International Review of Mathematical Sciences in the UK

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC

INTERNATIONAL REVIEW OF MATHEMATICAL SCIENCES IN THE UNITED KINGDOM

INFORMATION for the PANEL

PART I

EVIDENCE PREPARED by EPSRC

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Contents List of Annexes 3 Table of Tables 3 Table of Figures 4 1. Preface 5 2. Overview of UK research support structures and funding levels 7 2.1 Setting the scene – current context and recent history 7 2.2 Department for Business, Innovation and Skills (BIS) 9 2.3 The impact of ‘Full Economic Costs’ on Research Council budgets 10 2.4 The Technology Strategy Board 11 2.5 The Capital investment Framework 11 2.6 Other sources of research funding 11 3. The Research Councils 12 3.1 RCUK 12 3.2 RCUK Priority Themes 14 4. EPSRC 15 4.1 Overview 15 4.2 Governance 16 4.3 Defining programme priorities 17 4.4 Support for Research Projects 17 4.5 Sustaining Research Capacity 19 4.6 Support for Doctoral Training 20 4.7 Support for People 22 4.8 Support for Knowledge Transfer & Exchange (KTE). 23 4.9 Support for Public Engagement. 24 5. Mathematical Sciences Research in the UK 25 5.1 Introduction 25 5.2 Analysing the Evidence 25 5.3 The Character of Mathematical Sciences Research 26 5.4 The boundaries of the EPSRC Mathematical Sciences Programme 26 6. Funding for Mathematical Sciences Research in the UK 28 6.1 Overview 28 6.2 EPSRC support for Mathematical Sciences Research 30 6.3 Distribution of EPSRC Mathematical Sciences Funding 33 6.4 National and International Facilities and Services 35 6.5 Support for Mathematical Sciences PhDs 36 6.6 Destination of UK PhD students 40 7. Support for People 41 7.1 Non-EPSRC sources of support for People 41 7.2 EPSRC support for People 42 7.3 EPSRC Support for Established Researchers 47 7.4 Previous Schemes 48 7.5 Demographics 49 7.6 EPSRC Mathematical Sciences Programme Demographics 50 8. International Engagement 52 8.1 Overview 52 8.2 International collaboration with EPSRC-funded Mathematical Sciences Researchers52 9. Impact 53 9.2 Knowledge Transfer 53 9.3 Public Engagement 53 10. Bibliometric Evidence 54

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC

List of Annexes Annex A Making sense of research funding in UK higher education Annex B Main Recommendations to the IRM 2003 and the Review of Operational Research and Report on Subsequent Actions Annex C EPSRC Mathematical Sciences Programme Overview 2010 Annex D Additional Funding Data from Other Research Councils Annex E Research Assessment Exercise (RAE) Annex F Additional Bibliometric Evidence Annex G Evidence Framework Annex H List of Acronyms

Table of Tables Table 1 Research Councils and Funding Bodies actual research budget allocations for 2004/05 to 2010/11 (£000s) 8 Table 2 The Research Councils 13 Table 3 EPSRC’s core programmes post April 2008 16 Table 4 Main EPSRC PhD Training Funding Mechanisms 21 Table 5 Number of common Investigators between the Mathematical Sciences programme and other EPSRC programmes – all fEC grants awarded since 2005 (EPSRC data) 27 Table 6 Number of common investigators between the EPSRC Mathematical Sciences programme and other research councils on fEC grants awarded since 2005 (Research Councils data) 28 Table 7 UK university research funding - 'Mathematical Sciences' cost centres (£M) (excludes QR funding) (HESA data) 29 Table 8 UK university research funding - all cost centres (£M) (excludes QR funding) (HESA data) 29 Table 9 Research Councils’ contribution to UK university research funding - 'Mathematical Sciences' cost centres (HESA data) 30 Table 10 EPSRC research budget by programme: new investment (£M, 2005/6 - 2009/10) (EPSRC data). 31 Table 11 Contribution (in value and number) of other EPSRC research programmes to new Mathematical Sciences programme grants (years 2005/06-2009/10) (EPSRC data) 31 Table 12 Co-Funding by other public sector organisations to Mathematical Sciences programme grants awarded between 2005/06 and 2009/10 (EPSRC data) 32 Table 13 Contribution (in value and number) of the Mathematical Sciences programme to new research grants in other areas of the EPSRC remit (years 2005/06-2009/10) (EPSRC data) 34 Table 14 Destination of UK PhD students by subject area (HESA data) 41 Table 15 Number of Career Acceleration Fellowships per year (2007/8 – 2009/10), all EPSRC vs. Mathematical Sciences programme (EPSRC data) 46 Table 16 Number of First Grant applications per year (2005/6 – 2009/10), all EPSRC vs. Mathematical Sciences programme (EPSRC data) 47 Table 17 Number of Leadership Fellowships per year (all EPSRC vs. Mathematical Sciences programme) (EPSRC data) 47 Table 18 Science & Innovation Awards since 2005 in areas of Mathematical Science (EPSRC Data) 48 Table 19 Multidisciplinary Critical Mass Centres funded bythe EPSRC Mathematical Sciences programme 49

