Theses and Other Publications - Writing a Thesis

Mathematical Writing - Vivaldi Franco 2014

Theses and Other Publications
Writing a Thesis

Franco Vivaldi1

(1)

School of Mathematical Sciences, Queen Mary, University of London, London, UK

Franco Vivaldi

Email: f.vivaldi@qmul.ac.uk

The highlight of mathematical writing at university is the report on a final year project, called a thesis or dissertation. This document has a distinctive structure, in between a short book and a research paper. A thesis surveys a body of advanced literature and presents original work. At undergraduate level, usually the former outweighs the latter.

Writing a thesis is an irreplaceable experience in university education. This document can reveal a great deal about the author’s knowledge, understanding, curiosity, and enthusiasm. Examination scripts do not convey so much information.

One doesn’t learn how to write a thesis by reading an instruction manual. The thesis will take shape gradually, as a result of regular interactions between student and supervisor. The document’s high-level organisation—number of chapters, outline of their content—is normally considered at a relatively early stage of the project. Planning will also suggest appropriate headings and will aid the sequencing of the arguments. Periodic reviews of work in progress may alter priorities, or even re-direct the research. The writing process culminates with the exercise of proof-reading, in itself a valuable, if painstaking, experience.

In this chapter I give an overview of the structure of a thesis, and then discuss selected aspects of writing—choosing the title, writing the abstract, compiling the bibliography—where analysing examples may be helpful. I also include a brief introduction to LaTeX, the software adopted by the mathematical publishing industry. Most theses are now typeset in LaTeX.

10.1 Theses and Other Publications

A thesis is a substantial document. Its length varies considerably, depending on level, topic, and institutional requirements. An undergraduate thesis could be some fifty pages long. There are also theses at master and doctoral level, called MSc and PhD theses respectively.1

A good undergraduate thesis may rise to the level of a master thesis. A doctoral dissertation is longer, has greater depth, and more original research. The brief guidelines given in this chapter apply to any thesis. For further reading, see Higham’s book [19], which devotes two chapters to PhD theses and has an extended bibliography.

A thesis is subdivided into heading, body, and closing matters. The heading supplies essential information: title, abstract, table of contents, acknowledgements. One may also find lists of mathematical symbols, figures, and tables, which some institutions require explicitly. Title and abstract define the subject matter; we’ll consider them in some detail in Sects. 10.2 and 10.3. LaTeX will take care of the menial task of compiling the table of contents. The acknowledgements is a paragraph where the author thanks people for their guidance and support (the supervisor, typically), for useful discussions, for pointing out things, or for allowing the author to use their text in the thesis.

The body contains the bulk of the material, organised into chapters, sections, and, if appropriate, sub-sections. The first chapter is the introduction, written so as to make the thesis self-contained. Here we find motivations, background material and key references, enough details to be able to grasp the main results, and a description of the content of the rest of the thesis.

The length of individual chapters may vary, but each chapter must be a substantial component of the whole, with a clear identity. A brief closing chapter is desirable if not compulsory. This is an opportunity to re-visit ideas and results from an informed perspective, discuss limitations of the work, identify open problems, and chart directions of future research.

Any accessory material which should not interfere with the main text (data tables, computer programs, tedious proofs, etc.) is confined to appendices. The last item in a thesis is the bibliography, listing the sources cited in the text; it’ll be considered in Sect. 10.4.

There is a symbiotic relationship between a thesis and a research paper; the higher the level of the thesis, the stronger the bond. At doctoral level, research papers lead to a thesis, or are extracted from it after the thesis is completed. At undergraduate level this connection is necessarily more remote, yet a research element must be present in every respectable thesis.

But a thesis is not a research paper. A research paper communicates new results to specialists, invariably within tight space constraints. It is seldom self-contained: sketchy motivations, unexplained jargon, and terse citations are commonplace. A thesis and a paper also differ in format. A paper is shorter, divided into sections, not chapters, while its simpler navigation renders a table of contents unnecessary.

A thesis is not a book either. A book covers thoroughly a body of established material. It is longer than a thesis, and we expect it to be wholly self-contained. A book begins with a preface, which is neither an abstract nor an introduction, but a bit of both. It is rare to find a book without exercises in each chapter, and even rarer one without a subject index at the end. (Notably, the celebrated book [15] has neither.) A textbook, being conceived as teaching material, may offer solutions or hints to exercises. This is a modern trend: hardly any of the classic, timeless mathematics textbooks provided solutions or hints to exercises—a fact worth pondering on. Some advanced publications resemble theses: the review articles, published in dedicated research journals, and the research monographs, published as books, usually within a collection. These are a sort of advanced theses, which collect together previously published works. They usually appear when a new research area has reached maturity.

