Comparing top mathematics departments in universities in the UK 2008
Ranking, league tables and other data for university maths departments and courses in England, Scotland, Wales and Northern Ireland
At Shanghai Jiao Tong University they started their "league table" the Academic Ranking of World Universities [WP] to measure the gap between Chinese Universities and "world class" (mostly US) universities I was interested to get some idea of the difference between our School of Mathematics and other top mathematics departments in the United Kingdom. Since our formation from the merger of UMIST and the Victoria University on Manchester (VUM) in 2004 the School of Mathematics at Manchester has been on a generally upward trend, and while crude measures should be treated with extreme caution it helps us to get some idea of how we compare (for example on size and income). I was also interested in the reasons some of the newspaper league tables for university maths departments produce some results that are quite surprising relative to how we perceive research reputation of departments.
I was interested to note a survey by the Sunday Times of the opinions of headteachers about the top five universities for several subjects. The headteachers had this order for mathematics: Cambridge, Oxford, Imperial, Warwick, Bath. Lets see how this compares to some rankings using published data.
One starting point is the 2001 Research Assessment Exercise. Of course we did not exist in 2001 and at the time of writing the process of assessing the 2008 RAE has just begun. For the sake of comparison we can combine the UMIST and VUM entries, although I expect we have changed (upwards!) much more than most UK maths departments since 2001 so in most respects this gives a lower bound.
RAE 2001
The RAE 2001 grades were 5*, 5, 4, 3a, 3b, 2 and 1 in decreasing order. In terms of the RAE 2001 grades there are five Universities that gained at least two 5* and on 5 grade from the three Units of Assessment Pure Maths, Applied Maths and Statistics+OR. Cambridge gained top marks with three 5*s, while Oxford, Warwick, Imperial and Bristol all got two 5*s and a 5. For details see a table of all mathematics departments submitting to the 2001 RAE . One could perhaps make a distinction that some departments chose to leave people out of the RAE declaring them as not research active, this is indicated by a B or C (but it is not clear how to interpret this). I have sorted each university by a weighted average by the number of research active staff in each UoA (again of debatable merit!).
Does size matter? Certainly there are some benefits to a large department, for example the coverage of more specialisms which might be expected to give students a wider choice of optional courses. Economies of scale might be expected as well. It is hard to count the number of academic staff accurately from for example the department web sites. it is n't clear who might be emeritus, or on a fractional contract for example, or shared with another department. The 2001 RAE return gives the Full Time Equivalent (FTE) of research active staff. Some Universities split Pure and Applied Mathematics and Statistics in to two or three different departments. For the sake of comparison I have taken the union over the three units of assessment.
Number of research active staff in UK mathematics and statistics departments and weighted RAE score
University

Total Research Active Staff (FTE)

Weighted average grade by FTE in each unit. 5*=6, 3a=3.5,3b=3

Average grade over three UOA (taken as zero if no return)

University of Cambridge  114.5  6  6 
University of Oxford  81.6  5.59  5.67 
The University of Manchester (UMIST+VUM)  75.5  4.55  4.5 
University of Leeds  58.25  5  5 
Imperial College of Science, Technology and Medicine  54  5.84  5.67 
University of Nottingham  50.9  5  5 
University of Edinburgh  48  5.15  5 
University of Bristol  44  5.75  5.67 
University of Warwick  43.75  5.42  5.67 
University of Durham  42  5.22  5 
One difference between Manchester and most of the other universities with large mathematical communities is that we are all one big School, rather than separate Pure, Applied and Statistics departments. We seem to be the biggest "Maths and Stats" department all under one roof, as of 2008 we have about 80 academic staff.
The numbers of undergraduate maths students is harder to find in published data. However I have found the following:
 Manchester for 20052007 there were 350450 students starting as first years each year, with about half on a straight maths degree and the rest on maths with another subject as a minor. The 2007 intake was 280 straight maths and 170 with another subject total 450, but that is regarded as an exceptional year.
 Warwick helpfully publish online statistics each year. In 2007/08 they have 292 first year students enrolled at the Mathematics Institute, and 136 in statistics see table (pdf) giving a total of 428.

Oxford aims at 190 on straight maths, 30 on maths and CS, 20 on maths+philosophy, 28 on maths+statistics, a total of 268
 Bristol list total numbers of students. In 2006/07 Engineering Maths had 186 while Maths had 615 a total of 801, giving around 250 per year assuming a typically small fourth year.

