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Part III is a 9 month taught masters course, leading to an MMath degree for those students who are undergraduates at Cambridge, and to an MASt (Master of Advanced Study) for students who join from other universities. As a taught masters programme, the main emphasis is on learning through lecture courses. Assessment is based on written exams taken at the end of the academic year and on an essay submitted about a month before the written exams begin. The standard graduation dates for successful candidates are in June and July.

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Advice on writing Part III essays

This page contains some ideas that I have shared in previous years with students who I have supervised for their Part III essays.

I am setting two essays for 2015-16, on Accumulation and Caching Games., and Online Matching and Bin Packing.

Content of Your Essay

One possible way to think about your essay is to imagine that you have been asked to give a short Part III course (of perhaps 6 lectures). Your essay is the expanded set of lecture notes, or the book chapter, which you are writing to accompany your course. You should include both the notes that you might hand-out, plus what you would say verbally in the lecture room.

You can also think of your essay as preparatory work on a topic that you might like to pursue further in Ph.D. research.

An interesting essay will usually contain the following elements:

• description of a problem area or special field of mathematics;
• decription of one or more specific problems in that area;
• exposition of some relevant mathematical models, objects and definitions;
• well-chosen and interesting results, theorems and proofs;
• perhaps (though not necessarily so) some original ideas of your own;
• applications to problems;
• discussion of possible generalizations;
• conclusions, including pointers to related work, and to further possible work in the same problem area.

You essay should show "taste", by which I mean you should make discriminating choices as to which examples, results, and applications you think are the most interesting, or which you feel are most helpful in providing your reader with an attractive exposition of the essay's subject. (This is exactly the sort of task that a lecturer takes on when he or she prepares to teach a course.) It is unlikely that you would find the best collection of ideas for your essay from one source alone. Part of your project is to read in many places and make your own selection of what is best to include. Even if you do nothing else that is original in your essay, your selection of material and how you arrange it will be your original creation. It is in doing this that you can be impressive. It is more important to be motivating and interesting about your topic than it is to be comprehensive.