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC

Table of Figures Figure 1 UK higher education institutions’ income from research grants & contracts and funding council grants (Source: Research Information Network) 8 Figure 2 Structural changes to Government departments that support research 9 Figure 3 EPSRC budget 2004/05 to 2010/11 (source EPSRC, figures are #millions)) 11 Figure 4 Diagram of Government – Research Council hierarchy 13 Figure 5 EPSRC - The Big Picture (2008-11) 15 Figure 6 Distribution of requested support duration and value - Responsive mode applications 2009-10 19 Figure 7 People Support across the Career Path - 2009 Portfolio, headline numbers for all Programmes (EPSRC data) 22 Figure 8 Overlap between the Mathematical Sciences programme and other EPSRC programmes (EPSRC data) 27 Figure 9 EPSRC Funding (£M) to Mathematics and Statistics Departments – excluding Mathematical Sciences programme funding (EPSRC data, current grants) 32 Figure 10 EPSRC Funding (£M) to Mathematics and Statistics Departments (EPSRC data, current grants). Data referring to a restricted set of Departments. 33 Figure 11 Mathematical Sciences programme funding (£M) by department (current grants) (EPSRC data) 34 Figure 12 Top 15 institutions by value of funding from the Mathematical Sciences programme (current grants, includes research grants and fellowships; values in £M) (EPSRC data)35 Figure 13 Top 15 institutions by value of funding from EPSRC (current grants, includes research grants and fellowships; values in £M) (EPSRC data) 35 Figure 14 People Support across the Career Path - 2009 Mathematical Sciences Portfolio (EPSRC data) 37 Figure 15 Allocation of DTA budget by EPSRC programme over past 4 years (EPSRC data) 38 Figure 16 Recorded domicile of PhD students in mathematical sciences (HESA data) 39 Figure 17 Number of EPSRC project studentships starting per year, by programme (2005/06 to 2009/10) (EPSRC data) 51 39 Figure 18 Number of current Fellowships funded by EPSRC Mathematical Science Programme43 Figure 19 Number of current Fellowships funded all EPSRC Programmes 43 Figure 20 Number of EPSRC-funded PDRAs by programme (2005/06 to 2009/10) 44 Figure 21 Total number of PDRAs on grants funded by the Mathematical Sciences programme by area (from 2005 to 2010) (EPSRC data) 45 Figure 22 Total number of PDRFs funded by the Mathematical Sciences programme by area (from 2005 to 2009) (EPSRC data) 46 Figure 23 Distribution of staff number analysed by grade and gender between 2004/05 and 2008/09 (HESA data) 49 Figure 24 Distribution of staff numbers analysed by age between 2004/05 and 2008/09 (HESA data) 50 Figure 25 Distribution of staff numbers analysed by salary between 2004/05 and 2008/09 50 Figure 26 Age and gender of current EPSRC mathematical sciences portfolio principal and co- investigators, by proportion (July 2010) (EPSRC data) 51 Figure 27 Number of EPSRC Mathematical Sciences grants announced, by age group from 2005 to 2010 (EPSRC data) 51 Figure 28 Number and origin of Visiting Researchers on Mathematical Sciences projects (years 2005/06-2009/10) (EPSRC data) 52 Figure 29 Citation impact of Mathematics papers (Thomson Reuters ‘InCites’) 55

