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{\bf First Degree Mathematics
at Cambridge\\
For Overseas (Including EU) Students\\
An Extremely Unofficial {\tiny and only slightly helpful} View}
\end{LARGE}
\end{center}
\vspace{2\baselineskip}
\noindent
\begin{footnotesize}
\noindent
{\bf Small print}
This essay is a PERSONAL view and does not replace
the official advice which may be obtained from
the University and the Colleges
(see Section~\ref{Preface} below). Although I
have tried to be helpful the advice may be
INCORRECT, or OUT OF DATE
or MISLEADING or may not apply in your particular case.
I and my advice have NO OFFICIAL STANDING.
The advice, incorrect or otherwise, applies
ONLY TO MATHEMATICS.
I should be glad to be told of errors or of
any way in which these notes could be made more useful.
The guide is written in \LaTeX2e\  and is,
I hope, available by anonymous FTP as
\verb+/pub/twk/Over.tex+ at \verb+131.111.24.1+.
It can also be obtained by gopher or on the
Web, see the instructions at the
end of the guide.
My e-mail address is \verb+twk@dpmms.cam.ac.uk+.
Please write to the address at the end if you
cannot get this file electronically.
However if you require further information
help or advice
you must write to a college or the university
(see Section~\ref{Preface} below) and NOT to me.
\hfill T.W.K\"{o}rner (Trinity Hall).
\end{footnotesize}
\tableofcontents
\section{Preface}\label{Preface}
This note is not meant to supply
information but to make you think about what
the information means. There are three major sources
of information that I recommend.  All are on the web
but may also be obtained by post.

The quickest way of obtaining detailed information
on the content and structure of the Cambridge
mathematics course is via the web starting at
\begin{center}
\verb+http://www.maths.cam.ac.uk/undergrad/admissionsinfo/+
\end{center}
and following the links. If you can not
get the web write to
\begin{center}
Undergraduate Admissions,\\
The Faculty of Mathematics,\\
Silver Street,\\
Cambridge CB3 9EW,\\
United Kingdom.
\end{center}
(note that, in spite of the address, this merely
gives advice and does not handle admissions)
explaining that you can not get the web,
giving your address and asking for documents
about admission and course content.

Details of the colleges and courses are
found in the
`University of Cambridge Undergraduate Prospectus'
accessible  via the web on
\begin{center}
\verb+http://www.cam.ac.uk/CambUniv/ProspStudents.htlm+
\end{center}
Of course, this is mainly a recruitment brochure
packed with happy smiling faces and happy smiling
statements but it contains a substantial amount
of useful information and will help you choose
a college. In the opinion of the present writer
your main task is to decide whether you want
to do mathematics at Cambridge and the choice of college
is a secondary matter.

The pamphlet `Information for Overseas Students'
is accessible via the web on
\begin{center}
\verb+http://www.cam.ac.uk/CambUniv/Overseas/#heading1+
\end{center}
In typical Cambridge style it starts by telling
you that the University is known to have
been in existence since 1226\footnote{Equally
typically it neglects to tell you that
Cambridge was one of the smaller and less
prestigious mediaeval universities.}. However,
it is packed full of useful information
and should be read carefully.

The pamphlets 
`University of Cambridge Undergraduate Prospectus'
and 
`Information for Overseas Students'
can be obtained by post
from many different addresses but
the simplest way 
is to write to any college
asking for them.
The simple address
\begin{center}
Admissions Tutor\\
X college\\
Cambridge\\
England
\end{center}
will always work. If you feel unhappy about
choosing a college at random
you can always write to my college.
\begin{center}
Admissions Tutor\\
Trinity Hall\\
Cambridge CB2 1TJ\\
England
\end{center}
You do not commit yourself in any way by asking
for information.

In these notes `overseas student' is used
in the non-technical sense to mean a student
from outside the UK\footnote{For certain purposes to
do with fees `overseas student' means something
entirely different (see Appendix~1 of the
`Graduate Prospectus'). This particular lunacy
is forced on Cambridge by the British Government.}. 

\section{The place of the college} Those who are used
to the Paris Metro find the London Underground baffling.
Those who are used to the London Underground
are baffled by the New York Subway System
and the confident habitu\'{e}s of the NY system
are baffled in turn by Parisian public transport.
If you choose to study in a foreign country you
choose to study in a different culture where
even the simplest things may appear strange.

