to give a new proof of the Riemann-Roch formula. We introduce and investigate transfinite chip-firing, and chip-firing with respect to open covers in Chapters 3 and 4 respectively. Chapter 5 represents joint work with Arash Asadi, where we investigate Riemann-Roch theory for directed graphs and arithmetical graphs, the latter of which are a special class of balanced vertex weighted graphs arising naturally in arithmetic geometry.