# Course Guide

## Introduction

In these few pages we hope to give the prospective student some idea of the ins and outs and ups and downs of studying Mathematics at Trinity College. This can be done as aModeratorship in Mathematics, a Moderatorship in Theoretical Physics or a Two Subject Moderatorship with mathematics as one subject. The three degrees, all four year honour degrees,are described in full, including required courses and career options.

Word of mouth is often said to be the best advertising. Sharing the experiences of those who have gone before you may be the surest way of gaining a realistic picture of what to expect. With that in mind we will quote liberally from both current and former students of Maths at Trinity College.

If you're already reading this you must have an interest in mathematics, but be aware that mathematics studied at university level is totally different to maths at school. New students are often shocked at the new breadth and depth of a subject with which they have long been familiar.

What happened to sums?''
I expected the odd number or two.''
I didn't know it could be so theoretical.''

The surprise is that mathematics at university level is not just about solving problems but about understanding them - not always an easy task.

It is very hard work'',
It is very much work intensive''.

Like any other form of higher study, mathematics requires a serious commitment and unbridled enthusiasm. Satisfaction is its own ultimate reward, but gauge for yourself by the student comments sprinkled throughout this booklet, the challenges, delights and pit falls of a university degree in mathematics. This booklet went to print in 1998 but a more frequently updated version is accessible on the World Wide Web at http://www.maths.tcd.ie/. This site also includes more detailed course descriptions than those in this booklet, information on the School of Mathematics regulations and the Mathematics, Theoretical Physics, and Two Subject Moderatorship entries from the official University calendar. Please note that the Calendar is the only official document and not this booklet. Finally, the Mathematics and Physics Departments of Trinity College hold a joint open day each November/ December. As well as attending talks and demonstrations, the prospective student is afforded the opportunity to query staff and students about any and all concerns. We are always delighted to speak to students. Please feel free to phone the Maths Department for any other information, or ring to make an appointment for a visit. The phone number is 896-1949 or 896-1889, 9.15 to 1.00pm and 2.00pm to 5.00pm Monday to Friday.

## Mathematics as a Broad Subject

In Trinity College we just have one Department of Pure and Applied Mathematics. Just as Pure Mathematics covers a multitude of areas, such as Algebra, Analysis, Geometry, Topology etc, so Applied Mathematics covers Theoretical Physics, Computing, Numerical Analysis etc. Courses in all of these areas, plus Statistics and even Economics are available to mathematics students, so when we use the word mathematics' we do so in a very broad sense.

Mathematics has evolved as an abstract subject because of its usefulness as a tool in science and industry. The range of its applications is increasing all the time. New applications demand new mathematics, which often leads to further applications, and so on.

Theoretical Physics has always been a fertile ground for this interaction. Recently physical considerations have led to speculative results in the geometry of four-dimensional manifolds. These have now been proven by pure' mathematicians.

It is also true that over time mathematics seems to go through phases of divergence and convergence. In the divergent phase many seemingly independent theories are developed, motivated by particular problems. However, despite the fact that the problems have been totally different it often emerges that there is a unity underlying the mathematical theories that have been developed. This is the convergent phase, when the seemingly disparate threads are woven into a pattern.

Specialisation, long seen as dooming mathematics with a multiplicity of separate disciplines, led to deeper understanding and a new unity in mathematics.'' Bers

The main thing to understand in all of this is that as mathematics pushes back the boundaries of knowledge the subject is constantly in a flux. Theorems don't change, but the relative importance of theorems and even areas do, and so the need is for a broad background.

## Types of Mathematics

What is mathematics? Someone doing their Leaving Certificate would probably have numbers, maybe functions in their answer. Here are some descriptions from famous mathematicians - in most cases their descriptions would appear to have been coloured by their specialist area of mathematics.

Mathematics in its widest significance is the development of all types of formal, necessary, deductive reasoning.'' Whitehead, Logician
It is not the essence of mathematics to be conversant with the ideas of number and quantity.'' Boole. Boole who lived in Cork is famous for his Algebra of Sets.
Most mathematicians would regard Set Theory as an essential part of the language of the subject, but
Later generations will regard set theory as a disease from which one has recovered.'' Poincaré

Nothing about numbers so far. Even when they enter, it is as

Mathematics is not the art of computation but of minimal computation.'' Anonymous

In Computer Science, complexity theory is exactly about minimal computation. But most mathematicians will be driven by a desire not just to solve a given problem but to learn from their solution how they can most easily solve any similar problems. This is part of the intertwining of Pure and Applied Mathematics which is also well expressed in:

The Science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain with the aid of experience.'' Kant
Most of the best mathematical inspiration comes from experience.'' Von Neumann

Here we have the reasoning and inspiration of Pure Mathematics being driven by the experience of Applied Mathematics. On the same sort of point

The paradox is now fully established that utmost abstractions are the true weapons with which to control our thoughts of concrete facts.'' Whitehead.

But it can be very difficult to see this in first year.

While learned mathematicians wax eloquently on the beauty of their chosen field, others have expressed the frustration, enjoyment and humour of a life dedicated to reasonable thought.

I have had the results for a long time, but I do not know yet know how to arrive at them'' Gauss
If I only had the theorems, then I could find the proofs easily enough.'' Riemann

But who gets the most enjoyment?