One of the strangest things about applying to
Cambridge is that you apply not to the University
but  to a college. In practice this makes remarkably
little difference (though it does have the advantage
that instead of dealing with a faceless bureaucracy
which handles thousands upon thousands of applications
you are dealing with a small office which handles
a few  hundred applications and knows each of them as
an individual case).

I shall make a few remarks about the choice of college
later but I suggest that you ignore this matter for the time being.
\section{Main introduction}
At the graduate level Cambridge University is an international
university which happens to be in the UK. Our famous
Part~III one year graduate mathematics course has
more students from outside the UK than from inside
and we are similarly international at a higher level.
Our admissions procedure, scholarships, teaching
and so on are geared to an international clientele.

At the undergraduate level Cambridge is a national university
which happens to have some students from outside the
UK. Less than 10\% of our undergraduate
students come from abroad and, although most of
them seem happy to be here, they have had to adjust
to us and not us to them. Our admissions procedures,
teaching and so on are strongly geared to the British
school system. 


There are various questions you should ask yourself
before considering an application to Cambridge to
read mathematics.

(1) What is the date?

(2) Do you really wish to do mathematics at Cambridge?

(3) Can you afford to do mathematics at Cambridge?

(4) Are you prepared to make a `standard application'?

I will deal with these questions in turn.
\section{What is the date?}\label{date}
All application for undergraduate
entry have to be in by a fixed date sometime in the middle
of October for entry in the following October. Thus
applications for study starting in October~2000
will have to be in  by 15~October~1999.
(Exact dates are given in the pamphlets `Information for Overseas
Students' and the `Undergraduate Prospectus', see
Section~\ref{Preface}.) You MUST complete your
application by this date and the University suggests
that you should start 18~months before you hope
to begin your course by contacting the college to
which you wish to apply. Although the form filling
involved is no worse than that for most US universities
it does require a little time to get right
(you will need to fill in one set of forms for
Cambridge and a second set of forms for a body
called UCAS which is in overall charge of
all applications to British universities).
It is not the kind of thing which can be done
by fax in the last 24 hours before the deadline.
(Indeed the rules specifically exclude faxing
your forms.)
\section{Do you really wish to do mathematics at Cambridge?}\label{really}
When I ask overseas candidates this question they
look at me as though I have gone mad. Surely
everybody wishes to study mathematics at a world
famous university with excellent teaching
and pleasant surroundings. But further questioning
often shows that they want from the Cambridge
mathematics course things that it does not give.
Please bear in mind the following points.

{\sf Cambridge mathematics is a specialist course}
If you study mathematics in Cambridge you study
\emph{only} mathematics. You can not study
mathematics with a subsidiary course in literature, say.
The course consists of mathematics followed by
more mathematics varied by a little more mathematics.
(This statement must be modified in various ways,
see the `Undergraduate Prospectus' but is essentially
true.)

{\sf Cambridge mathematics requires you to take some
courses in every branch of mathematics} You
can not come to Cambridge just to do pure
mathematics or just to do applied. For the
first year and a half you have to do everything
though thereafter you have to specialise
(too much in my opinion).

{\sf Cambridge mathematics is exam orientated}
Teaching and lecturing are always directed towards
the end of year exams. The end of year exams consist
of four three hour exams and your grade for
the year is decided by those exams.

{\sf  Cambridge mathematics has a competitive flavour}
I do not wish to over-emphasise this point but it is
inevitable that a course which attracts many of
the best young mathematicians in the UK will
attract many of the most competitive young mathematicians
in the UK. (Moreover, since the number of awards
to do research for UK students is limited those who
wish to continue must aim for good examination results.)

{\sf  Cambridge mathematics is illiberal}  The Cambridge
course contains nothing on the philosophy of mathematics
or the history of mathematics. Your lecturers expect you
to want to learn lots and lots of mathematics and
not to be delayed by discussions of why you should
want to do so.

Briefly, the Cambridge course is a course run
by professional mathematicians for those who wish
to become professional mathematicians. If this
is the kind of course you want then it is,
I think, an excellent course. Otherwise, caveat
emptor\footnote{Let the buyer beware.}.

