Mathematical Writing - Vivaldi Franco 2014

Title
Writing a Thesis

The title provides an essential description of the content of a document. The titles of research papers compete for attention with other titles in journals, bibliographical lists, databases, etc. They help the reader decide whether to read on. A thesis has a captive readership—the examiners—so this problem does not arise immediately. Nonetheless, a good title can do a lot to give the thesis its unique identity, so it’s worth spending some time thinking about it.

A title is a short phrase without symbols. To get an idea of the wide range of possibilities, we now examine titles of various types of publications, and assess the suitability of the title for a thesis, possibly after amendments. Alongside each title we display the bibliographical data—see Sect. 10.4.

Introductory real analysis [A.N. Kolmogorov and S.V. Fomin, Dover Publications, New York (1970).] The term ’introductory’ immediately calls to mind a textbook, so this term is out of place in a thesis’ title. The shorter title Real analysis would be much too general for a thesis, as it defines a vast area of mathematics.

Making transcendence transparent. (Subtitle: An intuitive approach to classical transcendental number theory.) [E. Burger and R. Tubbs, Springer-Verlag, New York (2004).] Subtitles can be quite effective. In this example, the title has an appealing alliteration, while the subtitle spells out the title’s promise. However, the title’s pedagogical slant and broad take are again the signature of a textbook, not a thesis.

Algebraic aspects of cryptography [N. Koblitz, Springer-Verlag, Berlin (1998).] The subject area covered by this textbook appears to be narrower than in the previous examples, whereas in fact it is still rather broad. If this title were used in a thesis, an expert would probably expect a literature survey. A thesis with an original research component would require a more specific title.

The arithmetic-geometric mean of Gauss. [T. Gilmore, MSc thesis, Queen Mary, University of London (2012).] As the reference to Gauss suggests, this topic is classical, and an expert will know that the arithmetic-geometric mean is connected to many deep mathematical theories. So this is a good title for a thesis, even if there is no indication of what might be the original contribution.

On the zeros of polynomials with restricted coefficients [P. Borwein and T. Erdelyi, Illinois J. Math. 41 (1997) 667—675.] This is a rare instance of a research problem with a concise and non-technical description, which makes a good title. The preposition ’on’ lends an authoritative tone to it, promising some general result.

Toward a theory of Pollard’s -method [Eric Bach, J. Information and Computation 90 (1991) 139—155.] A very eloquent title. It suggests that the method being considered was not supported by a theory, and that some progress has been made. This would be an ideal title for a thesis. The use of the symbol is justified, because the aforementioned method is invariably associated with it.

A complete determination of the complex quadratic fields of class number one [H.M. Stark, Michigan Math. J. 14 (1967) 1—27.] The forceful, unambiguous title of this well-known publication announces the solution of a long-standing problem. Such an assertive style suits well a research paper with a strong result; the trimmed version Quadratic fields of class number one could be the title of a thesis.

Diophantine integrability [R.G. Halburd, J. Phys A: Math. Gen. 38 (2005) 1—7.] This title is short and effective. Each term has an established meaning in a specific area of mathematics, and the unexpected juxtaposition of the two terms suggests interdisciplinarity. This title would be perfect for a thesis.

Outer billiards on kites [R.E. Schwartz, volume 171 of Annals of Mathematics Studies, Princeton University Press, Princeton and Oxford (2009).] The title of this research monograph cleverly exploits double meanings. Both terms ’Outer billiards’ and ’kites’ are mathematical, but not many readers will be familiar with them. The unlikely significance that this phrase would have in ordinary English is bound to raise interest among a general mathematical audience.

Euclidean algorithms are Gaussian [V. Baladi and B. Vallée, Journal of Number Theory 110 (2005) 331—386.] Summarising a result with a sentence makes an effective title for a research article, but perhaps less so for a thesis. The variant Probabilistic aspects of Euclidean algorithms is not so specific, and would work better for a thesis.

Metric number theory: the good and the bad. [R.E. Thorn, PhD thesis, Queen Mary, University of London (2004).] An excellent thesis title, enlivened by a colon. After announcing the research area (metric number theory), a clever reference to an epic Western provides a hint to the specific topic (badly approximable numbers).

To persist or not to persist? [J. Holfbauer and S.J. Schreiber, Nonlinearity 17 (2004) 1393—1406.] This paper is about ’uniformly persistent vector fields’, and the author has transformed the subject matter into a Shakespearean question. Such a self-conscious title must be supported by strong content.

It is easy to determine whether a given integer is prime, [Andrew Granville Bulletin of the American Math. Soc. 42 (2005) 3—38.] Another title-sentence, with the added bonus of being provocative, because primality testing is a notoriously difficult problem. A fair deal of self-confidence is needed to choose a title of this kind. (This paper won the author the Chauvenet prize, awarded by the Mathematical Association of America to the author of an outstanding expository article on a mathematical topic.)