The most interesting moments are not where something is proved but where a new concept is involved.'' Kaplansky

On the usefulness of mathematics we might jokingly quote

God exists since mathematics is consistent and the devil exists since we can't prove this consistency.'' Weyl

Now here is a bit more information on what we mean when we talk about Pure Mathematics, Applied Mathematics, and so on. As we said earlier, mathematics in the broad sense develops as a continuous interplay between application and theory. The different sorts of mathematics arise as different stages of this interplay. At the risk of boring the reader we emphasise,

Mathematical Science is an indivisible whole, an organisation whose vitality depends on the connections between its parts. Advancement in mathematics is made by simplification of methods, the disappearance of old procedures which have lost their usefulness and the unification of fields until then foreign.'' Hilbert

Applied Mathematics is using mathematics to solve real world problems. In this field it is essential to be able to apply many different mathematical techniques, and be able to handle problems involving data where a knowledge of statistics becomes important. It is also necessary to be able to take a practical problem, from engineering for example, and turn it into a mathematical problem; this is referred to as mathematical modelling.

The number and rate of applications of mathematics is increasing and the equipment the students need to enable unforeseen applications, is not specialised mathematics but that core of the most general kind which will enable them to investigate new applications.'' Henkin

Pure Mathematics usually enters when the application has become abstracted. Now the basic concepts are the focus of attention. The subject is studied for its own interest, seeking out the inherent beauty, in the knowledge that the more deeply the subject is developed the better the applications will be later. In the pure mathematics courses a student will have an opportunity to see how centuries of widely varying applications have led to a number of abstract theories. These could include Differential Geometry, Algebraic Topology, Abstract Algebra, Functional Analysis and more.

Theoretical Physics is interested in physical systems. The objective is to understand nature at its most basic level and thus be in a position to predict the future behaviour of any system in which one is interested. To do this a physicist must have a sound grasp of basic physical laws and be able to unravel the implications of the laws using mathematics. Some of the basic tools for a theoretical physicist are: classical mechanics, electromagnetic theory, fluid mechanics, statistical mechanics, quantum mechanics and relativity.

Computing - at a theoretical level - tries to identify which tasks can be automated on a computer and which tasks can not. This involves looking for good solutions to, or theoretical difficulties in, computational problems; typically it involves establishing the correctness and efficiency of computer programs. Although the area deals sometimes with abstractions like formal logic, it requires diverse methods from discrete mathematics and difference equations for example, and therefore some of it is applied rather than pure mathematics. At a practical level, computer science deals with the design and implementation of modern computer software. The design and implementation of computer hardware is more a branch of electronic engineering than a mathematical topic and does not play any significant part in the curriculum of the computing courses on offer to mathematics students.

Numerical Analysis is concerned with numerical calculations of all kinds. The growth of numerical analysis as a branch of mathematics has been especially rapid in the last fifty years because of the development of electronic computers. It is concerned with the mathematical and experimental study of robust and efficient algorithms for obtaining approximate numerical solutions to problems in all areas of science and engineering. Since the early 1980's it has become a very exciting field to work in because of the availability of powerful computers at reasonable cost, and recent developments are based on a range of new parallel architectures. The efficient exploitation of these new computers will be a major concern of numerical analysis for the foreseeable future.

Statistics involves the use of modern techniques in mathematics and computing to analyse the data that arise in many areas of business and science. Recent trends involve a resurgence of interest in methods for informal exploratory analysis of data. This relies increasingly on friendly computing environments supported by efficient algorithms. The first year course discusses such procedures. The second year course lays down a more formal modelling probability framework. Later courses involve more advanced applications to time series, management science methods, and linear statistical models.

No matter which course option is chosen, care is taken to make sure that all students are computer literate and are exposed to important ideas from the different areas.

If you would like more information you could usefully consult the Encyclopedia Brittanica which has excellent essays on most mathematical subjects.

## Moderatorship in Mathematics

CAO Number TR031
Entry Quota: 30
Entry Requirement: B in Honours Maths Leaving Certificate
Points Cutoff in 1998: 485.
Sex: The male/female breakdown in first year in October 1997 for mathematics was 19 male/11 female. The complete first year class, Mathematics and Theoretical Physics and Two Subject Moderatorship was 43 male/19 female.

The aim of this course is to provide students with a firm foundation in all the basic areas of mathematics and then allow them to specialize in those areas that most suit their interests and their talents. As we tried to explain in the earlier section , Mathematics as a Broad Subject, what we are dealing with is a subject constantly in flux. So we attempt to provide a programme that is flexible and will produce flexible students with a broad range of skills who can quickly acquire new skills.

First year (called Junior Freshman)

There is no choice. Students take courses 111 (Algebra), 121 (Analysis), 131 (Mathematical Methods), 141 (Mechanics), 151 (Statistics), and 161 (Computing). There is also a course 061 to introduce students to the computing system in the department.

Course 111 is currently examined by continuous assessment, while all of the other courses are examined principally by an end of the year exam. There are repeat exams in September in all subjects.

Second year (called Senior Freshman)

Here there are three compulsory courses 211 (Algebra), 221 (Analysis) and 231 (Mathematical Methods). Students choose three more courses from among 212 (Topology), 241 (Theoretical Physics), 251 (Statistics), 261 (Numerical Analysis), and 262 (Computer Science).

All the courses are examined principally by an end of the year exam. There are repeat exams in September in all subjects.

Third and Fourth years (called Junior and Senior Sophister)

In each of the final years, students choose five courses from a wide range of group III courses in Pure Mathematics, Applied Mathematics, Theoretical Physics, Statistics, Numerical Analysis, Computing and recently we have added a course in Mathematical Economics taught by the Economics Department. In addition in the final year it may be possible for a student to undertake a project in place of one of the five courses.

Most courses are examined by end of the year exams, but some may have a continuous assessment portion. There are no September repeats.

The moderatorship degree is awarded on the basis of students' performance in all ten of the student's courses/project in Junior and Senior Sophister years, all ten counting equally.

### Student Reaction

Early in the academic year '97/'98, second, third and fourth year students were surveyed for the booklet. Here are some of the results and comments for Mathematics.