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC

1. Preface 1.1.1 This document has been produced in preparation for the International Review of Mathematical Sciences 2010. The review is being organised by EPSRC and aims to inform stakeholders about the quality and impact of the UK science base in mathematical sciences research. 1.1.2 The purpose of the Review is to benchmark UK mathematical sciences research in relation to the best in the world. The Terms of Reference for this review are that the Panel will: • Assess and compare the quality of the UK research base in the mathematical sciences with the rest of the world; • Assess the impact of the research base activities in the mathematical sciences internationally and on other disciplines nationally, on wealth creation and quality of life; • Comment on progress since the 2004 Reviews of Mathematics and Operational Research (including comment on any changed factors affecting the recommendations); and • Present findings and recommendations to the Research community and Councils. 1.1.3 The purpose of this document is to provide the Panel with a broad overview of how support for research in general is organised in the UK together with detailed information about various aspects of mathematical sciences research to help inform the Panel’s deliberations. The document presents evidence relating to the composition of the mathematical sciences research community, as well as other contextual information with a bearing on the review, and it is hoped that it will prove useful to the panel when considering the questions in the evidence framework (see Annex G). 1.1.4 The s cope of t his r eview is t he ent ire br eadth of r esearch i n t he m athematical s ciences, rather than the research funded solely by the EPSRC Mathematical Sciences programme or even the Research Councils in general. Given the broad scope of the review, the data contained in this document is not and cannot be complete. This data should be considered only as an element of the evidence available to the Panel and is complemented both by the other documents provided to the panel (including those listed below) and the Panel’s visits to the different Institutions. 1.1.5 Data in t his document has been collected from a v ariety of sources, of which the three main ones are: • The Higher Education Statistics Agency (HESA); • Research Councils’ and other administrative data sources including other funders of mathematical sciences research and training; • Thomson Reuters (bibliometrics data). 1.1.6 This document is supplemented by the following additional sources of information: • a set of 15 ‘Landscape Documents’ covering the full remit of the review prepared by respected members of the research community; • collated responses to a public consultation seeking evidence to assist the panel to address the framework questions; the full responses are available together with summaries of the main points they contain; • department-level information prepared for the panel by each institution which the panel is expected to meet; • A number of case studies prepared by the IMA, the Industrial mathematics KTN and the EPSRC Public Engagement programme. These documents are available securely to panel members at ftp://ftp.rcuk.ac.uk/. 1.1.7 Due to the pervasive nature of mathematics no data source has been identified that accurately captures the full extent of supporting activities or resources; in particular, no information on how QR funding is spent is available1, and care should be used in drawing

1 For an explanation of ‘QR funding’ see 2.1.4 5

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC conclusions solely from the data provided. The panel is asked to bear in mind at all times that the data presented is generally an underestimate of the true situation. 1.1.8 Due to the variety of sources some data are more recent than others. However, wherever possible, t he data c overs UK f inancial years 20 05/06 t o 200 9/10 ( UK f inancial years run from 1 April to 31 March). Unless otherwise specified Research Council data refers to the EPSRC’s ‘Mathematical Sciences’ programme, and the ‘research topic’ definitions are as used internally by EPSRC. 1.1.9 With the exception of data provided in confidence, this document will be published on the EPSRC web site on completion of the Review. No data from this document may be quoted or used publicly without written permission from EPSRC. 1.1.10 It is not the purpose of this document to steer the opinion of t he Panel, but to present information which the Panel may interpret as it sees fit. Therefore the authors have tried to avoid drawing inferences or conclusions from the data.