\section{Can you afford to do mathematics at Cambridge?}
(Or more usually, can your family afford for you
to do mathematics at Cambridge?)
The cost of studying in Cambridge is substantially
less than
than that of studying at a comparable US university
but is still very high for non-EU students
and high for EU students. Because the course is so intense
you are forbidden to take paid work during term time
and there is no possibility of working your
way through college. The University estimated
the cost of fees and maintenance for a non-EU
student in 1997/1998 doing mathematics at \pounds11000
and the cost rises year by year. (Consult this year's
`Undergraduate Prospectus' for the latest figures.)

Cambridge University and its colleges are committed 
to try and ensure that no UK student of the appropriate
ability is prevented from studying in Cambridge
for financial reasons. This policy is likely to
strain the resources of the colleges to the utmost
in the next few years and there is no way whatsoever
that we could extend this policy to cover students
from the rest of the world. Unless you are exceptionally
brilliant (in which case look at section~\ref{Scholarships})
you (or your family) will have to raise the money
yourself. (There are possibilities \dots
some of our students have industrial sponsorship,
some foreign students are supported by their own
governments, some US university loan schemes cover Cambridge
\dots but it is up to you
to follow them up, not us.)

Even if you can find the money it is important that
you use it to buy something that you want rather
than something that you think you ought to want (or worse
that your family thinks you ought to want).
Ask yourself: `If Cambridge was a collection of
Nissen huts in the Outer Hebrides, but gave exactly
the same courses with the same teachers would I
still wish to go there.'


\section{Are you prepared to make a `standard application'?}%
\label{standard} A college will find your application
easiest to deal with if it satisfies the following conditions

(1) You will be 18 years old in the October
that you wish to start the course. (If you would be under
18 look at section~\ref{under age}, if you would be over~18
look at section~\ref{over age}.)

(2) You make it clear that you are prepared to take
papers~2 and~3 of the STEP mathematics exam. (For
details of STEP see section~\ref{STEP}. If you are
not prepared to take STEP see section~\ref{no STEP}.)

(3) It is clear that you (or your family) can
support you for three years. (The details can be
worked out later but remember that you will not
be allowed to start the course unless your support
is guaranteed.)

(There are other less important matters which are
dealt with in Section~\ref{Matriculation}.)

Such an application will be welcomed at any college
since it can (and will) be treated almost exactly
the same way as that as that of a UK applicant.  
If you can not make such an application then
you are still welcome to apply but the
college will have to consider your application
specially.

\section{References} Most British students will have
references written by their mathematics teacher
or by their head teacher in collaboration
with their mathematics teacher. The key parts
of the reference will read something like this.
\begin{quotation} X came to us from one of our feeder
schools with excellent GCSE results in 
science and mathematics
and respectable results in other subjects.
Her mathematics teachers say that she is
the one of the two best students in her year
and as good as two of our previous students who got firsts
at Warwick and KCL. In her previous school she took
part in the national mathematics contests obtaining
a silver and a gold. Although she is very quiet in class
she thinks about everything that is said. When she
knows a method she applies it efficiently but when
working in novel situations she often comes up
with solutions of her own. Her physics teacher
says that though her approach is that of mathematician
he could confidently support an application to any
physics department in the country. We expect
her to obtain 
straight $A$'s in Mathematics, Further Mathematics
and Physics.
\end{quotation}

You will see that the key portion of the reference
is concerned with providing evidence of the candidate's
mathematical ability. The best kind of reference
you can provide will come from someone like
your mathematics teacher who is in good position
to judge your mathematical ability and will
do their best to assess your suitability for
a very hard and intense mathematics course. Naturally,
the reference will deal with your other personal qualities
but your mathematics is our primary concern\footnote{In
particular, the Cambridge mathematics faculty is not
in the business of selecting or producing English gentlemen.
Many of my colleagues are not English, many are not gentlemen
and even the few who are both English and gentlemen
are not, for the most part, English gentlemen in the proper
sense of the term.}.

\section{Less important matters} There are other
matters which you will need to agree with the college
to which you apply.

{\sf Matriculation}\label{Matriculation}
UK candidates must have passed
certain examinations in other subjects (this
is called matriculation). You and your college
must be able to show the University that you
have achieved appropriate equivalent standards.
Since most of the candidates who apply for Cambridge
mathematics have done well in all their school subjects
this is rarely a problem but the requirement
is not a mere formality.