In response to the question Rate the course', where Perfect=10 and Terrible=0, the results were as follows:

 Rating: 1 2 3 4 5 6 7 8 9 10 Number: - - 3 - 5 7 14 15 7 -

The average rating was 7.

You must have a strong interest in mathematical theory, and not just in problem solving, or it will not be much fun''
It is very hard work''
It is akin to having a vocation for the priesthood, if you really enjoy maths you will love this''

I am not totally taken by the last analogy. The lecturers certainly don't see ourselves as High Priests, and notice the words fun', enjoy' and love' above, words not always associated with the priesthood. These occurred a lot as did hard work'. These go together. You will only work hard and well at anything if you are enjoying yourself. Interest and enjoyment should be key words in deciding on what subjects and/or career to choose.

The first two years are very theoretical - somebody who wants to do a lot of applied work would probably be disappointed by this''
If you cannot accept and grasp new ideas quickly, do a different course''
If you want to become a mathematician don't just learn every theorem thrown at you, appreciate it and understand what it means and what it can be used for. Else do some business course as you won't get anything from maths.''
Talk to someone who has done or is doing the course.''

You can talk to students during the mathematics Open Day or contact the Mathematics Department and we will put you in touch with someone.

What the students liked were:

The course is more varied, interesting and challenging than I expected.''
I like the way it feels as if we are the best or the most rigorous course available. The pure maths is brilliant.''
It is a difficult course so doing well is very satisfying.''
The computing facilities are the best in College.''
The way the various strands of the course entwine at various times is a particularly appealing aspect and shows the perceived disunity of maths to be incorrect.''
It is a very open and friendly department.''

See section on careers for more benefits.

Things that people disliked about the course were:
Examples are very thin on the ground in some courses.''

Each year two or three students will transfer from the mathematics course - usually because they find the course, as taught, too abstract. Computer Science is the most common destination. The more applied maths courses only proliferate in third/fourth year. First and second year attempt to give a broad theoretical background.

141 is very hard.''

This is the Mechanics course. Many of the students suggest that applied maths in the Leaving Certificate is almost a prerequisite.

The early morning lectures.''

Between a lack of lecture theatres, due to overcrowding, and timetable clashes, lectures can commence any time between 9.00am and 5.00pm, Monday to Friday. So don't count on arriving late on Monday or getting a bus home early on Friday.

## Moderatorship in Theoretical Physics

CAO Code: TR035
Entry Quota: 20
Entry Requirements: B in Maths & B in Physics
Points Cutoff in 1998: 535 in first round. 530* last offers.
Sex: The male/female breakdown in first year in October 1997 for Theoretical Physics was 18 male/2 female. The complete first year class, Mathematics, Theoretical Physics and Two Subject Moderatorship was 43 male/19 female.

Theoretical physics is the study of nature at its most fundamental level. It strives to uncover the unifying principles from which all observed phenomena can be described. Remarkably, the language of nature happens to be that of mathematics. Increasingly at the very frontier of physics extremely sophisticated mathematical ideas are being used. A theoretical physicist utilizes mathematics as a logical means of getting from basic principles to specific predictions which are susceptible to experimental verification. Therefore, a good theoretical physicist must not only be at ease with modern mathematics, but must also be fully aware of the physical significance of his/her calculations. This requires that the early formative years also expose him/her to the experimental side of physics and so the course combines large components from the Mathematics Moderatorship course and from the physics course of the Science Moderatorship programme.

In the first year, students take mathematics 111, 121, 131, 141, 161, 061 and the full first year physics course, including laboratory work, from Science. This means a course load roughly 7/6 of that of someone in pure mathematics or in natural science. So it is demanding. But some students also take 151!

Course 111 is currently examined by continuous assessment, while all of the other courses are examined principally by an end of the year exam. There are repeat exams in September in all subjects.

In the second year students take mathematics 211, 221, 231, 241 and the full second year physics course, including laboratory work, from Science. This corresponds to a normal load.

All the courses are examined principally by an end of the year exam. There are repeat exams in September in all subjects.

In the third and fourth years, students take theoretical physics courses given by the mathematics department, and most of the physics courses, (including introductions to topics such as high temperature superconductivity, the quark model, chaos and astrophysics). In the third year they are only required to take half the laboratory classes, and in the fourth year they have the option of doing some laboratory work or submitting a small computational project.

The degree mark is determined by the work of the final two years, each of the mathematics years counting for 25%, third year Physics 10%, and fourth year Physics 40%.

Some students find that their interest or proficiency is greater in Mathematics or in Physics, and want to change their course. For these it is possible to transfer from Theoretical Physics to Mathematics at the end of the first or second year. It is also possible at the end of first or second year to transfer from Theoretical Physics to Experimental Physics.

Theoretical physics is one of the most demanding intellectual disciplines. Those who successfully complete the four-year honors degree have a wide variety of career options open to them. Normally, those gifted in the subject continue as postgraduate degree students. Because the training is fundamental, one may also pursue advanced studies in engineering, quantum chemistry, economics, pure and applied mathematics, meteorology,geology, genetics and operations research, to mention only a few of the possibilities.

Another not so obvious option is to study law. As a result of the rising number of legal questions requiring sound scientific analysis, lawyers with a background in science are in demand. Theoretical physics with its rigorous mathematics allied to its science is a particularly suitable background.

See the Careers section for more details on job prospects.

### Student Reaction

Early in the academic year '97 '98, second, third and fourth year students were surveyed for the booklet. Here are some of the results and comments for Theoretical Physics.

In response to the question Rate the course,' where Perfect=10 and Terrible=0, the results were as follows:

 Rating: 1 2 3 4 5 6 7 8 9 10 Number: - - - - - 5 9 16 8 1

The average rating was 7.8.