The authors wish to thank the large number of individuals in EPSRC, other Research Councils and on the review Steering Committee who have provided assistance during the production of this document

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC

2. Overview of UK research support structures and funding levels

2.1 Setting the scene – current context and recent history 2.1.1 Since 2004, UK government support for research has been provided within t he overall context of a t en-year Science and Innovation Investment F ramework 2, which set out a long-term vision for science and innovation, including an ambition for total pu blic and private investment in R&D to reach 2.5 per cent of GDP by 2014. The framework places particular emphasis on increasing the exploitation of the outcomes of research. 2.1.2 The recent past has seen substantial growth in real terms public spending on research, accompanied by substantial structural changes. For example, the government department responsible for the science budget has changed twice since 2007, new approaches to f unding research have been ( and are still b eing) introduced and a new public body (the Technology Strategy Board3) has been created to stimulate innovation in those areas which offer the greatest scope for boosting UK growth and productivity. 2.1.3 Against this background the primary channels used by government to fund basic research have remained relatively stable. These are: • four Higher Education Funding bodies (serving England, Scotland, Wales and Northern Ireland); and • seven Research Councils4 which between them cover the full spectrum of research ranging across the Arts and Humanities, Engineering, and the Economic, Social, Environmental, Biological, Medical and Physical sciences. 2.1.4 Figure 1 illustrates schematically how research is funded in the UK. Under the ‘ dual support s ystem’ f unds are channelled v ia the Research Councils (mainly in the f orm of research grants) and the four funding bodies5 which provide untied ‘QR’ (quality related) research funds as well as grants to contribute to the costs of academic research staff and infrastructure. The ‘ QR’ s tream makes up a substantial proportion; its level for each institution is based on a combination of the volume of research activity and quality ratings which, since the mid-1990s, have been determined through a series of Research Assessment Exercises (RAEs). The total funding being provided for research through the QR and research council streams is close to £5.4 billion this financial year. Table 1 shows the budgets allocated to the research councils over the last seven years and the volume of QR funding allocated by the funding councils over the same period. Table 1 also shows that there has been a relative shift from funding council allocations, to the Research Councils’ allocations. This is primarily a result of the introduction in 2005 of a new approach to funding basic research known as ‘full Economic Cost’ (fEC). For a more comprehensive description of how research is funded in the UK, and of fEC, please refer to Annex A ‘Making sense of research funding in UK higher education’ 2.1.5 In June 2007, the government signalled an intention to replace the RAE with a proposed new ‘Research Excellence Framework (REF). The 2008 RAE was the final such exercise to be held, and it is currently planned that the first REF assessment will take place 2014 to c over r esearch undertaken dur ing t he per iod 20 08–2013 i nclusive. The allocation of QR depends on the outcome of whatever process is used to assess research quality and volume, and the proposed REF, which is a more metrics-based approach and includes an element of impact assessment, has attracted a good deal of attention from the research base. The or iginal proposals have since evolved to a significant degree, and t he REF now includes a substantial element of Peer Review intended to place metrics-based quality indicators in context.

2 This was updated in 2006 and is available at http://www.bis.gov.uk/files/file29096.pdf , 3 See also section 2.4 and http://www.innovateuk.org/aboutus.ashx 4 For further information on the Research Councils se section 3 5 The Higher Education Funding Council for England (HEFCE); the Scottish Funding Council (SFC); the Higher Education Funding Council for Wales (HEFCW); the Department of Education & Learning in Northern Ireland (DELNI). These four bodies have similar, but not identical, policies. Examples of where they differ include the conversion of Research Assessment Exercise (RAE) results into cash, or in the imposition of student fees. 7

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 1 UK higher education institutions’ income from research grants & contracts and 6 funding council grants (Source: Research Information Network)

Table 1 Research Councils and Funding Bodies actual research budget allocations for 2004/05 to 2010/11 (£000s) 2010/11 increase against 2004-05 2005-06 2006-07 2007-08 2008-09 2009-10 2010-11 2004/05 AHRC 67,746 80,536 91,379 96,792 103,492 104,397 108,827 61% BBSRC 287,571 336,186 371,644 386,854 427,000 452,563 471,057 64% EPSRC 497,318 568,193 636,294 711,112 795,057 814,528 843,465 70%

