University of New South Wales - School of Mathematics

Introduction

Each Mathematics Honours student must undertake, as part of their course, an Honours Project, which counts for (usually) 30% of their Honour grade. The project involves:

Independent study under the supervision of a member of staff.
The writing of a thesis giving an account of what was studied.
The presentation of some of this material in a 50 minute seminar which will be attended by staff and other students.

The objectives of the project are:

To give the student an opportunity to engage in some study which is driven more by their curiosity than by a given syllabus.
To expose the student to mathematics beyond the standard syllabus and to get some view of what is happening at the research front.
To help develop some of the basic research skills including use of the library and computer databases.
To develop the skill of writing technical material well.
To develop the skill of presenting technical material orally.

Many of these skills will be of great benefit whether or not the student progresses to further study in mathematics. Of course, for the student who does go on to do mathematical research, these skills are vital. It should be noted that although the project is an important part of a student's research training, the aim of the project is not to produce original new theorems. Occasionally a project will contain some original results, but this is certainly not a required component.

Remember that different projects can have quite different natures. Some are surveys of a particular area of mathematics, some look at the history behind a famous problem, others may require calculating some examples, or filling in gaps in published works.

You and your supervisor

Before you start your Honours year, you should speak to a number of members of staff about possible projects. Find out who works in the areas that you are interested in Find out who you find it easy to talk mathematics with. If at all possible, settle on a topic and supervisor before the start of session.

Most students see their supervisor about once a week, although this is usually open to negotiation between the student and the supervisor. Even if you haven't done much between visits it is a good idea to have a regular chat so that your supervisor can keep track of how you are going. You can expect your supervisor to:

Help you select - and modify - your topic.
Direct you to useful references on your topic.
Explain difficults points.
Provide feedback on the direction of your research.
Read and comment on drafts of your thesis.
Help prepare you for your talk.
Give general course advice.

The thesis

Your thesis is a report of what you have been studying in your project. Write it as if you were trying to explain the area of mathematics that you have been looking at to a fellow Honours student.

Include an introduction that explains what the thesis is all about, and what its contents are. (It is sometimes better to leave writing this to the end!) For some theses, a conclusion or summary is appropriate.
Your thesis should be a coherent, self-contained piece of work.
Your writing should conform to the highest standards of English. Aim at clarity, precision and correct grammar. Start sentences with capital letters and end them with full-stops. Don't start sentences with a symbol. (Otherwise you can get things like `... a function f. g is sometimes used ...' which is hard to parse.)
Take great care with bibliographic referencing. Wherever some material has an external source, this should be clear to the reader. Don't just write in the introduction: this thesis contains material from [1],[2] and [3]: give the references for the material wherever it is used. Don't gratuitously pad your reference list with references that are not referred to in the text. Check current journals for acceptable referencing styles.
Be careful not to plagiarize. What constitutes plagiarism is perhaps a little different in mathematics compared to some other subjects since there is a limit to how different you may be able to make a proof (at least in its basic structure). We do however, expect the thesis to be written in your own words. A basic rule is: if you put a fact or an idea in your thesis which is not your own, the reader should be able to tell where you got this fact or idea.*

These days theses are almost always typed in LaTeX. If you are going to type the thesis yourself, you should allow a certain amount of time to become familiar with this software. Indeed starting to learn LaTeX well before you actually want to write the thesis is a very good idea.

You should not underestimate the time it takes to produce a polished document. You will almost certainly need several drafts. It is very difficult to concentrate on getting the mathematics, spelling, grammar, layout, etc, all correct at once. Try getting another student to proofread what you have written - from their different viewpoint they may pick up on lots of things that you can't see.

You should be beginning to plan the `shape' of the thesis before the start of your second session. You supervisor should have a fairly mature draft by the end of the midsession break, but you should probably give them a chapter rather sooner than this so they can check your writing style.

Typically a thesis should be between 50-70 pages in length. If you think that you have a good reason to write a shorter or longer thesis, discuss this with your supervisor. Theses are not judged by their weight! It is better to write a shorter thesis in which you understand everything than a longer one where you are rather hazy on the details.