Asked to give advice to prospective students the two words that occurred most were love' and hard' as in:

You must really love maths and physics and be willing to work hard.''
You must get an adrenaline rush when maths is mentioned - you must love it for its own sake.''
The intensity of the course and the range of subjects covered will soon weed out those who are not exceptionally keen on all aspects of maths and physics.''

The next most common advice was to expect change from second level maths.

Be prepared to adjust your perceptions of maths, it is totally different to what you have done.''
You must have a mathematical intuition rather than just being able to apply techniques.''
Be prepared for fast moving courses, with plenty of new methods and concepts to tackle.''

This last is where the hard work comes in. There is much less time for practising technique. Understanding theory is what is stressed. See good points and bad points below.

You should have an inquisitive but disciplined mind.''

It is a very good idea to try and understand what sort of mind you have. Try thinking of what sort of learning do you most enjoy/love. Interestingly, mathematics students say you should enjoy maths. The Theoretical Physics students say you should love maths.

You should consider all three options, studying Maths, Physics, or Theoretical Physics.''
Be prepared for studying more maths than physics. Maths is the harder part.''
Consider just doing maths and specialising in the Theoretical Physics courses if you don't like labs.''

Many first year students find the physics component boring and the laboratory work tedious.

See the part about transferring courses.

How did they find the course: Rewarding', Interesting', Challenging'.

What were the good points:

The broad range of courses.''
The computing facilities and the access to them.''
The insights afforded into the fundamentals of maths and physics.''
The rigour and exactness that permeate every course.''
The completeness.''
The elegance of the absolute.''
The broad basis in mathematics is great - gives the opportunity to pursue a myriad of activities after the course - most doors lie open.''
I always felt like I was doing the most intellectually demanding course in College.''
T.P.'s get respect.''

Not enough choice, not enough computing in the course.''
Applications of what is learned are not always pointed out.''
Early on the maths seemed very abstract and difficult.''

There is not much time for nights out.''

## Two Subject Moderatorship

CAO Code: TR001
Entry Requirements: B in Honours Mathematics and any requirement for the second subject
Points Cutoff in 1998: 540.
Sex: The male/female breakdown in first year in October 1997 for Two Subject Moderatorship was 6 male/6 female. The complete class, Mathematics, Theoretical Physics and Two Subject Moderatorship was 43 male/19 female.

As the name implies, you study mathematics and another subject. Mathematics may be combined with Economics, Geography, Philosophy, Psychology, English Literature, French, German, Latin or Music.

When combined with Psychology, both subjects are studied equally for four years. When combined with Economics, Philosophy or Geography, the student can choose at the end of the second year whether to study both subjects equally for four years or to specialize in Geography or Philosophy or Economics for the fourth year.

When combined with English Literature, French, German, Latin, or Music, then Mathematics is studied for three years only and the fourth year is devoted entirely to the other subject.

Consistently the most popular choice is Mathematics and Economics. Due to the interconnections between the subjects, we have just changed the course so that it now has a study programme different from the other combinations. Here is a rough outline.

Mathematics + Economics

First year: Maths 111, 121, 131, 151, Economics and either 161 or a social sciences subject

Second year: Maths 211, 221, 251 + 3 Economics subjects

Third and Fourth year: courses are chosen from a wide choice of Mathematics and Economics courses.

The idea behind this scheme of things is that students take more mathematics courses in their first and second year than a normal TSM student, but may if they wish take more economics than mathematics in their final years.

Mathematics + any other course

First year: Maths 111, 131, 151

Second year: Maths 121, 141, 161

Third year: Maths 211, 221 and a choice of one course from six.

Fourth year: (If applicable) In the fourth year they have a wide choice (three courses total in most cases).

### Student Reaction

Early in the academic year '97/'98, second, third and fourth year students were surveyed for the booklet. Here are some of the results and comments for Two Subject Moderatorship.

In response to the question Rate the course', where Perfect=10 and Terrible=0, the results were as follows:

 Rating: 1 2 3 4 5 6 7 8 9 10 Number: - - 1 2 - - 1 2 - -

The average rating was 5.66. (The sample was unreliably small.)

The TSM courses other than mathematics with economics were generally found to be less satisfactory in structure than the maths and theoretical physics options. Students in those courses take first year maths courses in first and in second year, so they are with a different class in second year. Quite a lot of students dislike this, and we have managed to avoid this disadvantage in the mathematics/economics combination. At present we are working on a revised programme for all other T.S.M. students. Hopefully it will be implemented before long.

Also the two subjects being studied don't usually have programmes that are planned to complement one another, which means that a student has to be able to switch mentally from one subject to an entirely different one. Some people relish this sort of change but it poses problems for others. If one is immersed in a single subject be it a language or mathematics or music or whatever, one crosses a certain threshold after which learning becomes easier.

For all this there have been many outstanding students in TSM.

Think carefully, as options go T.S.M. maths is very tough. Only do it if you are interested in the subject, not just to get a piece of paper.''
Talk to somebody who has done your combination of subjects.''

This last is very good advice. Phone the Maths Department and we will try to put you in touch with somebody.

For the good points, they said:

The challenge, the completeness.''
It is very satisfying when you actually do some work and start to understand things.''
The maths complements the Economics component.''

The bad points concerned doing a subject other than economics:

As some of the first year maths courses overlap - only doing some of them you have to work harder.''
Our second year is spent in first year with a new class.''

As said earlier, we hope to dilute these bad points.

## Overseas Option - SOCRATES

In 1995 the Socrates Programme replaced the ERASMUS Programme. It is an EU supported scheme whose aim is to promote European co-operation and to improve the quality of education through partnerships across national boundaries.