{\sf Command of English} It would be foolish to
attempt the course without a good command of English.
(In particular, at least one
of the lecturers uses the same
kind of English that this note is written in.)
Where necessary you and the college will
need to agree on an appropriate test.
Again this is rarely a problem but it is
not a mere formality.

{\sf Interview} Almost all UK candidates are
interviewed. The colleges appreciate that
it is inconvenient and expensive for 
candidates from overseas to came for
interviews which typically last about
an hour in total. If a college thinks that
it can come to a fair decision without an
interview it may dispense with interviews.
If an interview is required then it
may be possible to offer you a choice
of dates.
\section{Choice of College}\label{College}
You should not spend too
much time worrying which college to apply too.
Mathematics is a jewel in the Cambridge crown
and all colleges maintain a high standard of
mathematics teaching. A network of formal and
informal contacts between colleges and between
colleges and the departments
means that you can expect to be taught
the same things in the same way whichever college
you go to. In the same way an informal community
of interest and the formal mechanism of the
Winter and Summer pools (see the section on
the process of applying in `The Undergraduate Prospectus')
means that the standards required for
entry remain the same for all colleges.
Almost all undergraduates are happy with
the college they end up in (and I suspect
that most of the rest would be unhappy anywhere).

As an overseas candidate you need to make your
application early so that any problems can be
sorted out. If you do not specify a college
you will only be assigned a college quite
late. I therefore advise strongly against
making such an `open application'.
Instead make a list of the colleges you
like and choose one at random.
(Then check the details of your chosen college
and if you have second thoughts, choose another.)

There are some obvious differences between colleges
(for example New Hall and Newnham are women only)
but, in my opinion, you may safely ignore any differences which
are not obvious.
\section{Scholarships}\label{Scholarships}
Because of the particular historic
path taken by higher education in Great Britain,
Cambridge has only recently begun to build up
scholarship funds for students from outside the UK.
The awards available are mainly for postgraduate
study and competition for undergraduate scholarships
is very fierce indeed. Details are available
from
\begin{center}
The Cambridge Overseas Trust\\

[or The Cambridge Commonwealth Trust]\\
Trinity College,\\
Trinity Street\\
Cambridge CB2 1TQ
\end{center}

In my opinion you should only follow this path
if you \emph{really} need a scholarship
and if you \emph{really}  are brilliant
(say as good as those in the top 10\%
of students doing mathematics at Cambridge).
Remember that UK students are now expected
to pay their own living costs (they receive
a government loan of about \pounds3000 a year
with very favourable terms and must borrow
any further amount commercially). If you
can raise part of your costs yourself
through sponsorship, other awards and borrowing
the Trusts may be more willing to close
the gap.
\section{Young candidates}\label{under age} If
you will be under~18 on 1~October when
you wish to start your course then your
College will need permission from the
University to admit you. If you are
only a few months younger there will
normally be no problem. If you are younger
than this the University and the College
will need to be convinced that you are
not only not merely \emph{academically}
but also \emph{socially} ready for university
life. It is contrary to University policy to
admit students much younger than~18
and it is very rare for exceptions to be made.
\section{Older candidates}\label{over age}
There is no upper age limit for entry to Cambridge.
However, if you are going to be substantially
older than the usual candidate (certainly
if you will 20 or older at the start of the course)
the college will wish to know why you are
starting so late. The number of older students
doing undergraduate mathematics at Cambridge
is very small and my impression (though others
may disagree) is that many of them find the
experience very stressful. If you are a
mature student (the British government in
its wisdom classifies any one who is~21
or over at the start of the course as `mature')
you should think very carefully about what
you want from a mathematics course and whether
Cambridge with its hectic pace and macho approach
is going to give you what you want.
You could also consider applying to
one of the four colleges (Hughes Hall, Wolfson,
St~Edmunds and the women-only Lucy Cavendish)
which specialise in mature and graduate students
and are more likely to provide the support
and back-up that you need. (Again this my personal
view, some of my colleagues disagree.)
\section{What is STEP?}\label{STEP}
Many more applicants for
Cambridge mathematics get the highest grades
in the English public exams than there are
places. Most UK mathematics applicants are therefor
asked to take two three hour papers in mathematics
(STEP II and STEP III) which cover the same syllabus
but ask more searching questions.
If you are prepared to take these papers
this will enable the college to

(a) check that you have the mathematical background
to cope with the course,

(b) compare you fairly with its UK applicants.