Cambridge aims to take 220 straight maths and 20 maths + physics each year, total 240

Imperial aims to take 240
As far as we can tell from this data Manchester has the biggest population of undergraduate mathematical sciences students (judged by the 2007 intake) by a very small margin over Warwick. Going by the 2001RAE figures for staff it seems unlikely that any other departments would make it in to the top six by size, but Leeds (about 500 total  see brochure), Nottingham (total approx 500 ) and Edinburgh have fairly large undergraduate populations.
I wrote a very crude bash [WP] script for our own purposes that extracts data on the currently held research grants by mathematics (and statistics) departments.
It runs every month on a cron job so should update automatically (and will break if EPSRC change their web site). The latest version of the table here. Scroll down to see the results with the various departments in each university combined. As the script is to some extent written by hand I had to type in a couple of lines for each university. If one department not listed suddenly gets loads of grants it will not show up unless I add it! You can check the results individually on the EPSRC Grants on the Web site. Go to institution and the select maths department.
On January 7th 2008 the table is as follows, MORE RECENT VERSION
EPSRC grants held by mathematical sciences departments in the UK 2008
N  University  Total pounds 
1  Bristol  £19,366,708 
2  Warwick  £9,604,880 
3  Oxford  £9,515,222 
4  Cambridge  £8,573,353 
5  Imperial  £6,604,829 
6  Manchester  £6,568,348 
7  Sheffield  £4,612,261 
8  Edinburgh  £4,494,493 
9  Nottingham  £4,411,390 
10  Bath  £3,119,602 
11  Glasgow  £2,993,405 
12  UCL  £2,987,122 
13  Southampton  £2,502,640 
14  Liverpool  £2,127,533 
15  Queen_Mary  £1,626,088 
16  Cardiff  £1,467,353 
17  York  £1,444,890 
18  Royal_Holloway  £1,396,629 
19  KCL  £1,360,514 
20  Strathclyde  £1,275,419 
21  Leeds  £1,251,566 
22  Birmingham  £934,290 
23  Aberystwyth  £886,386 
24  Durham  £845,066 
These results should be taken with a very large hand full of salt. Of course they indicate some merit as at least getting a grant means that referees and panel members approved the proposal. Here are some caveats
 There is plenty of really good mathematics done without grants
 There are other sources of grant funding such as Leverhulm Fellowships and other research councils not listed
 A joint grant where the coinvestigator is in maths but the PI is in another department will not show up
 Older grants that were not "Full" economic cost are still running and are much smaller
 Industrial funding, either on an EPSRC project or contract research does not show up
Royal Society, Wolfson Merit, Leverhulme
The Royal Society administers a number of prestigious research fellowships. These are the Royal Society Research Professorship, Wolfson Research Merit Award, Leverhulme Senior Research Fellowship, as well as the Royal Society University Research Fellowship, Industrial Fellowship and Dorothy Hodgkin Fellowship. The RS web site lists all current fellows on a single web site. On 6th January 2008 this was the whole list. I extracted this to a text file and then used grep to count those held in Mathematics and Statistics Departments. The results as follows:
RS grants and Fellowships
University 
Royal Soc. Research Professorship 
Wolfson Research Merit Award 
Leverhulme Trust Senior Research Fellowship 
University Research Fellowship 
Dorothy Hodgkin Fellowship 
Industry Fellowship 
Cambridge 
0

2

0

6

0

0

Oxford 
0

2

0

4

0

0

Bristol 
0

3

0

0

0

0

Warwick 
0

0

2

0

0

0

Manchester 
0

2

0

1

0

0

Imperial 
1

3

0

4

0

0

Sheffield 
0

0

0

1

0

0

UCL 
0

0

0

1

0

0

Queens Belfast 
0

1

0

1

0

0

St Andrews 
0

0

0

1

1

0

Bath 
0

0

0

0

0

0

Nottingham 
0

1

0

0

0

0

Glasgow 
0

0

0

1

0

0

How many Fellows of the Royal Society currently work in each maths departments? Its pretty hard to get this data, but probably having a few is some kind of sign of a healthy research active department! Here is the data I have (not counting emeritus and retired). There is no systematic list so I may have missed some people. Also at Cambridge DAMTP arguably includes some people, Hawking for example, better known as physicists than mathematicians. However even with a tight definition of mathematician Cambridge still comes top if the FRS table.
Fellows of the Royal Society in UK maths departments (incomplete)
University  FRSs  Notes 
Cambridge 
Barlow,
Turok,
Coates,
Gowers,
Kelly,
Hinch,
Huppert,
ShepherdBarron,
Spiegelhalter,
Pedley,Wills +etc