Single Honour Mathematics students may opt to spend their Junior Sophister year at one of the following institutions:

 A.U.E.B., Athens; Technische Hochschule, Darmstadt; Katholieke Universiteit, Leuven; University of Durham, Durham.

The following should be noted:

• Theoretical Physics and T.S.M. students are unfortunately unable to participate.
• Resources and places are limited and must be applied for. Applications should be made to the office of International Student Affairs before April 1st of the Academic Year prior to that of the overseas option.
• The typical student grant to help defray a part of the extra expense is 600 to 800 pounds.
• No fees are payable to the host university.
• Support grants, eg. Local Authority grants, continue as if the student was in Dublin and Dublin students who qualify become eligible for a living away from home grant.
• There is unfortunately no money currently for linguistic preparation.
• This year will be accepted by Trinity College as the Junior Sophister component of their degree.

We have had more students coming to the Mathematics Department than leaving but the reaction of those that have gone is summed up by one of them,

It was a wonderful year, culturally, socially, and academically.''

## Overseas Students

As mentioned earlier the Mathematics Department has a strong commitment to the SOCRATES programme. In addition we welcome one year students from outside the programme and from non EC countries. It is a policy of the University to encourage a certain proportion of foreign students and the diversity created is seen as enriching the life of the College and contributing to the experience of all students. Those not very familiar with the University might find a few basic facts helpful. Trinity College was founded in 1592 and is situated on a 40 acre site in the heart of Dublin. There are over 8100 undergraduates and 2100 postgraduates. About 6% of the students are from outside Ireland. The academic year is divided into three terms October to December (9 weeks), January to March (9 weeks) and April to May (6 weeks) with a three or four week study period between the teaching terms. Examinations start one week after the end of the third term and continue for up to four weeks. Although small by present day international standards, with a population of just one million, Dublin has nevertheless the resources of a capital city with a full and varied cultured and intellectual life. Situated on an attractive bay, it is also close to the scenic mountains of Wicklow - a county known as the Garden of Ireland. Application forms for one-year Undergraduate Admissions may be obtained from the office of International Student Affairs and should be returned by March 1st in the year of entry. It is also possible for people from outside Ireland to apply to do a full degree course and for this they should apply to the Central Applications Office, Tower House, Eglinton Street, Galway, Ireland.

## Career Opportunities

Most people who study mathematics initially have very little idea of the sort of career they wish to pursue. A survey of third and fourth year mathematics students at TCD in 1992 found that twelve had started with a definite career plan while thirty-two had not and many of the twelve changed their plan while at College. Recently, a student about to start first year who wanted to become an actuary, was seriously concerned as to whether to accept an offer of direct entry to an insurance company from his Leaving Certificate or to do a degree first. We advised the degree and to keep his options open. He has since graduated and gone on to a well paid position in the Financial Analysis department of a London bank. Should he decide this is not for him, he still has the option of studying to be an actuary but with a mathematical maturity that his earlier starting colleagues will never attain. He is also considering obtaining a postgraduate qualification in applied statistics. The best advice is:

Unless and until you are very sure, keep your career options open.

Careers chosen by mathematicians fall into three categories.

Those directly related to mathematics. These include all forms of teaching and academic work, and a variety of positions in industry or the Civil Service which involve using the mathematical, statistical, and computing knowledge they have acquired. Increasingly this includes positions in information technology and in finance where deep mathematical results are used and even derived.

It is widely accepted that for such careers the path should be mathematics in depth first and then finance, information technology, or other applications.

Those that involve thinking logically and quantitatively. Typically these include actuarial, accountancy, banking, etc.

Careers that are open to graduates of many areas.

What gives mathematics graduates a head start in competition for posts in categories two and three is that employers are aware that regardless of the particular courses that have been taken, all such graduates have highly developed problem-solving skills. It is recognised that success in maths requires the ability to master complex and difficult problems; a characteristic that gives mathematicians an edge in acquiring other skills quickly and efficiently.

Here are some quotes from a U.S. survey of mathematics graduates about their degrees and their career choices.

I think the keys to success in any mathematics job are a love for problem solving, an inquisitive mind and willingness to learn new skills.'' A mathematician with the National Science Agency.
Experience in mathematics allows for a way of thinking in a structured format. Corporations realise that many types of problems can be solved using the analytical thought processes that mathematics requires. A degree in mathematics provides the analytical skills and methods of decision making that are necessary in the working place today.'' Financial Analyst.
I could have pursued the same career with a degree in Computer Science. Since Computer Science is constantly and rapidly changing, you always need to learn new things. These you can learn on the job. Studying mathematics gives you the tools to analyse problems and think logically, which helps in whatever profession you choose. People have great respect for a degree in mathematics.'' Self employed Computer Marketing Consultant.
The real astonishment about my career is how deeply the study of mathematics has informed its every turn. Studying poetry or philosophy I find myself thinking in a mathematical way, not arithmetical, but full of love of analogy, mapping, abstraction and clarity.'' Cultural Critic, New York Times.
I majored in mathematics because I liked the subject but I have used my training in more ways than I ever imagined possible.'' Marketing Manager, I.B.M.
In the end you must do what really turns you on and not necessarily what is the most lucrative.'' Applied Mathematician, Rockwell Science Center.

Many referred to their mathematics degree as a gateway or jumping off point to a wide variety of careers. Disciplined study leading to unlimited opportunities is what a mathematics degree at Trinity College is all about. We encourage students to diversify their course choices. Opening doors in your mind will enable you to open many other doors in your life.

Next is a list of where all of our graduates went in the last four years.

