TeX and LaTeX - Writing a Thesis

Mathematical Writing - Vivaldi Franco 2014

TeX and LaTeX
Writing a Thesis

The programming language TeX, developed by D. E. Knuth [23], and its user-friendly offspring LaTeX, written by L. Lamport [25], are the standard software in mathematical and scientific publishing. This book was typeset in LaTeX.

This typesetting system is intended for text containing mathematical formulae. It also supports the high-level features needed to manage a complex document: generating tables of contents, numbering of chapters and equations, cross referencing, handling bibliographical data, etc.

Let us inspect a very short LaTeX source file5:

The information is stored in plain text. We must type it in a file, and then run the LaTeX program, which generates a new file suitable for display, typically in PDF format.6 The end result is the following text, placed at the top of an empty page with the page number 1 at the bottom:

In 1735, Leonhard Euler proved the following remarkable formula:

$$\begin{aligned} \sum _{n=1}^\infty {\frac{1}{n^{2}}}=1+{\frac{1}{4}} +{\frac{1}{9}}+{\frac{1}{16}}+\cdots ={\frac{\pi ^{2}}{6}}. \end{aligned}$$

(1)

The contrast between the awkwardness of the LaTeX source and the neatness of the output may be off-putting at first. On closer acquaintance, it will instead become clear why this logical and powerful language has become so widespread.

In the source file we note frequent occurrences of the distinctive backslash symbol ’$$\backslash $$’, followed by a word. These are LaTeX commands, which instruct the program to do something. We see that, after an initial preamble, a pair of commands delimit the beginning and the end of the document. Within the document, another pair delimits a displayed equation, and within the equation various commands generate mathematical symbols and structures. Certain things occur automatically, such as the centering and numbering of the displayed equation, and the output is insensitive to the layout of the source code.

Of course, the software does a lot more than typesetting. Thus LaTeX may be expanded by incorporating auxiliary libraries of commands for specific tasks, such as BIBTeX, which manages bibliographical information. Similarly, the appearance of a document may be altered by means of macros. These are libraries transparent to the user, which reset the layout parameters (font type and size, line and paragraph spacing, style of bibliographical list, etc.). Publishers use this facility to convert standard LaTeX source files into documents with the desired appearance.

Learning LaTeX takes some time, but if you plan to write a thesis this is a worthwhile investment. There is no shortage of introductory material: the Cambridge University LaTeX page is a good place to start [27].

Exercise 10.1

The following abstracts are poorly written. Identify the problems, hence write an improved version.

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Exercise 10.2

This is the longest opening sentence of a mathematical abstract I managed to find in the literature [39].

By Morava’s point of view on the stable homotopy category, the quotient in some sense associated to the filtration related to the height of formal group laws is studied by the category of modules over the function ring of the deformation space of the Honda group law of height $$n$$ with the lift of the action of the automorphism group on the closed fibre through the Adams-Novikov spectral sequence.

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Exercise 10.3

Explore the statement (4.31), and write a one-page report on your findings.

Exercise 10.4

This exercise proposes some MICRO-PROJECTS in experimental mathematics. You are given a sequence to study; your task is to generate numerically a sufficient number of terms, analyse the data, and then write a succinct report—three/four pages, say.

In the report you must define the problem, give a concise account of your findings, and synthesise them in a precise mathematical conjecture. Invariably, some plots will be necessary. You should provide at least one bibliographical item, and include the numerical code in an appendix.

For an introduction to experimental mathematics, see [42]; the book [5] is a good source of computational problems suitable for projects.

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Footnotes

1

Acronyms for Master of Science and Philosophy Doctor.

2

The suffix ff to a page number means ’and the following pages’.

3

This is the oldest mathematics journal still in existence, founded in 1826.

4

Abbreviation for the Latin et alii, meaning ’and others’.

5

The source file for the Mathematical Writing book contains over half a million characters.

6

This is an established format for document representation.