\noindent If you do well in STEP this should also
reassure you that you should be able to cope
with with the work here. (Most Cambridge mathematics
students go through a phase when the work seems
impossible and everybody else seems much cleverer
than they are.)

You can obtain the syllabus for STEP together with
past papers from the STEP office
at the  address
\begin{center}
STEP Office\\
OCR\\
1 Hills Road\\
Cambridge CB1 2EU
\end{center}
If there is time, it would be a good idea to
get hold of these before making your application
(there is an sample paper on the Mathematics Faculty Website
referred to in Section~\ref{Preface}).
Since UK students cover different courses
\emph{you are not expected to cover all the topics}.
In the exam you will be given a choice of 14~questions
and only your best 6~questions are taken into account.
We are not interested in what you can not do
but in what you can. A candidate who can
complete 4~questions is normally considered to
have done well.

If you wish to use the syllabus as a guide to
the background knowledge of our UK students
you should note that very few will have done
both the mechanics (Part~B) and the probability (Part~C)
and some will have done neither. On the
other hand, most will have covered most
of the pure mathematics (Part~A). Previous
knowledge of mechanics (Part~B) is generally
considered more helpful in tackling the
Cambridge course than previous knowledge
of probability (Part~C) though both are
useful and neither is necessary.

The technical details of taking the exam are
relatively simple.
You should ask your school (or some similar
institution) to contact the STEP office (see the address
given earlier).
They will arrange for your school to be registered
as an `examination centre' and send instructions
on how the exam is to be conducted.  
\section{What if you do not wish to take STEP?}\label{no STEP}
Most colleges
ask UK mathematics applicants to do STEP. This
does not necessarily mean that they will
not consider non-UK applicants who do not
wish to take STEP but it does mean that
you should write \emph{early} and
\emph{before making a formal application}
to the college (or colleges) that you are
thinking of applying to, explaining which
exams you will be taking and asking whether
they would be prepared to consider you on that
basis.
\section{A note of caution} All colleges try
to select the best mathematicians they can.
(What else would it be sensible to do?)
However we know that we make many mistakes
even when selecting UK students. Sometimes
we admit students who do badly in the course,
more often we reject students whose performance
at other universities show that they would have done
well here. If we make mistakes when dealing
with students whose educational background 
we know well it follows that we will make
still more mistakes when dealing with
students from educational systems which we are
not familiar with. Even if Cambridge is your
first choice university \emph{you must have
alternative plans} in case your application is not successful.

If your College accepts you subject to a
specified performance in some examination
you must still have an alternative plan
ready. However confident you are of
success, you might be ill when the exam is to
be taken or the exam may be harder than
you expected.
\section{What if you change your mind?}
If your college accepts you or accepts you
subject to certain conditions it is committed.
Provided that you satisfy whatever conditions
the College has made it \emph{must} take you.

You are not committed in the same way.
We do not want unwilling students and
if you decide not to come to Cambridge
you should simply write to the College
telling them so. All we ask is that you
\emph{write at once} so that your place
can (if possible) be offered to someone else.
Please remember that if you turn down
a place after the middle of August you
will probably have deprived someone
of a place at Cambridge.
\section{Visas} Unfortunately we live in a world
whose politicians believe in the free movement of
goods and the restricted movement of people.
It is up to you to apply for visas and so on
(although your College will write in support).
\section{A Year in Cambridge?} Sometime people
ask if they can spend a single undergraduate 
year in Cambridge.
This does not seem to me academically advisable
(at least in mathematics)
since the Cambridge course consists of three
linked years in which lectures for one year depend
strongly on those in the previous year. Occasionally
absolutely outstanding students have made 
a success of of such a year (though even they
find the experience pretty tough) but there have
also been serious problems. Most colleges do
not  accept such `occasional students'
but a few accept a very small number
(see the pamphlet `Information for Overseas
Students').