ESRC 105,252 123,465 142,468 150,136 164,924 170,614 177,574 69% MRC 455,279 478,787 503,461 543,399 605,538 658,472 707,025 55%

search NERC 314,256 334,047 359,367 372,398 392,150 408,162 436,000 39% Councils Re PPARC 274,037 293,916 306,540 n/a n/a n/a n/a n/a CCLRC 127,940 167,004 182,256 n/a n/a n/a n/a n/a STFC n/a n/a n/a 573,464 623,641 630,337 651,636 62% total 2,129,399 2,382,134 2,593,409 2,834,155 3,111,802 3,239,073 3,395,584 59% HEFCE 1,078,750 1,249,254 1,340,843 1,413,014 1,458,447 1,582,636 1,603,000 49%

SFC 174,220 181,408 201,958 217,444 228,890 240,543 242,696 39% HEFCW 66,480 68,690 70,257 75,016 76,965 84,738 84,905 28%

Bodies DELNI 37,158 40,301 44,155 47,264 49,251 53,932 56,994 53% Funding Funding total 1,356,608 1,539,653 1,657,212 1,752,738 1,813,554 1,961,849 1,987,595 47% Overall total 3,486,007 3,921,787 4,250,621 4,586,893 4,925,356 5,200,922 5,383,179 54% 2.1.6 The Review is taking place at a time of considerable uncertainty due to t he prevailing financial climate and the a pproaching end of current f unding levels (last established f or

6 The difference between the figure shown coming from Research Councils in Figure 1 and the budget allocation shown in Table 2 is accounted for principally by Research Council Institutes. The figure shown coming from Funding Councils in Figure 1 is higher than the QR allocations shown in Table 2 because it includes non-recurrent (i.e. capital ) grants. Further information on the sources for the data in Figure 1 can be found at http://www.rin.ac.uk/our-work/research-funding-policy-and-guidance/making-sense-research- funding-uk-higher-education 8

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC the three financial years 2008/9-2010/11). The government has recently announced that the resource e lement of the ‘science bud get’ w ill be maintained i n cash terms over t he four years 2 011-14: which equates to approximately a 10% r eduction in r eal t erms. A separate d ecrease of 50% ov er t he p eriod in c apital i nvestment (currently ar ound £1. 8 billion per annum) was also announced, together with a decision to maintain the value of investment in medical research in real terms, both of which are factors that may affect the profiling of allocations to individual research councils. However, t hese allocations have not yet bee n an nounced (they are expected t o be published b y BIS in December), and therefore neither the magnitude nor the immediacy of any impact on EPSRC’s budget is yet known. The panel will be briefed with the latest available information during the review week.

2.2 Department for Business, Innovation and Skills (BIS) 2.2.1 BIS has overall responsibility for research and the health of the research base across the UK, as well as for higher education in England ( responsibility for higher education in Scotland, Wales and Northern Ireland i s devolved). A core B IS obj ective is t o promote world-class research in the UK, with a research base responsive to users and the economy, with sustainable and financially strong universities and public laboratories, and with a strong supply of scientists, engineers and technologists. The department was created in June 2009 by merging the Department for Innovation, Universities a nd Skills (DIUS) and the Department for Business, Enterprise and Regulatory Reform (BERR), which had themselves been established in 2007 following the abolition of the Department of Trade and Industry (which incorporated the Office of Science and Innovation to which the Research Councils reported). These structural changes are illustrated in Figure 2. Figure 2 Structural changes to Government departments that support research

2.2.2 BIS is headed by a Secretary of S tate (currently Vince Cable), with matters r elating t o universities a nd science being t he responsibility of a Minister of State ( currently David Willetts), both of whom attend Cabinet meetings. The Research Councils and the Technology Strategy Board are directly overseen by the Director General for Science & Research (DGSR), currently Professor Adrian Smith. Professor Smith was f or 10 years Principal of Queen Mary, University of London, and has been Professor of Statistics and Head of the Department of Mathematics at Imperial College. 2.2.3 The Treasury allocations to government departments reflects the Government's priorities and resource l evels ar e set, t ypically f or t hree years at a time, in t he light of i ndividual departmental bids setting out their needs and future plans. The research element of the BIS bid is based substantially on cases made by the Research Councils through delivery 9