Style of Your Essay

• Your ideas are obviously the most important thing, but an attractive presentation is also essential. A sloppy presentation will predispose the examiner to thinking that your mental processes are also sloppy and that your essay is likely to contain errors of fact. So aim to bring to "the look" of your essay whatever you have of an artist's eye and make it as visually pleasing as you can.
• Go to the Moore Library and carefully study a few papers in a premier journal, such as Mathematics for Operations Research or Journal of Applied Probability . Study the way the journal formats the abstract, section headings, lists, figures, tables, captions, references, etc. By taking a journal's style as your model for formatting you can give your essay a professional look. Observe how the sections are organized. Notice how academic authors of mathematical arguments write their English. There are some conventions that most good academic writers usually follow. Use words like "I" or "will" only very occasionally. Academic papers are normally in the present tense. E.g., write "In Section 4 we describe ...", rather than "In Section 4 I will describe ...". Always write in full sentences. Every sentence needs a verb. Most sentences should contain just one idea. If a sentence contains two ideas there should be a reason they are in the same sentence. Avoid run-on sentences, such as: "The Nash equilibrium is a fundamental concept of game theory which is used for routing problems discussed in Section 3 and which one always exists". (This is an actual sentence quoted from an essay.) Re-read each of your sentences. Try expressing the same idea in a sentence with a different word order to see if that sounds better. You might want to reorder the sentence structure by placing the verb at a different point. It can help to read your essay out loud. Your ear will often detect poor grammar that you have failed to catch by reading only on paper. Displayed mathematics should be centred on the line, and end in a full stop. This is because it usually ends a sentence, such as in: We use the fact that \begin{equation} S= \sum_{i=1}^n = (1/2)n(n+1). \end{equation} Tables and figures should be centred between right and left margins. Captions go under figures. Be consistent. If you decide to use "nonnegative", then don't sometimes also write it as "non-negative". If you use "maximize" rather than "maximise", then also use "optimize" rather than "optimise". If $G$ is a game, remember that LaTeX sets $G$ in mathematical italics. Don't elsewhere write it as G (without italics). If you call participants in game "customers", don't randomly switch to also calling them "players" or "agents" in an unnecessary way. Make sure every bit of notation is defined before it is used for the first time. Don't be afraid to remind the reader what an obscure symbol means if you have used it last on page 2 and you are now using it again on page 12. Try to help your reader by minimizing the quantity of special notation and language that she must hold to the forefront of her mind while reading your essay. Pick sensible symbols. It is better to say "game $G$" than "game $\phi$". The reader will find it much easier to remember. By the way, in mathematical writing one tries to avoid starting a sentence with a symbol. Don't write: "x and y consist of only of ..." but "The vectors x and y consist of only of ..." You will know from your own experience that it can be hard work to read multiple lines of algebra (say, in a proof). This is particularly true if lots of new and strange symbols are being used. Please surround chunks of mathematics by plain language commentary that tells the reader why this mathematics is here, why it is important, and what part it plays in your story. Proofs should contain hints that help the reader undertstand how one line follows from a previous one, when this is not obvious.
• The examiner of your essay should be convinced that you really understand what you are writing about. It should be obvious to the examiner that you are not simply copying things verbatim. The University has a very strict policy on plagiarism . Read and understand it. Be careful not to accidentally commit plagiarism by failing to fully mention sources. The golden rule is that the Examiners must be in no doubt as to which parts of your work are your own original work and which are the rightful property of someone else .
• Unlike a Ph.D. dissertation, a Part III essay does not have to make an original contribution to scholarship. However, if you think you have done something that is a bit original, then please flag that up to the reader. You might say something like, "In the following proof I have combined the best ideas from the two approaches that are used in [3] and [5]". (This is the sort of sentence in which "my" is appropriate.) Or say, "Consider the following numerical example which we have invented to illustrate the effectiveness of this algorithm".
• The ordering of material is usually: title page, abstract, foreword, acknowledgments, introductory section, main body, concluding section, references, appendices. (Not all of these must appear.)
• As regards referencing style, compare these styles "[2] proposes that ..." "Smith and Wilson [2] propose that ..." (or if there are more than two authors: "Smith et al. [2] propose that ...") The second form reads like proper English, and is more informative. The first version forces the reader to look at the end of the essay to see exactly what paper is being mentioned. You might anticipate that some of your readers are familiar with research in the field and so by mentioning author names, as does the second version, you can prompt readers to make a mental connection with something they already know about. Another nice style can be "Smith and Wilson (1999) propose that ..." This can be even more helpful to a knowledgeable reader, since he or she probably recgnizes papers by author names and dates, and may know from memory the difference between Smith and Wilson (1999) and Smith and Wilson (2001). However, the author-date form becomes difficult when you want to give a list of papers and it would start to look messy to list all the author names. Then it is easier to say, "This issue has been addressed in many papers (see, for example, [2], [6], [8], [11] and [13])."
• LaTeX treats ' , " , and ` differently. Usually, you will want something like ``this is in quotes'' .
• You can get brackets of the right size around larger bits of mathematics, such as those created by \sum_{i=1}^\infty , by using \left[ and \right] (or \left\{ and \right\} , etc.) Here is an example. Define \begin{equation} k^* = \inf\left\{ i : \sum_{i=1}^k a_i \geq 1 \right\}. \end{equation}
• Note the difference between \cdots and \ldots (or \dotsc ). The first is used in something like $x_1<x_2<\cdots<x_n$ , whereas the second is appropriate for $x=(x_1,x_2,\ldots,x_n)$ .
• \usepackage{amsmath,amssmb} can be useful. The package amsmath gives the helpful environoment \begin{align}..\end{align} .
• You might like to investigate how to use the package psfrag to put nice text into figures.