On the other hand the Cambridge Faculty of Mathematics
runs a one year course called Part~III which
is taken by students from all over the world as
a link between undergraduate studies
and the PhD. If you want to spend a year
doing mathematics in Cambridge (and you
are able and hard working) why not look
at Part~III?
\section{Transfer from other universities?} 
Sometimes people wish to change in mid-course
from other universities to Cambridge. Such an
application is very unlikely to succeed 
and University regulations mean that
you would have to start from scratch.
In Cambridge nothing is impossible
and you can make such an application to college
but I would expect you to get a polite brush off.
\section{A Second First Degree?} The Cambridge mathematics
degree is very intense and students cover a great deal
of ground. If you are very able but have a first degree from
a university where the mathematics degree
is not very advanced it may make sense to
apply as an `Affiliated Student' taking the
second and third year of the Cambridge mathematics course
(or possibly the third year together with Part~III)
to obtain the Cambridge first degree.
(See the pamphlet `Information for Overseas
Students'). However,
financial considerations are likely to loom very
large in such a case.
\section{Some final warnings} The reader of these
notes may find them cautious to the point of being
unwelcoming. `Surely' the potential student
will say, `even if there is some risk, it is
a risk that I am prepared to take and 
no business of yours.' To this there are 
two answers. The first is that it is our
business. If you fail, or even if you do badly
then it causes a substantial number of problems
for your College and your teachers. Worse, if
you are unhappy at Cambridge you have denied a
place to someone who might have been happy here.

The second answer is perhaps more to the point.
There is all the difference in the world between
someone who takes a risk recklessly without 
thinking things through and someone who
after carefully weighing the alternatives
takes a calculated risk. If you come to
Cambridge from overseas to be a mathematics undergraduate
you will be making three transitions

(1) from the adolescent, protected world of school
to the adult, unprotected world of university,

(2) from the easy pace of school mathematics
to the breakneck speed of one of the world's
hardest courses, 

(3) from one culture to another.

\noindent Experience shows that students find
each of these transitions very hard and that
some fail to make the jump. You will
be attempting all three. Why should you
succeed where others fail?

I have never known an applicant who did not
say that that he or she would work as hard as necessary
if accepted and who did not believe what he or she said.
On the other hand I have known quite a few 
students who failed to work as hard as necessary
when accepted. To see why this might be so,
observe that most of our students were the
best mathematicians in their previous schools
(and if you are not in this happy position,
you should think yet again before applying)
but now find themselves merely average.
Sometimes they are simply caught unaware by
the amount of work required to keep up.
Sometimes they have other interests ---
chess, mountaineering, playing the violin, \dots
--- and are unwilling to cut down on
these pursuits when mathematical work requires it.
Sometimes, faced with a situation where
no matter how hard they work there will still
be others better than they are, students consciously
or unconsciously decide that if they can not be
first then they would prefer to be nowhere.
You should ask yourself how you would cope
with being just one average student among many.
\section{Modified by a few words of encouragement}
If you love ballet this may mean that
you love watching it from a comfortable
place in the stalls. If on the other hand
you love ballet so much that you wish
to be a ballet dancer then you must go to
ballet school where you will spend a great
deal of time doing in exercises, a little
of your time performing and hardly any time
discussing the beauties of ballet.
Not all the pupils at ballet school go on
to be ballet dancers and, in the nature of
things very few go on to be great dancers --- but
if you want to be a ballet dancer you must
go to a ballet school.

Cambridge is one of the great mathematics
schools of the world. A look at the
incomplete list at
\begin{center}
\verb+http://www.cms.cam.ac.uk/mathematicians.html+
\end{center}
shows that Cambridge is both a leading research
centre and a leading teaching centre.
(Some universities have eminent teachers,
we have both eminent teachers and
eminent pupils.) Moreover we still link
teaching and research. The great Professor X
not only adorns our list of staff, he also
teaches first and second year courses.

We can not promise that if you come to Cambridge
you will be happy. Most of our students seem
reasonably happy but a few are deeply unhappy.
We can not promise you lectures 
or courses perfectly
attuned to your needs. If there are 200 students in the
audience then the lecturer must be going too
fast for some and too slow for others. Most
of our staff are too busy doing mathematics to pursue
the
\emph{ignis fatuus} of pedagogic perfection.
There is no royal road to mathematics
even at Cambridge.