### Mathematics

Further study (10)
Cambridge Maths dept., Manchester Maths dept., Trinity Maths dept. (2), U.C.D. Business dept., D.C.U. Accounting dept., Oxford Engineering dept., T.C.D. Computing Science dept. (2), T.C.D. Statistics dept.

In Employment (6)
Computer related (4) [3 in Dublin, 1 in UK], Currency trading (Dublin), School teaching (Dublin)

Travelling in Australia

Unemployed (1)

Further study (9)
Berkeley Statistics, Trinity H.Dip (3), Trinity Economics dept., D.I.T. Legal Studies, Trinity Maths dept., D.C.U. Maths dept., Portugal.

In Employment (15)
Actuary (5) [Dublin (3), U.K. (2)], Computing/I.T. (5), U.S.A. (1), Banking/Accounting (3), Teaching, Fishing,

Travelling in Australia (2)

Unemployed/unknown (3)

Further study (10)
Trinity Maths dept. (2), Trinity H.Dip. (2), Cambridge Maths dept. (2), Trinity Statistics dept., U.S.A. Statistics, U.C.D. Maths dept., Massachusetts Maths dept.

In Employment (10)
Computing/I.T. (4), Finance/Accounting (4), Bookshop Manageress, Medical Statistics Officer, Construction Australia

Unknown (2)

Unemployed (1)

Further study (13)
Trinity Maths dept. (2), Columbia Maths dept. (2), D.C.U. Maths dept. (2), Trinity H.Dip (2), Darmstadt Maths dept., Trinity Computer Science dept., Trinity Environmental Science dept., University unknown (2)

In Employment (5)
Computing/I.T. (3), Banking, Hotel.

Unknown (2)

### Two Subject Moderatorship

Further study (8)
U.C.D. Maths dept., U.C.C. Maths dept., Abroad Maths dept., U.C.D. Business dept., Information Technology (2), Unknown (2).

In Employment (7)
Software Engineer, Banking, Business Analyst, Actuary, Trainee Manager, Clerical, Research Assistant.

Unknown (2).

Further study (6)
Trinity H.Dip (2), Maynooth H.Dip., Vienna Architecture, Abroad Actuarial, Unknown.

In Employment (5)
Tax Consultant, Science Communicator -- Edinburgh, Administrator -- Germany, Accountant -- U.K., Maintenance Officer.

Unknown (4).

Further study (7)
Trinity Economics dept., Trinity English dept., U.C.D. Economics dept., Maynooth Social Policy dept., Abroad Economics dept., Unknown (2).

In Employment (1)
Teaching.

Unknown (1).

Further study (1)
U.S.A. Economics dept.

In Employment (4)
Banking (2), Teaching, Entertainment Agent.

Unknown (1).

### Theoretical Physics

Further study (4)
Trinity Maths dept., Cambridge Maths dept., Trinity Physics dept., Trinity Genetics dept.

Further study (5)
Trinity Maths dept. (2), Darmstadt Physics dept., Cambridge Maths dept., Oxford Maths dept.

In Employment (1)
Computing

Further study (2)
Trinity Maths dept., DCU Maths dept.

Further study (5)
Trinity Maths dept. (2), Harvard Physics dept., Trinity Physics dept., Trinity Engineering dept.

Many modern careers involve further study beyond a Bachelor's degree, sometimes done while in employment, but frequently involving further study at a university or research institute. Academic employment in any form of third level college, almost always requires a Ph.D. By checking the above data you will notice that the proportion at study is very high, but far more interesting are the areas of study and universities chosen.

It should also be noted that the very high rate of Theoretical Physics students going for further study is not a reflection of their career prospects but their uniformly intense interest in their subject. It is also noteworthy that the low numbers graduating do not reflect a high dropout/failure rate as the quota was only recently raised from ten to twenty.

Graduates with any of the three mathematics degrees who show promise for teaching are usually welcomed into the H.Dip courses, and with a genuine shortage of well qualified mathematics teachers, they are far more fortunate than graduates of other subjects in landing permanent jobs.

The 1988 U.S. publication, Jobs Rated Almanac', rates 250 jobs according to the following criteria: income, outlook, physical demands, security, stress, and work environment. Using these six components of job quality the top five in descending order were actuary, computer programmer, computer systems analyst, mathematician, statistician; all of these careers are open to our graduates.

Trinity College has a Careers Advisory Service which students are encouraged to contact in their final years. Much of the information contained here was furnished by them, and the Maths Department is very grateful for their support to the Department and to our students.

The combined Careers Services offices for all Irish Universities produce an annual survey of careers which includes a section on mathematics. Your School Careers Guidance Councillor should have a copy. Your degree in Mathematical Sciences, what next?' published by the British Association of Graduate Careers Advisory Services, Crawford House, Precinct Centre, Manchester M13 9EP, is another helpful booklet. We gratefully acknowledge these sources in compiling this entry.

## Scholarships, Prizes and Financial Assistance

### Foundation Scholarship

The most prestigious and valuable awards at Trinity College are the Foundation Scholarships. These Scholarships are awarded annually on the basis of an examination held at the end of the second term. In mathematics the exams cover the course material up to the end of the second term of the second year. Anyone attaining a very high standard is proposed for Scholarship. Scholars are entitled to free tuition, plus free accommodation and free evening meal in term, for up to five years. The number of scholars elected in any year is not fixed. In past years, the number elected in mathematics, all three degree courses, has varied from one to eight. In addition the first unsuccessful candidate in the mathematics scholarship exam is awarded the Roberts Prize of £500, provided sufficient merit is shown.

### Continuing Students

The department is fortunate to have been the recipient of various endowments and bequests to provide prizes for students. Most recently a fund was established in honour of, and named after John L Synge the world renowned relativist and geometer, who has a long association with this College.