Most scholarly publications are specialised; they are not intended for the general public. So how narrow should the target audience of a thesis be? This is largely a question of personal taste. Conscious decisions must be made from the very beginning, because the title, the abstract, and the first paragraphs of the introduction will set the tone for the rest of the document.

Let us inspect the opening sentences of some theses and research papers, to see how strongly the character of the work is established with just few words. Later on, we shall analyse titles and abstracts from the same perspective.

Our first example is a master thesis (from [35], with minor editing).

Let $$GL(d,\,q)$$ denote the general linear group of invertible $$d\times d$$-matrices over the finite field $$\mathbb {F}_q$$ with $$q$$ elements. In computational group theory, it is of interest to calculate the order $$n$$ of an element $$A\in GL(d,q)$$, i.e.,

$$ n=\min \{j\in \mathbb {N}\,:\, A^j=I\} $$

where $$I$$ is the $$d\times d$$ identity matrix. The integer $$d$$ can be large, e.g., $$d>100$$.

The style is formal, with definitions and symbols appearing straight away. The jargon (general linear group, finite field, order of a matrix, etc.) and the corresponding notation ($$GL(d,q)$$, $$\mathbb {F}_q$$, etc.) belong to standard introductory algebra. The author still provides brief explanations, which convey an impression of helpful, considerate writing. This document promises to be accessible to any mathematically mature reader, a desirable feature of any undergraduate thesis.

Our next example, from a thesis in geometry [16], illustrates a completely different expository strategy.

In 1984, Schechtman et al. [SGBC84] announced that the symmetry group of an aluminium-manganese alloy crystal, produced by rapid cooling, was that of the icosahedron. Such a symmetry is not possible for a periodic structure in three dimensions. This discovery brought down a long-held assumption in crystallography, that the only structures with some sense of long-range order were periodic.

The thesis begins with an account of the surprising outcome of a physics experiment. (We shall learn about literature citation in Sect. 10.4.) The author tells a story, skilfully building some drama. The near complete absence of mathematical terms—formal definitions and symbols are given much later—makes this excerpt very inviting. An engaging, accessible style is welcome in any publication, at any level.

A thesis can be highly specialised. One could use the established dictionary of a particular research area, without explanations or apologies, from the very beginning. This choice undoubtedly simplifies the writer’s task, but it creates problems for the reader, thereby reducing the readership basis. This is legitimate, as long as the jargon is used wisely to sharpen the exposition, not gratuitously to impress the reader.

The following opening passage is taken from a PhD thesis in the area of ergodic theory [3].

Ergodic optimisation is a branch of the ergodic theory of topological dynamics which is concerned with the study of $$T$$-invariant probability measures, and ergodic averages of real-valued functions $$f$$ defined on the phase space of a dynamical systems $$T:X\rightarrow X$$, where $$X$$ is a compact metric space.

This sentence explains the meaning of the expression ’ergodic optimisation’, which features in the thesis’ title. To achieve this within a limited space, the author relies on the reader’s familiarity with half a dozen advanced concepts. Three important symbols are also defined in the same sentence. Undeniably, this thesis is for specialists.

In research papers, extremes of specialisation are accepted, or at least tolerated. Some editors insist—wisely, in my view—that the author provide clear motivations and a minimum of background, but this requirement is by no means universal. Papers that begin with a dry, unmotivated list of definitions are not uncommon. The following opening paragraph of standard definitions appears unchanged in three related publications [28—30].

Let $$K$$ be an algebraic extension of $${\mathbb {Q}_{p}}$$, and let $${\fancyscript{O}}$$ be its integer ring with maximal ideal $$\fancyscript{M}$$, and residue field $$k$$. If $${\overline{K}}$$ is an algebraic closure of $$K$$, we denote by $$\overline{\fancyscript{O}}$$ and $$\overline{\fancyscript{M}}$$ the integral closure of $$\fancyscript{O}$$ in $$K$$ and the maximal ideal of $$\overline{\fancyscript{O}}$$, respectively.

In the following extreme example, the author disposes of definitions altogether [7].

1. Introduction. The notation used is that of [2]; in particular ...

One cannot proceed without the cited publication (by the same author). Only a committed reader will accept such a blunt treatment.