What we can promise is the presence of able
staff and the company of talented students
in an atmosphere imbued
with mathematics.

\section{Another British university?} Perhaps
you want to study mathematics at a British
university but you would like a course which
is less intense, or more open (possibly combined
with another subject) than that at Cambridge.
The cost of study at other British universities
is comparable with that of study at Cambridge
(some will cost more, some less, the information
should be obtained from the university concerned)
so it is unlikely to be a determining factor.

There are several guides which give details of
all British universities (PUSH, Virgin, Times, \dots).
In my view, none of them can be relied on
for detailed comparisons (ask yourself how you
would collect information on one hundred or so
institutions and remember that what is said
of a university as a whole may not apply
to the mathematics department in particular) 
but they do give you a rough
idea of what is on offer. Once you have decided
on what you want --- campus/non-campus, London/non-London,
methods of examination, industrial links, \dots
--- you should draw up a short list and investigate
each individually. A letter addressed to
\begin{center}
Undergraduate Admissions,\\
Department of Mathematics,\\
University of X,\\
Xville,\\
United Kingdom.
\end{center}
will certainly find its way to the correct
office which will be delighted to send you
all the official information that you need.
Most mathematics departments maintain a
web page and these pages are accessible
via
\begin{center}
\verb+http://www.ma.hw.ac.uk/uk_maths.html+
\end{center}
 
Most universities have an `Alternative Prospectus'
written by students (for the `Cambridge University
Alternative Prospectus' see the section
of useful addresses in the official `Undergraduate
prospectus'). In practice this is less useful than
it sounds since four years out of every five the alternative
prospectus
will be written by students who are every bit as keen
on the place as the writers of the official prospectus
and the fifth year it will be written by students
so disaffected as to be unable any good whatsoever
in the system.

There exist lists purporting to rank UK universities
in order of merit but they suffer from three defects.
The first is that a one dimensional scale of
merit is impossible (if university A has better student
housing than university B but university B has better
social facilities than university A then which you
prefer will depend on the weight you attach
to each factor). The second is that all the
British lists I have seen are compiled by rather
foolish methods (you may disagree but before doing so
you should check how the ordering is obtained).
The third is that they refer to the universities as
a whole and not to the mathematics departments
within them.

In my opinion (and it must be emphasised that the
following paragraph is \emph{only my opinion})
the most important single indicator you can
use is the offer level for each department.
This is given in Brian Heap's 
`How to choose your degree course'.
(Try to get a reasonably recent edition
from a library.)
Of course, nothing can be read into small
variations in offer level but it is obvious
that \emph{on the whole} students at
an institution with a high offer level (say, AAB
at A-level) will be abler and start the course
with a higher level of knowledge than those
at institutions with a lower offer level
(say, CCC at A-level). The courses at a high offer
level department will thus \emph{on the whole}
start at a higher level and proceed at a faster
pace than those at a department which
makes lower level offers. Note that this says
nothing about the quality of teaching (which I
suspect differs rather little between British
universities) but only about the nature of
what is taught.

Applications to British universities are through
a body called UCAS whose address is
\begin{center}
UCAS\\
Fulton House\\
Jessop Avenue\\
Cheltenham\\
Glos\\
GL50 3SH
\end{center}
The rules governing applications
are fairly complicated but there are
two essential points:-

(1) You you can apply to several universities at
the same time. (Though you can not apply to \emph{both} Oxford
and Cambridge at the same time.)

(2) You need to apply almost a year in advance.
(The exact date will vary from year to year
but in essence if you wish to start in Autumn
of 1999 you should try and get things done by
mid October 1998.)


\vspace{2\baselineskip}

\begin{footnotesize}
\noindent
[Printed out \today. These notes are
may be accessed via my web home page

\begin{center}
{\tt http://www.pmms.cam.ac.uk/home/emu/twk/.my-home-page.html}.
\end{center}
Also available:-

`Dr K\"{o}rner's Helpful Guide For Mathematicians Seeking A
Cambridge Research Fellowship',

`In Praise of Lectures' (how to listen to a mathematics lecture),

`An Unofficial Guide To Part~III',

`How to Write a Part~III Essay',

`A Supervisor's Primer'.]
\end{footnotesize}

\end{document}

 