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC plans which set ou t each council's f unding pr iorities and outline the ac tivities that t hey intend to undertake over next spending period. Once Treasury has established the overall funding l evel f or BIS t he DGSR and the Science M inister are responsible f or al locating the B IS “Science B udget”. Over t he three f inancial years 2008/09 – 2010/11, this has amounted to £11.24 billion, with around 80% allocated to the Research Councils and the balance allocated to the National Academies and other focussed areas of investment7. In addition to overseeing the research councils, DGSR’s remit covers: • Knowledge transfer, covering work to promote transferring good ideas, research results and skills between universities, other research organisations, business and the wider community to enable innovative new products and services to be developed; • the Science and Society programme, covering its work to improve science communications, boost the skills base and build public confidence in science, delivered primarily through the Science and Society Secretariat; • Science and innovation analysis, providing professional advice and evaluation to develop the evidence base underpinning science, technology and innovation policy. 2.2.4 An excellent overview8 of the nature, roles and responsibilities of the different UK Government bodies involved in supporting research in the UK has been prepared by the Research Information Network.

2.3 The impact of ‘Full Economic Costs’ on Research Council budgets 2.3.1 As not ed above, t he ‘ full E conomic C ost’ (fEC) funding m odel w as i ntroduced in 2005. Under t his approach r esearch c ouncils pr ovide 80% of t he fEC of successful research grant and f ellowship applications; a further 10% of t he c ost of undertaking r esearch is provided through the Science Research Investment Funding (SRIF) which is contributed to by the Research Councils but distributed by the higher education funding bodies. 2.3.2 The full economic cost of research varies by discipline and university, and the additional cost to the Research Councils varies accordingly. However, the Research Councils estimate that post-fEC grants cost on average 45% more than pre-fEC grants. This has meant that despite the growth in funding since 2005, there has been a slight reduction in the volume of new projects that can be supported, as illustrated by the data for EPSRC shown in Figure 3.

7 The Royal Society, the Royal Academy, the Royal Academy of Engineering, the Large Facilities Capital Fund, University Capital, the Higher Education Innovation Fund, Public Sector Research Establishments, the ‘Science & Society programme and other programmes. 8 See http://www.rin.ac.uk/our-work/research-funding-policy-and-guidance/government-and-research-policy- uk-introduction (also available in the ‘Research Council Overviews’ folder on the MSIR FTP site (login required). 10

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 3 EPSRC budget 2004/05 to 2010/11 (source EPSRC, figures are #millions))

2.4 The Technology Strategy Board 2.4.1 The Technology Strategy Board’s purpose is to promote, accelerate and invest in technology-enabled innovation in the UK. The Research Councils have an on-going and increasing portfolio of engagement with the Technology Strategy Board to help them achieve their objectives. The Technology Strategy Board has a budget of c£200 million a year to invest in research and development projects, and technology demonstrators. This is planned to include co-funding of at least £120 million from the research councils, and co-funding of £180 m illion from t he R egional D evelopment A gencies 9. T he T echnology Strategy Board is a non-departmental government body analogous to the Research Councils and is located on the same site.

2.5 The Capital investment Framework 2.5.1 Over the past decade the government has used successive funds to distribute additional capital to universities to support physical research infrastructure. The Joint Infrastructure Fund (JIF, 199 8-2000) was replaced by the Science Research Investment Fund (SRIF, 2002-2008), which has in turn been replaced by the Capital Investment Fund (CIF). The funding has been used to refurbish and/or build new research premises and to replace, renew and/or up grade of research equipment; there are notable examples of direct benefits to the mathematics research base, e.g. the Centre for Mathematical Sciences at the University of Cambridge and the Bristol Laboratory for Advanced Dynamic Engineering (BLADE).