Gowers is a Fields Medallist 
Oxford 
Ball, J. Ockendon, Silverman, Segal, Birch, Kirwan, James, Lyons HeathBrown, Hitchin, Donnelly, Trefethen 

Warwick 
MacKay, Reid, Stewart, Preiss
 
Imperial 
Donaldson, Hayman, Atkinson,
 Donaldson is a Fields Medallist. 
Manchester 
Higham, Taylor, Wilkie
 Paris is a Fellow of the British Academy
Buchstabber and Shiryaev are Corr. Memb. Russian Acad. Sci. 
Bristol 
Green, Wooley,


Liverpool 
Rees

Mazya is Member of the Royal Swedish Academy of Sciences

Fields Medalist[WP] are very rare so not really useful for league tables. Probably more important taking a long view is that a department has a culture that will produce one. For example Donaldson and Atiya got their Fields Medals while at Oxford. That said the research culture of a department is likely to be improved by having one there.
It is good for ones research papers to get cited and ISI Highly Cited researchers are ones that have got cited a lot, an indication that their work is valued by other researchers. Some topics in mathematics, for example numerical analysis, seem to generate more citations than others. But maybe that means these areas are more useful anyway?
ISI Highly Cited researchers in UK Maths Depts
University  Highly cited researchers 
Oxford 
Ball, Cox, Quillen, Silverman,Trefethen,

Cambridge 
Clayton, Spiegelhalter, Fokas, Lickorish,

Imperial 
Donaldson, Liebeck

Bristol 
Goldstein, Green

Manchester 
Higham, Hammarling (hon post), Dongarra (fractional post)

Warwick 
Roberts

LSE 
Atkinson

King's College London 
Davies

UCL 
Dawid

Open University 
Jones

Queen Mary 
Smith

Glasgow 
Titterington

The Times Good University Guide has listings by subject. (What follows was accessed Jan 2008) Here is their table for mathematics. They have used RAE 2001 scores, entry qualification and something called "graduate prospects". You can reorder them to rank by one of the criteria. For example according to them the top 20 by ranked by entry qualifications (UCAS points). There is some explanation of the methodology is given indicating that it is based on some very old UCAS figures! No wonder this and other newspaper tables give very strange results for mathematics. Some also use the results of the teaching quality assessment the government carried some times ten years ago.
More up to date information on entry requirements can be found on the UCAS web site, e.g. by a search for course code G100 (Maths). Here are some examples as of Jan 2008 (some are slightly more complicated than stated, follow links for more details). Most of the top universities do not use UCAS points tariff as their entry criterion, in England and Wales the grades on three Alevels is the usual standard. The top maths departments find that this is not sufficient to select on mathematical ability and use things such as STEP and AEA results, module grades, for example on the core mathematics modules C3 and C4, and Further maths A or AS grades. UCAS tariffs for students who are accepted may include things like Alevel general studies that are often exluded from entry requirements, as well as standalone AS scores.
Alevel entry grades for G100 Maths degree courses in the UK
Cambridge 
AAA + STEP 

Oxford 
AAA (varies by college) 
Warwick 
A(M),A(FM),B,2STEP
A(M),A(FM),A,A

Imperial 
A(M),A(FM),A
A(M),A,A, STEP
A(M),A,A,B

Leeds 
A(M),A,A
A(M) B 3STEP
A(M)A a(FM)

Bristol 
A(M)AB

Bath 
A(M),A,A or A(M),A,B

Manchester 
A(M with AB in C34)AB

Nottingham 
A(M)AB (A in FM if taken)

UCL 
A(M),A,B(FM)

Durham 
A(M) AB (FM at AS if not A2)

