This is now outof date. See 2009 version
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.
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 
The numbers of undergraduate maths students is harder to find in published data. However I have found the following:
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.
Note added September 2008. I asked one of our librarians and he found the data (published by HESA) on the number of combined undergraduate and postgraduate student for the academic year 2006/2007. This is the latest data they publish. The Open University comes first, but it is not clear how to compare as their undergraduates are part time. The next few are Warwick Oxford Cambridge and Manchester, which were quite close, then Bath and Imperial. Complete table is here
On January 7th 2008 the table is as follows, MORE RECENT VERSION
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
Organisation 
% of Total Maths EPSRC Doctoral Training Account 
University of Oxford 
10.994 
University of Warwick 
8.191 
University of Cambridge 
7.780 
Imperial College London 
7.548 
University of Bristol 
6.066 
University of Nottingham 
4.378 
The University of Manchester 
3.916 
University of Bath 
3.598 
University of Edinburgh 
3.478 
University of Durham 
3.071 
University of Southampton 
2.992 
University of Leeds 
2.803 
Lancaster University 
2.568 
University of Glasgow 
2.250 
Queen Mary, University of London 
2.101 
Kings College London 
1.828 
HeriotWatt University 
1.808 
University of Sheffield 
1.703 
University of Strathclyde 
1.647 
University of York 
1.560 
Newcastle University 
1.477 
University College London 
1.467 
University of Birmingham 
1.433 
University of Liverpool 
1.420 
University of East Anglia 
1.401 
University of St Andrews 
1.357 
University of Leicester 
1.197 
Loughborough University 
1.116 
Cardiff University 
1.051 
University of Aberdeen 
1.001 
University of Surrey 
0.951 
Royal Holloway, Univ of London 
0.892 
University of Exeter 
0.875 
University of Kent 
0.797 
LSE 
0.775 
University of Sussex 
0.690 
University of Salford 
0.627 
Brunel University 
0.624 
University of Wales Swansea 
0.268 
Aston University 
0.065 
Kingston University 
0.055 
Durham University 
0.050 
Coventry University 
0.050 
University of Wales, Aberystwyth 
0.035 
University of Reading 
0.029 
University of Essex 
0.011 
Goldsmiths College 
0.005 
University of Bradford 
0.002 
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 
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?
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, Stewart, MacKay 
LSE  Atkinson 
King's College London  Davies 
UCL  Dawid 
Open University  Jones 
Queen Mary  Smith 
Glasgow  Titterington 
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 excluded from entry requirements, as well as standalone AS scores.
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 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.
URL  Page rank 
Cambridge www.maths.cam.ac.uk  7 
Oxford www.maths.ox.ac.uk  7 
Warwick www.maths.warwick.ac.uk  6 
Imperial www.ma.ic.ac.uk  N/A 
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.
Added Sept 2009. The Google Directory has a list of uk maths departments sorted by page rank (but the page rank still does not vary very much between th top 30).
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 noticed 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.
University  No. of Pubs in MSN in 2007 
Manchester  163 
Cambridge DPMMS+DAMPT  151 
Oxford MI+OCIAM+Stats  151 
Imperial  140 
Warwick M+MI+S  124 
Queen Mary  114 
Leeds (several)  98 
Bath  73 
Edinburgh  73 
KCL  70 
Bristol M+EngM  68 
Glasgow  62 
St Andrews (several)^{*}  61 
Birmingham  51 
Durham  51 
Loughborough  49 
LSE M+S+OR  48 
Southampton  48 
UCL  48 
Nottingham  43 
HerotWatt  25 
Sheffield  24 
* St Andrews has seven institution codes 4STAN? for maths and stats departments so results may be inaccurate.
**Updated 23th Sept 2008, including all the departments with at least 30 staff in the 2001 RAE + Birmingham. Numbers gradually increase for 2007 as the indexing catches up.
I have not yet got around to working out papers per individual, but you can do that approximately from the 2001 RAE figures.
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 university 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.
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.