Writing a thesis is daunting:

1. you have to understand complex maths (or develop some new maths),
2. then you have to write it all down.

Where to begin?

Fortunately for us mathematicians, we spend our whole lives training to do the maths.

Fortunately, the writing up of said maths is highly structured and governed by conventions that you can mostly learn in this lecture.

Also once you’ve properly written your first thesis, writing any subsequent long pieces of maths will be comparatively easy. (Of course, the mathematics does not necessarily get easier—just the writing process!)

These notes are primarily aimed at students writing their BSc or MSc thesis. However most of this advice is also to applicable to PhD theses, and papers in general.

### The Preparatory Work

To have something to write, you have to have something to say.

That something has to be of the correct level of detail and understanding. In other words, the first step towards writing a good Thesis is doing the correct kind of maths (reading, rewriting proofs, researching new proofs, proving new theorems).

##### One extreme: understanding everything

You might think that the only way to write a thesis is to understand everything first.

This is commendable, but is likely to lead to lifelong learning and an incomplete degree.

Accept that some aspects of your topic or research will remain unclear.

##### The other extreme: understanding the minimum

Alternatively, you might think it is sufficient to draw a straight line from your current state of knowledge to your desired written thesis.

There is no such straight line from here to there.

Accept that you will spend hours reading up on material or struggling with a new proof that will not make it into your thesis.

Think of these detours as learning opportunities—they ultimately lend you a broad view of the subject and contribute to your mathematical maturity.

##### How a thesis is similar to an exam

Like on an exam, you wish to present your knowledge in the best possible light.
So it is best to organise your work around your stronger points and away from your weaker ones.

Moreover, like every exam, every Thesis has a well-defined purpose—to showcase your knowledge—and thus you can treat it as a piece of work that you should be able to defend as being “true” in the best mathematical sense.

##### How a thesis is not similar to an exam

In an exam you may be expected to provide an answer building on the first principles of the subject.

In a thesis you are expected to start at a much higher level and choose your first principles (the results you can cite and use without proof).

The balance of citations and proofs is something you should discuss with your adviser.

##### Understanding the purpose of your thesis

Before you can think about writing, you have to understand, clearly and specifically, the purpose of your work—what it is you are aiming for.

Here are some possible aims:

• Any old Thesis.
• An impressive Thesis.
• A well-written, concise Thesis that does some serious maths.
• A Thesis that shows you've put in the hours.
• A Thesis that displays your knowledge.
• A Thesis that will get you an excellent recommendation from your adviser.
• A Thesis that will get published.
• A Thesis that you can use to impress your parents/siblings/partner.
• A Thesis that you will be proud of.
• A Thesis that you will enjoy writing.
• A Thesis that you will enjoy 🔥 burning 🔥 after you are done.

All of these are acceptable goals, but what you need to understand are the concrete requirements each type of thesis is meant to fulfil.

• The aim of a BSc thesis is to show that the student has read up on a certain subject in sufficient depth and breadth, and is able to produce a clear, well-structured exposition on the topic of their thesis that shows they understand the material.
• The aim of an MSc thesis is to show that the student is capable of independent original research. This means displaying skills beyond those necessary for a BSc thesis, by offering new insight into well-established problems (usually through proving existing theorems using different methods than the original proofs).
• The aim of a PhD thesis it to show that the student has completed original research. This means displaying skills (way) beyond those necessary for an MSc thesis, by proving sufficiently many interesting new results.

Remember the concept of an Ideal Reader?

Your Ideal Reader will be judging you against these aims, so your job is to write a Thesis which ticks all the expected items in a conventional way.

The remainder of today's notes discusses what conventional means.

### The Local Structural Elements

Each page of maths comprises some or all of the following elements:

1. named structural elements, such as Definitions or Theorems, that have strict, precise, minimalist form,
2. graphic elements, such as Tables or Figures,
3. free-form exposition between these structural elements.

You can approach 1. and 2. more or less independently, focusing on getting a proof right or drawing an appropriate graph. However, 3. requires you to have the other elements prepared before you can proceed to bind them together and smooth the transition between them.