At the time of writing the total amount available for prizes exceeds £9,000 per annum. These prizes include three Lloyd Exhibitions awarded to students on the basis of marks obtained in the Junior Sophister honor examinations in mathematics: Value £1,000, £750 and £500. The Rowe Prizes are awarded to the candidates who obtain the highest marks at the Senior Freshman examination: Value £500 and £500. The Townsend Memorial Prizes are awarded annually in three parts on the result of the Junior Freshman Honor examination in mathematics: Value, first part £600, second part £500, third part £400. The eight Arthur Lyster Prizes are awarded on the basis of examination performances in mathematics in any of the Junior Freshman, Senior Freshman or Junior Sophister years. (A candidate who has been awarded a Townsend or Rowe prize or a Lloyd exhibition will not be eligible to receive a Lyster prize in the same year): Value four at £300, two at £200, two at £100. The Minchin Prize is awarded annually in two parts in Michaelmas term to students who performed with particular merit in the work of the previous Junior Sophister year in mathematics and / or physics: Value of each part of the prize £900.

The John L Synge Prize value £200 is awarded biennially to a candidate or candidates who distinguish themselves at the examination for moderatorship in a subject in mathematics or theoretical physics related to Professor Synge's interests.

The Bishop Law Prize is awarded each year to the first moderator in mathematics: Value £50.

### Entering students

A number of Entrance Exhibitions are awarded to beginning students. These are based on Leaving Certificate/A level marks and are book prizes worth £200. But for people of limited means, the award can also include the free evening meal.

There are other awards for beginning students also, and for these one should see the University Calendar. Some carry restrictions such as the Reid Exhibition, which is restricted to natives of County Kerry.

### Financial Assistance

The College operates a student aid scheme, whereby students who have successfully completed their Junior Freshman year, and who are in difficult financial circumstances, may obtain a small grant.

## The Department and its Facilities

Trinity College is justly proud of its long tradition of excellence in mathematics. Most famous is William Rowan Hamilton, renowned for his work on dynamics, his prediction of conical refraction and for his invention of quaternions. It was Hamilton's ideas that led Schrödinger to the discovery of wave mechanics in 1926, ideas which continue to pervade our view of physics. There are many other distinguished mathematicians who have, through the centuries, made important contributions to the advancement of the subject, E.T. Whittaker and J.L. Synge, two more recent examples.

Research interest in the Mathematics Department is enormously varied; ranging from the abstract ideas of differential geometry and analysis to practical ideas of numerical analysis; modelling and computer algorithms; the nature of fundamental particles and general relativity; non-linear systems and fluid mechanics. This departmental diversity is reflected in the specialist sophister courses available to students.

At present there are fourteen full-time permanent academic staff members, about two hundred undergraduates and approximately thirty postgraduate students in the department. In addition, there are a number of research, administrative and part-time teaching staff. Besides the four year degree course leading to the B.A.(Mod) degree, the department also offers M.Sc. and Ph.D. degrees. Details regarding post-graduate degrees are contained in a separate booklet.

### Computing Facilities

The department has its own computing network which is linked to that of the College and hence to international networks. The departmental facilities are all based on the UNIX operating system with approximately a dozen machines acting as servers for undergraduate and postgraduate students and staff. In total there are around 60 graphics X-window work places on the network, with around 30 of these for undergraduate student use including general facilities such as email and World Wide Web browsers, various compilers and more specialised computational and graphical software for mathematical purposes. There are also research machines and newly initiated innovative research projects which use the joint TCD/Queen's University of Belfast parallel computer. The College Informations Systems Service provides PC and Macintosh facilities in many College locations.

### The Library

The department boasts the finest mathematics research library in Ireland, with over sixteen thousand books and a current subscription to over one hundred journals. Students also have access to the College library which is a copyright library with over three million volumes.

### The Dublin University Mathematical Society

The Maths Society is, as its name implies, a student society with a room in the department at their disposal. Perhaps the best description of the Society is one given by the members themselves...

The Dublin University Mathematical Society (D.U.M.S.) room is populated with eager students intent only on the study of maths and the pursuit of a clean healthy lifestyle. No, honest. Really.
No free coffee and tea all year round, those are pernicious lies spread by our many enemies who would honestly have you believe that we are willing to talk to anyone because all our friends think we're too strange. Nope.
Interesting talks are not given every 2 weeks on topics of interest to the average science student. That's the Physical Society.
We don't have two computer terminals, comfortable chairs, a library of over 1500 mathematical, science and computer books. Nosiree bob. That's someone else too.
We don't have the only society rooms within staggering distance of the Hamilton Building. I'm not sitting here now.
We do admit that we never hesitate to go for the cheap laugh when a serious summary is called for, though, and that our society has been active since 1923, when Ireland's only Science Nobel Laureate helped to found it.''

They may be too modest to admit it, but the membership is usually about 300, including many non-mathematics students, so they must be doing something right.

## Course Descriptions

The following is an informal guide to the courses which form part of the moderatorships in mathematics. Necessary prerequisite courses are shown in parentheses. More detailed descriptions may be had on application to the Department of Mathematics, or on the World Wide Web at http://www.maths.tcd.ie/.