The talk

Towards the end of your Honours year you will give a seminar on your project. The temptation is to try to include everything that you have learnt. Although 50 minutes might sound like a long time, you will be surprised at how little you can fit in a 50 minute talk.

Your talk should be self-contained. As with the thesis, it should be aimed at the level of your fellow Honours students.
Discuss with your supervisor which aspect of your project would make a good talk. The hard thing is deciding what to leave out while still telling a coherent story.
Start the talk with a non-technical introduction so that everyone understands what your project is. Pitching the talk at a higher level may be necessary later in the talk, but try not to get to the stage when only you and your supervisor understand what is going on. Remember that it is very easy to overestimate what the audience understands (haven't your lecturers been doing that to you for years!?).
Go to other seminars during the year. If you can, go to the seminars of Honours students who are ahead of you. Take notes of what works and what doesn't work in their seminars. If you get annoyed by, for example, them taking overheads on and off too fast, make sure that you don't repeat the mistake.
Have a run through with your supervisor at least a week before the talk is scheduled.
Don't worry too much. This is not an examination, and we are not trying to catch you out. We all remember how nervous we were before our first talk, and will take that into account

There will usually be some questions asked at the end of the talk. Don't feel that you have to know the answer to all of them. Questions are asked because the questioner wants to know the answer, not because they want to see whether you know.

What we are looking for

The basic question that needs to be answered for the staff is `Can this student independently take in some advanced mathematics and then clearly present this in written and oral form'.

The thesis will be assessed for quality in four major areas:

Exposition:
This includes expression, grammar, layout, organisation, communication skills. Is it easy to follow what is written? Is the notation sensibly chosen and consistent?

Literature coverage:
Does the student understand the context and history of the subject area? Is the bibliography and referencing adequate?

Critical analysis and insight:
This includes technical competency, understanding and the level of the material. Does the student understand what they are writing/saying?

Originality:
Are there new results, insights, methods or applications? Does the thesis contain a critical evaluation of work on a topic, or is it just a list of results?

The weighting of the various assessment components will vary depending on the type of thesis. For example, the literature coverage will be more important in a survey type project than in one which contains original results. All theses are expected to address all four areas however.

A timetable

The following is a general guide to how work on your project should progress. If you think that a major variation is warranted, please discuss this with either your supervisor or the Honours Coordinator.

Select supervisor and topic	Before the start of your Honours year
Research, reading, discussion, understanding	First session & mid year break
Start work on first draft	Early in second session
Give draft to supervisor	End of second session break
Talk	Week 10-13 second session
Final submission	End of second session

Honours Thesis Format

This section provides some guidance as to the physical presentation of an Honours thesis at UNSW. To quote the rules:

Students are required to submit four copies of their thesis, typed and in a protective binder or cover. Students are responsible for the production of their theses. They may choose to type their thesis themselves, possibly using one of the word-processing facilities available in the School (e.g. LaTeX). Another option is to have the typing done by a professional typist. Some of the secretarial staff in the School are prepared to type theses for payment outside normal working hours.

These days, almost all students type their theses using LaTeX. The School runs periodic classes on how to get started in LaTeX and you should take advantage of these as early as you can.

Getting a good looking thesis can be helped by having a good style or `document class' file and a decent example to copy. Below are links to a UNSW thesis class which has been set up to produce a nice front page which says all the right sort of things and has a copy of the UNSW crest on it. There is also a cut down example of a past thesis.

UNSWthesis.cls. To use this the start of your LaTeX document should include the line \documentclass[honours]{UNSWthesis}
crest.eps. A Postscript version of the UNSW crest.
Sample thesis. If you put the three files in a directory and then latex this file, you'll be able to look at what it produces. This sample is of course much shorter than a real thesis should be! The file contains lots of macros and special environments if you want some examples of how these work.

Past Theses

For some role models, see the page of recent theses in the School.

This document is based on a number of earlier documents for Pure Mathematics Honours students at UNSW and Sydney University.
Ian Doust

* This quote came from an internet source at Mansfield University, which has since disappeared. Be wary of internet sources unless they appear to be part of a stable site.