##### Named elements
• Definitions are used for introducing important new notation and concepts. Less important notation, or definitions that are being recalled rather than introduced are usually given in free-form exposition.
• Theorems, Propositions, Lemmas, and Corollaries signal noteworthy results.
• Theorems either used to be, or currently are the pinnacles of current research; they are the weightiest and most important results on which further theory is built. You get theorems that are named, such as the Fundamental Theorem of Calculus, Fermat's Last Theorem, the Arzelà–Ascoli Theorem etc.
• Propositions are results that have lesser importance beyond the scope of the text. They do not become famous enough to be named.
• Lemmas are usually technical statements used to prove more important results. Nonetheless, some lemmas do become famous enough to merit a name, such as Dehn's Lemma, Kronecker's Lemma, the Morse Lemma.
• Corollaries are results of varying importance, but they must always follow fairly obviously from some previously stated result.
• Proofs normally follow immediately after the statement of the result, but can sometimes be delayed by exposition or, indeed, by Lemmas, Propositions, and other elements.
• Remarks signpost a variety of observations, side notes, caveats. Remarks are essentially elevated bits of exposition that the writer wishes to highlight.
• The less commonly encountered Conjectures and Questions signpost speculation.
• Conjectures, like the Riemann Hypothesis, are predictions of results left to others to prove.
• Questions are open-ended. You can either pose questions to others or your can cite other people's questions as motivation for your own work.
• Footnotes contain short asides that are unimportant for the main results of your work, but are still noteworthy in the context. Footnotes are rare in modern mathematics[1].
##### Referencing other work

If you are not proving your own theorem, but are writing up someone else’s in your own words, you should indicate the degree of modification.

1. This proof comes from Artin’s book: means that you virtually copied out the proof given in Artin’s book.
2. This proof follows Artin’s: means that you are paraphrasing Artin’s proof and preserving its structure.
3. This proof is modified from Artin’s proof: means that your proof is, in detail, significantly different from Artin’s proof, but in structure comes from or follows that source with slight modification.
4. This proof is based on Artin’s proof: means that your proof follows the general outline of Artin’s proof.

1. Although I personally am fond of them. ↩︎

### The Global Structural Elements

These are the larger structural elements of a thesis.

• Title Page. Every thesis has one and it includes the university logo or header, the title of your Thesis, your name, the name of your adviser, and the date. Papers do not have title pages.

• Abstract. The abstract of the thesis should appear either on the title page or on a separate page.

The abstract should complement the title, by expanding on it and summarising the gist of your work. It should be no more than two paragraphs long, and preferably less than 200 words. Any new results should be mentioned.

Especially when it comes to papers, most people will only ever read the paper after they have read the abstract and judged it to be sufficiently “interesting”.

Therefore to entice readers, in the abstract you should phrase your results in an accessible, quick-to-read fashion; this means that you should avoid using technical jargon and introducing notation unless absolutely necessary.

Even though the abstract appears first on the page, it is written last: after you have completed everything else.

Please avoid the following common mistakes when writing the abstract:

• Not including the abstract.
• Including the abstract, but not titling it, and leaving it as a standalone paragraph to float about on a page.
• Calling the abstract anything other than Abstract, e.g. An abstract, The Abstract, My Abstract, Preface, Summary, Introduction.
• Including the roadmap (see below) in the abstract.
• Including words of thanks in the abstract.
• Treating the abstract as the Introduction.
• Introducing symbols in the abstract that are not used again in the abstract.
• Including long equations or specialised technical jargon.
• Using expressions such as:
• In this paper we would like to,
• Down below we will attempt to,
• Here I shall go on to show that, and so on.
• A roadmap is a brief summary of the thesis or paper that explains the main purpose and content of each section. Sometimes it is only a paragraph long; at other times, when the work is more complex, each section on its own may take up a whole paragraph.

Here is an example of a short roadmap appropriate for a BSc thesis:

1. In Section 2, we give a brief introduction to Knot Theory. Section 3 defines the Jones polynomial, discusses its properties, and computes it for a few examples. Finally, in Section 4, we use the Jones polynomial to prove a result about alternating links.
• Acknowledgements. There are two kinds of acknowledgements: the personal (relatives, friends, pets) and the professional (adviser, institution, foundation).
• Personal acknowledgements do not appear in papers, but are quite common in PhD theses and are then likely to appear on their own page after the abstract.
• Professional acknowledgements are always welcome and should appear as the last element of the Introduction, or the last element before the appendices; in BSc and MSc theses, the students are expected to write a single line thanking the adviser[1].
• Table of Contents. Shorter texts usually do not have a table of contents, but anything more than seventy or eighty pages will likely have one. In that case, you should let $\mathrm{\LaTeX}$ generate this page automatically.