So the whole there is still basically "COWI" Cambridge Oxford Warwick Imperial asking for three As and some, and then the next bunch who will let you get a B as long as you do well enough in maths. In contrast to the Times GUG based on 2001 entry requirement the current entry requirements are broadly in line with the other rankings..
The hilarious Guardian University Guide
The Guardian compiles a league table noted mainly for it's comic value. The biggest gaffs where the listing of maths departments at Bangor and Hull with quite high league table positions after they had closed down. The table continues to produce some quite unexpected rankings. The 2008 edition is out already. The current top 10 are Cambridge, Oxford, St Andrews Portsmouth Warwick Imperial Aberdeen Leicester Lancaster Kingston. Well Portsmouth will be pleased! The methodology is given. The PDF seems to indicate that they are also using out of date information. The student data is taken from 2004/2005, and the leavers destination from 2004/2005. The entrance qualification are the 2005/2006 figures. It makes one wonder why they bother making a new league table each year! One other interesting feature of the Guardian table is the inclusion of spend per student. So any university that wants to go up the league table just needs to burn money, not spend it on anything useful. You would think that Manchester would do quite well on that having recently invested £ 43m on the new Alan Turing Building,[WP] which is at least half for maths, however I suspect again old figures are used and they may not attribute capital project to subject "cost centres". However such investments clearly have an impact on maths students' experience.
Good University Guide
The Good University Guide produces table by subject including Mathematics(choose maths on the menu). As other tables it is based (again I am looking in 2008) on old data including the 2001 RAE and the 20042005 HESA entry qualification data and destination data. The current version lists the top ten as Cambridge Oxford Bristol Bath Durham Warwick= Imperial LSE UCL and St Andrews. As I understand it a university that did not submit RAE returns in all three of Pure Applied and Stats was not penalised (eg LSE only submitted in Stat which seems a bit strange.
The GUG is compiled by a consortium called Mayfield University Consultants, largely current or former university administrators, and the University of Sheffield. It seems this group used to be responsible for the Times Good University Guide, but have split from them. The Times GUG are now produced by Exeter Enterprises, owned by Exeter University.
UniStat
The on line university statistics site UniStats would seem to be a way to check league table data. However it seems to use the same out of date HESA data. For Manchester it says our average UCAS score is 420 and only 42% of our graduates get a job after six months. Both of which sound totally fictitious to me. The details for the University of Manchester even list the web site as the Victoria University of Manchester's now defunct URL. Looks to me as if this site is pretty much pointless unless you are interested in historical data.
Google Page rank
The Page rank [WP] is a number in the range 0 to 10 used by Google to sort search results and is uses an eigenvector of a graph of web links to decide which pages are most important. So presumably important maths department's web pages have lots of important links to them. The following should by updated when you reload this web page. From time to time they all show zero, not sure why, just come back after a few hours and it seems to work again! The numbers are provided by PR checker.
N/A
URL  Page rank 
Cambridge www.maths.cam.ac.uk 