### First Year

111 Algebra
Algebra has its foundations in the study of solutions of equations. It has developed, (and still is developing) in many different directions. In this course the elementary properties of Groups, Rings, Fields and Vector Spaces are studied and applications to combinatorics, geometric constructions and coding theory revealed.
121 Introduction to Analysis
This course sets out to lay the theoretical foundation for calculus, with the emphasis on one-variable calculus, covering number systems, limits, convergence of series, differentiation and integration.
131 Mathematical Methods
This course introduces and develops the basic mathematical and computational techniques as a preparation for applications. Of all the Junior Freshman courses, this one is the one which follows on most directly from school mathematics.
141 Mechanics
This course presents the principles of Newtonian mechanics. In particular, students learn how to translate physical problems into mathematical ones via Newton's laws and to solve the equations of motion in a variety of cases.
151 Statistics
Statistical methods are of fundamental importance in the collection and analysis of data in, for example, medical research, industrial process optimisation, market research, forecasting, genetics, environmental science, and a host of other areas. Probability theory is the mathematical basis on which most of these methods are built. Statistics and probability are of interest in themselves as mathematical fields of study, as well as being tools for other sciences. This is the first in a series of courses in these areas which will be available to students over the four years of the degree programme.
161 Computer Science
Course 161 provides an introduction to computer programming for Mathematicians and Physicists. It begins with a discussion of computer architecture, memory organization and data formats. The C programming language is used as the model compiler, and is described in some detail. Simple numerical techniques to find roots, maxima and minima of functions, to solve differential equations, and to calculate integrals are described. Finally, Fortran and Pascal compilers are discussed.

### Second Year

211 Linear Algebra and Differential Forms
In this course the properties of linear operators on finite dimensional vector spaces are studied in some detail, up to the Jordan form. This is followed by a study of bilinear forms and more general tensors. Differential forms give an application of linear algebra to calculus of several variables, and are important in many areas of geometry, analysis, and theoretical physics. (111, 131)
212 Topology
Topology is a subject with at least two faces. On the one hand it provides a language and a basic background which is essential for modern treatments of many branches of mathematics; on the other it deals with the use of algebraic tools for isolating the essential shape of an object (such as a surface, a curve or a knot). (121)
221 Real and Complex Analysis
This continues the subject matter of course 121 and deals with the calculus of several real variables and of a single complex variable. Topics included are: the derivative as a linear operator and the inverse function theorem, the Lebesgue integral, and Cauchy's integral theorem for complex functions with applications. The course is of fundamental importance for an understanding of the theoretical basis for all applications of the calculus. (121, 131)
231 Mathematical methods
This course is a continuation of 131 and deals with more advanced methods such as calculus of variations and Fourier series. (131, 141)
241 Mechanics
This course starts with Lagrangian and Hamiltonian dynamics, building on course 141, and then provides a brief introduction to modern (20th century) theories of quantum mechanics and (special) relativity. (141)
251 Probability and Mathematical Statistics
This is a sequel to course 151, but it pays more attention to the underlying probabilistic concepts than is done in the first year course. In the last term, some of the more subtle concepts of practical statistics are considered. (151)
261 Numerical Analysis
This is a first course and not directly a continuation of any first year course, although it requires a background in several different topics. Numerical analysis is the branch of mathematics on which computers had the earliest effect. It is concerned with the mathematical analysis and development of good numerical methods for finding answers to practical problems which arise in many different areas of science, engineering and finance.
262 Computer Science
This is a continuation of course 161 with the emphasis on the software side of computing. (161)

### Third and Fourth Year

The sophister courses are more specialised and students have the opportunity to choose courses in areas they find most interesting. Some courses are available every year, but many are available only in alternate years. A sophister course may either be a `core' course central to advanced work in several areas or it may reach the frontiers of current research in some field.

Although a certain amount of choice is allowed in the senior freshman year, sophister students in Mathematics have been prepared for a wide range of the sophister courses by the basic common programme of the Freshman years. While many students will have developed an interest and aptitude in some area of mathematics, and pursue courses mainly in that area, in all cases some breadth is required, and students consult with a course adviser to plan a coherent programme.

The following is an informal guide to the advanced courses. Almost all may be taken in either third of fourth year.

Pure Mathematics courses:

 311 Abstract Algebra 321 Modern Analysis 412 Applied Probability 414 Complex Analysis 421 Algebraic Topology 424 Group Representations 425 Differential Geometry 428 Prime Numbers

Applied Mathematics/Numerical Analysis/Statistics courses:

 381 Mathematical Economics 413 Partial Differential Equations 431 Fluid Mechanics 451 Applied Linear Statistical Models 452 Stochastic Processes in Space and Time 453 Multivariate Analysis 454 Statistical Inference 461 Numerical Solution of Differential Equations I 462 Numerical Solution of Differential Equations II

Computing courses:

 361 Algorithms and Geometry 362 Advanced Programming (372) 363 Algorithms and Complexity 364 Computer Aided Design 371 Computation Theory and Logic 372 Microprocessor Architecture 373 Finite Fields and Coding Theory

Theoretical Physics courses:

 342 Numerical Simulation of Physical Systems (one term) 343 Topics in Theoretical Physics (one term) 432 Electromagnetic Theory 433 Statistical Mechanics 441 Quantum Mechanics 442 General Relativity and Cosmology 443 Statistical Physics 444 Astrophysics

Economics:

 381 Mathematical Economics

Project:

 499 Project

Trinity College, Dublin 2, Ireland.
 School of Mathematics,
 Phone: (01) 896 1949, or 896 1889 FAX: (01) 896 2282 Email: mathdep@maths.tcd.ie WWW: http://www.maths.tcd.ie/
Admissions Office, West Theatre, Trinity College,
Phone: (01) 896 1039 or 896 1532
Information Office, West Theatre, Trinity College,
Phone: (01) 896 1724 or 896 1897
Careers and Appointments Office, East Chapel, Trinity College, Dublin 2, Ireland.
Phone: 896 1705
Dean of Overseas and Visiting Students, Arts Building, Trinity College, Dublin 2, Ireland.
Central Applications Office (C.A.O.) Tower House, Eglinton Street, Galway, Ireland.

Closing Date for CAO applications: 15th December, for non Irish Applicants, and 1st February, for Irish applicants, for entry the following October.