• Introduction. The first chapter of your thesis should be called Introduction. The introduction is there to—unsurprisingly enough—introduce the problems you have tackled. It consists of two parts:

1. the free-form exposition;
2. the structured segment containing the conclusion (optional), the roadmap (mandatory), and the professional acknowledgements (optional, though expected).

The free-form part gives some historical background that leads up to the statement of the problem (or the other way around: you state the problem then explain its history).

Then follows a discussion of a few notable attempts to solve the problem or a few notable (partial) solutions to the problem. Finally, there comes a summary of your approach, culminating in your own results.

For a thesis that has no new results (which is invariably the case for your BSc/MSc thesis!), you can still go through all of these steps except for the last one; you simply end your Introduction after describing “your” approach to the problem, where “your” is most likely qualified as a modification of someone else's work.

Like with the abstract, the goal of the first few sentences of the introduction is to entice the reader to keep going (when possible).

Here are a few sentences that we have modified from real-life papers. They illustrate the layout patterns of information that you might expect in the introduction of a professional paper.

1. The history of Raindrop algorithms goes back at least forty years to the work of Cloud and Thunder, and their application to the Rainfall Problem.
Aiming for a historical context; probably quite extensively.
2. The first Tree-type equation was proposed by Yellow Seeds in 1953.
Aiming for a briefer historical context.
3. Labyrinth sets were introduced by A. House in [GH45].
A similar, slightly more technical approach; no one would mistake this for a non-technical paper.
4. Let $D$ be the Diamond Polygon of Matrices, also defined as $$D: = \{ \diamond \triangleleft \Diamond \mid \diamond^2=\diamondsuit\}.$$ Then it is well-known that $D$ is not a Valuable Diamond.
Jumps straight into defining the key concepts. This usually occurs with highly technical results that require many concepts to be defined before the problem can at all be stated.
• Notation and Conventions. For longer texts, or texts that use nonstandard notation, you should add a section specifying this notation, like a key on a map telling the reader which symbols means what. This section would go before or after the Introduction, or before or after the References. It is not necessary to state all of your notational conventions, particularly the standard ones. (Do not patronise your readers by reminding them that $\mathbb{Z}$ denotes the integers, for instance.)
• Preliminaries. This second chapter should recall the basic material you will then build on.
• The other chapters!
• Conclusion or Outlook. This would usually be a short discussion of future work or ideas. It is not common to include such a section in a Thesis, and it would only be included in a paper if the authors had something they felt was significant to point out.
• Appendix. If your work calls for technical proofs or material that does not naturally fit in the flow of your Thesis, you may want to consider relegating it to one or more appendices. The appendices are usually labelled: Appendix A, Appendix B, etc. A Thesis is unlikely to have more than a few appendices[2].
• References The list of all sources should be formatted to contain the name of the authors, the title, and the precise location of the source. As previously discussed, I recommend the biblatex package in $\mathrm{\LaTeX}$.

1. Even if what you really mean is “I hate my adviser for giving me a terrible topic and never having enough time to meet with me”! ↩︎

2. If you find you have reached Appendix M then you have shifted too much material into the appendices! Put some of it back into the main body of the paper. ↩︎

### A Few Tips on the Process of Writing

There are many ways to write a thesis or paper, but none of them involve starting with the abstract or the introduction—those you write last.

One method involves the top-down approach, where you start with what you wish to prove (or have proved). You then look at what results you needed for your proof, then at what results those results used, and so on until you reach the level at which you can state things without proof. This level is something you determine by thinking about the purpose of your work, the expectations, and the Ideal Reader.

Before you sit down to do some actual writing, you might think about outlining your work: decide on your preliminaries, decide on your notation, decide on a nearly complete list of all the other named elements, decide on any figures or tables you may need.

Play around with the ordering in your outline until you find a flow that takes the reader from page 1 to the end in the most logical, least choppy way.

If you are happy with this order, then you can start filling out the various structural elements and leaving placeholder text for the free-form exposition.

Once a draft is complete, read it through, focusing mainly on the free-form exposition.

The exposition is what gives your work a feeling of unity, and what makes a Thesis or paper more than just a list of results.

Leave your work aside for a few weeks, then return to it with a fresh mindset—this should help you identify any problems either with the writing or with the maths.

1. The instructions “leave your work aside for a few weeks” and “hand you thesis in before the deadline” are mutually incompatible if you finish your thesis at 4am in the morning on the day before you are due to hand it in. Pro Tip: Don’t do this.

Get a friend or colleague in roughly the same field to look over your work, or at least, to look over the beginning.

Ask them whether the Abstract and Introduction make sense (both should).