Oxford www.maths.ox.ac.uk 
7

Warwick www.maths.warwick.ac.uk 
6

Imperial www.ma.ic.ac.uk 

Bristol www.maths.bris.ac.uk 
6

Manchester www.maths.manchester.ac.uk 
6

Durham maths.dur.ac.uk 
5

Leeds www.maths.leeds.ac.uk 
6

Heriot Watt www.ma.hw.ac.uk 
6

St Andrews wwwmaths.mcs.standrews.ac.uk 
6

Southampton www.maths.soton.ac.uk 
6

Birmingham www.mat.bham.ac.uk 
6

UCL www.ucl.ac.uk/Mathematics 
5

Aberdeen www.maths.abdn.ac.uk 
5

Edinburgh www.maths.ed.ac.uk 
6

At the time of writing that does not look very interesting. Most of the reasonably sized maths departments at the time of writing (Jan 2006) have a Page rank of about 6, so it is currently not a very discriminating measure. A list of all maths departments in UK Universities is maintained at HeriotWatt, if you want to check others. There are some quite respectable departments with Page rank zero. Perhaps because they changed their URL recently. In time there will no doubt be more useful tools to asses the impact of departments on line. Already Webometrics ranks Universities by the impact of their on line publications.
More citations nonsense
There are not many world league tables of mathematics departments. One based on citation was published in 2002 by ScienceWatch. It lists the top 25 in the world by the number of citations and impact.
On number of citations no UK departments even made it in to the table! On the "impact" UK departments in the table were 9th Imperial, 11th Cambridge, 15th Oxford, 18th Warwick. Agreeing with the traditional rating as COWI being the top four (but in a different order). Of course many pinches of salt here needed as usual when interpreting citation data. That said it is certainly a good thing to write papers that other researchers cite.
I have recently notcied a paper by Bollen et al that suggests the use of something more like the Page Rank could be a better indicator of quality rather than popularity in networks of journal citations. No doubt such a thing will be implemented and if it is used, eg to allocate funding, people will start to optimise against that criterion. In mathematics citation based measures of the importance of work suffer from the effect that the importance of some mathematical result only shows a considerable time later. For example Fritz John's 1938 paper on the ultrahyperbolic equation had to wait half a century before it found use in 3D xray tomography, which had not been invented at the time he did the work.
MathSciNet (MSN) lists and reviews publications in mathematics. It takes a fairly narrow view of mathematics, certainly including everything in pure mathematics but missing some applied papers. I wrote a script that extracts publications for the year 2007. I expect not all 2007 papers have made it in to MSN this early in the year (12th Jan 2008). The 2006 totals are greater. I only checked the institutions listed and only the codes for maths and statistics departments (see MSN Institutional codes) Some people might list their address as some interdisciplinary centre in which case their paper might be missed. Certainly the numbers of papers published in the year is going to be much bigger. Forexample any of my own papers are not in engineering journals and therefore do not show up. As it counts the number of papers on the MSN database with at least one author from each institution two authors from one institution counts as one paper. However an author from each of pure and applied departments in one university counts as two for that university (I should have used OR), similarly a paper authored by an author from Oxford and Warwick would count as one for each.
University  No. of Pubs in MSN in 2007 
Oxford MI+OCIAM+Stats  114 
Cambridge DPMMS+DAMPT  110 
Manchester  103 
Warwick M+MI+S  90 
Imperial  84 
Bath  66 
Leeds (several)  62 
Bristol M+EngM  44 
St Andrews (several)^{*}  40 
Durham  34 
LSE M+S+OR  31 
UCL  31 
Nottingham  17 
Sheffield  11 
* St Andrews has seven institution codes 4STAN? for maths and stats departments so results may be inaccurate.
I have not yet got around to working out papers per individual, but you can do that approximately from the 2001 RAE figures.
Overall rankings of universities
"The Times Higher Education" provides an annual league table of universities but does not go down to individual subjects However some of these measures are only likely ever to be applied to whole universities or large subsets of them and are never the less of interest, for example, to a student choosing where to study maths.
The 2007 Lists of Top 10s includes the top 10 on a survey of employers. Out of the UK universities they list Cambridge 1st, Oxford 2nd, LSE 3rd, Manchester 5th and Imperial 8th. It is interesting that the UK does so well so there may be some bias. On the whole large companies concentrate their recruitment effort on a few (usually large) universities and this might be a useful figure for prospective students to choose a univerity within the UK. The scores are given for each university in the Top 200 Universities in the World list.
US universities slot in the gaps with Harvard 4th, MIT 5th and Stanford 6th. By contrast the table of top 10 citations per staff member contains no UK universities, 8 US and one from France and one from Switzerland. However this is not likely to be especially relevant score for mathematics.
Conclusions
Before I started this exercise I shared the view of most of the UK mathematical community that "COWI" (but the order debatable) are the top four mathematics departments in the UK, but that the ranking of the next few is very much a matter of opinion. At the time of writing one would certainly have to rank Bristol up with COWI on the basis of research grants and RAE and Manchester close at least on grant income, research activity and sheer size. That said there is excellent research work and teaching going on in smaller departments, most notably Bath, many others excellent in their specialisms. Also before looking at some of the details some of the newspaper league tables and the GUG puzzled me as to their mathematics rankings, especially how Manchester seemed to rank surprisingly low and some relatively unknown maths departments rise up their rankings. This seems to be down mainly to the use of some very old figures from RAE2001 and HESA, and in the case of Manchester probably some confusion in the statistics caused by merger. Also tables such as the Guardian ignore research quality. However for students with a passion for mathematics a research active departments would be essential for a satisfactory undergraduate experience. I expect I will revisit some of these issues in 2009, in the light of RAE2008 and other new data.
There is an increasing tendency in Higher Education to employ "metrics",that is crude numerical values as a substitute for an analysis of quality. I think it is useful for academics and heads of maths departments especially to look at some of the above figures to try to understand if they have created an atmosphere in which excellent mathematics can be done, if they are applying for and being awarded a reasonable number of grants, if the work is getting published and if the academics get to be recognised as leaders in their field. From the point of view of students (undergraduate and postgraduate) and individual academic staff we would like to know if a department has a culture and "climate" in which we can achieve our mathematical potential, and to enjoy ourselves doing so. These are of course very difficult to quantify!
Please feel free to post comments, corrections etc, on this blog entry.
I also wrote some thoughts on advising prospective undergraduates on where to study mathematics in the uk
Bill Lionheart School of Mathematics University of Manchester. January 2008.