# Honours in Mathematics general information

The convenor for the fourth year honours program is Dr Joan Licata.

The honours degree in mathematics is the initial qualification towards becoming a professional mathematician. However, the rigorous training in mathematics provided by the honours degree will be of great use to the students intending to enter a wide variety of careers. In recent years, honours graduates from the Mathematical Sciences Institute (MSI) have gone on to higher degrees in biology, computer science, economics, linguistics, philosophy, physics and theoretical physics, as well as further mathematical studies, and others have easily found attractive positions in industry, financial institutions and the public service. The honours degree is taken over four years, with the final year consisting of mathematics courses and thesis work (Double honours, in which two complete honours programs are undertaken in successive years, are possible; the most likely combinations with mathematics are computer science, philosophy, physics, statistics, or theoretical physics).

Admission to the honours program is granted at the end of the third year, but students intending to take honours should have discussed this with staff at an early stage of their course.

For entry into the fourth year in mathematics, the students should have completed

- at least 48 units of B and C level mathematics courses with a minimum grade of credit (completion of units in a cognate subject may be taken into account), including at least 24 units of C level mathematics courses.
- the students must also have sufficient prerequisites to enable them to pursue an approved course of study in their fourth year. For instance, the students wishing to pursue a course of study in fundamental mathematics should include MATH3116 (or MATH2320) and MATH3104 (or MATH2322) in their program.
- the College rules concerning entry into honours must be satisfied.

Students who enter the ANU with direct entry to honours will be admitted to honours in mathematics if they meet the relevant College requirements. However, our strong recommendation is that students entering fourth year mathematics honours do four mathematics courses in the THIRD year. Failure to do so will make it difficult to achieve a high level in the honours degree.

Part-time enrolment, with completion of the honours program over two years, is possible; the fourth year honours coordinator and the College office should be consulted for advice. It is also possible to commence the program in mid-year. Students wishing to undertake further studies in mathematics after completing a pass degree some time ago should consult the College office for advice.

## Honours in Mathematical Applications and Computations

The Honours specialisation in Mathematical Applications and Computations differs from the traditional Honours in Mathematics specialisation in that it focusses more on the application of mathematics. This program aims to be flexible enough to allow students to work on application areas from other subjects such as physics, chemistry, environmental studies or mathematical finance etc. The interested students should contact Dr Linda Stals (linda.stals@anu.edu.au) for more details.

## Organisation of the Honours Specialisation

The duration of the full-time fourth year honours program is approximately ten months; the students will normally be required to be in attendance from late January-early February to mid-November (late June-early July to mid-June for mid-year entries). After consultation with staff members, each student will provide a plan of study for course- and project work, to the satisfaction of the convenor. Any subsequent changes must be approved by the convenor.

Assessment of fourth year honours will be on the basis of a thesis (47%), coursework (50%) and two seminar talks (3%). Assessment of the coursework will be made in the usual manner. The thesis will be assessed by two examiners in accordance with the criteria listed below. The seminar talks will be assessed by the convenor jointly with other members of the MSI (Note that the ten month duration of the full-time honours course includes the time needed for assessment).

## The Thesis

The topic for the thesis should be chosen in consultation with the supervisor during February (second half of June-first half of July for mid-year entries), or even earlier if possible. The student’s supervisor may be expected to give close guidance throughout the year; it is a good idea to schedule a set time each week to see one’s supervisor, even if there is nothing in particular needing discussion at certain times. Note that the supervisor should not be expected to help with the coursework or general problems not connected with the project; general problems should be referred to the convenor, coursework matters should be discussed with the lecturers. The thesis should normally have the form of an in-depth survey article. Perhaps the most effective way to find out what is acceptable is to view some of the essays which have been presented in previous years; they are located in the MSI library, and (from 2005 onwards) can be accessed through the MSI website (MSI access only).

The points upon which the essay will be judged are as follows:

- the topic chosen should involve mathematics mainly outside the available coursework and standard texts;
- the depth of understanding, not only of individual results but also of their relationship to the subject as a whole;
- familiarity with the literature and use of the library in background reading;
- the independence of outlook, usually in the area of synthesis of the literature into a cohesive account. Original mathematics is NOT required, and indeed the supervisors should discourage their students from attempting original work at the beginning as being too uncertain an undertaking. However, evidence of original ideas about the subject matter, its development and importance, will be considered crucial. Where direct quotes and close paraphrasing of other authors is considered desirable, the corresponding references must be given;
- style; mathematics is a human endeavour and the point of the essay is to communicate ideas. The essay must therefore be readable, with discussion illuminating the technicalities. The article by Halmos
*How to write mathematics*may be found useful.

The length of the essay will necessarily depend on the nature of the topic; however, 60 to 70 pages should be regarded as normal, less than 50 pages or more than 100 pages is unusual. The introduction to the essay should contain a statement indicating in general terms the sources of the results given, and any claims to originality in either results or methods. Three hard copies and a pdf file are to be submitted, with the pdf file submitted online through Wattle. The manuscript will usually be prepared using the TEX typesetting system, which is available on University computer systems. (The MSI will be able to provide access to word-processing equipment for this purpose). To allow time for assessment, the project must be completed and submitted approximately by the end of October (approximately by the end of May for mid-year entries); for the precise dates see the Student Guidelines page. The College milestones require a draft to be submitted to the Supevisor for feedback by the beginning of October. Possible honours essay topics can be found at our projects page.

## Coursework

Meetings will be held on the first Monday of each semester at 10 am in Room G35 to discuss the third/fourth year courses for the year. Courses totalling 24 units must be taken during the honours year, with a least 2 courses in the first semester.

## Seminars

The students will give two seminar talks during the course of the program. The first talk is 25 minutes long and may be on any mathematical topic. The second talk is 50 minutes long and must be related to the project. The purpose of the seminars is to give practice in delivering a lecture on a mathematical topic to a non-specialist mathematical audience.

The points for judgement are as follows:

- the organisation of the material, the clarity and suitability of the presentation for the audience mentioned;
- the timing — appropriate choice of the material for the allotted time, including the time for questions;
- attention to mechanical details of lecture techniques, such as audibility, board technique, judgement of audience comprehension, etc;
- the depth of understanding of the material chosen for the seminar.

Your supervisor, or perhaps your colleagues, should audit a “dry run”: you will almost certainly find that you initially overestimate the amount of material that can be covered in your talk.

The seminar talks will be combined into student conferences; we will have two such conferences during the year. Honours students are expected to attend both conferences; students with significant conflicts should discuss this with the convener as early as possible.

Further, you should note the weekly Mathematics Seminars held by the MSI; occasionally these may be specialised in nature and not appropriate for you, but frequently they will give you an opportunity to learn of the latest research methods in fields of interest to you, to hear an expository survey from an expert, or simply to meet a famous mathematician.

## General

The fourth year students will be provided with a desk in the John Dedman building and with computer accounts. The ANU e-mail accounts (not just private accounts) should be checked frequently for messages and seminar notices. Part-time tutoring positions are often available to the fourth year students: contact the first year coordinator if you are interested.