2.6 Other sources of research funding 2.6.1 ‘Research users’ based mainly in industry and sometimes i n government departments are a major supporter of UK university research. Universities, departments or researchers may hold contracts with ‘research user’ partners as part of normal business. 2.6.2 The Wellcome Trust is a major charitable trust which spends over £600 million a year in the medical, biological a nd chemical areas, sometimes in partnership with t he funding and Research Councils. Other trusts and charities, such as the Wolfson Foundation (£35M per annum), the Nuffield Foundation (£9M per annum ), the Leverhulme Trust

9 The government has recently decide to abolish the RDAs; it is not clear at the time of writing where the responsibility for meeting this obligation will fall in the future. 11

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC (£40M per annum) and Cancer Research UK (£330M per annum) support research in a variety of areas. 2.6.3 The EU Framework Programmes (e.g. FP6, FP7) are also a significant source of funds for UK researchers, providing over £200 million in 2004-05. During FP6, which ran from 2002 to 2006, UK academia received approximately €1.4 billion; to date, under FP7, UK academia has been awarded approximately €1.2 billion. 2.6.4 European support is a lso available to researchers through Marie Curie funding and the ERC. 2.6.5 Various learned societies and professional bodies also sponsor research.

3. The Research Councils

3.1 RCUK 3.1.1 Research Councils UK (RCUK) is the strategic partnership of the UK’s seven Research Councils, which are listed (with brief descriptions of the scope of and scale of research and training they support) in Table 2. 3.1.2 The Research Councils are the UK’s principle vehicles for public funding of research and postgraduate training, and they use peer review extensively to ensure that funds are directed to the highest quality research. They also fund numerous research training opportunities, and work closely with business and on public engagement. Through their investments Research Councils seek to ensure that the UK is one of the most attractive locations i n the world f or science an d innovation. T he relationship of RCUK with BIS is illustrated schematically in Figure 4. 3.1.3 The shared priorities of the research councils are: • Research excellence - the delivery of independent, world class research with impact in globally competitive, networked institutions. • Impact - knowledge and expertise through investment in people, creativity and innovation, enabling the UK to maintain a technological edge, to build a strong economy, to exploit its unique cultural heritage and to improve the quality of life for its citizens. • Public engagement - to help ensure that young people are increasingly attracted to research-based careers. • Skilled people - maintaining a strong supply of skilled people for the research base, business and society. • Facilities and infrastructure – ensuring the UK research base has the access it needs to the full range of world-class research facilities in the UK or abroad. 3.1.4 Research c ouncil f unding (for w hich there is i ntense competition) is ge nerally i nvested through Universities and Research Council Institutes. Co-funding agreements between all Research Councils are in place to guard against proposals “falling between Councils” and thus encourage interdisciplinary research. 3.1.5 RCUK m aintains of fices i n C hina, I ndia and t he U SA to f acilitate international r esearch collaboration and to present a consolidated picture of UK research resources and expertise. Through a Brussels office10 RCUK also provides information and advice on EU research funding and promotes the involvement of UK researchers in EU research programmes.

10 The UK Research Office (UKRO) – see http://www.ukro.ac.uk/ 12

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Table 2 The Research Councils 2010/11 Research Council & Remit Investment budget Overview Arts and Humanities Research Council (AHRC) www.ahrc.ac.uk Languages, law, archaeology, Approximately 700 research awards £110m AHRC English literature, design, and around 1,500 postgraduate (ftp site login creative and performing arts. awards annually required) Biotechnology and Biological Sciences Research Council (BBSRC) www.bbsrc.ac.uk Life sciences, agriculture, Supports around 1600 scientists and £470m BBSRC healthcare, food, chemical and 2000 research students in (ftp site login pharmaceutical research. universities and institutes in the UK required) Economic and Social Research Council (ESRC) www.esrc.ac.uk The study of society and the Supports over 2,500 researchers and £180m ESRC manner in which people behave more than 2,000 postgraduate (ftp site login and impact on the world students. required) including longitudinal studies, economics and development studies. Engineering and Physical Sciences Research Council (EPSRC) www.epsrc.ac.uk Chemistry, physics, Research and training, with a £850m EPSRC mathematical sciences, portfolio of around 5400 grants (ftp site login materials science, information supporting approximately 8000 required) and communications technology, researchers and 7,500 doctoral engineering and high students annually. performance computing. Medical Research Council (MRC) www.mrc.ac.uk Mission to improve human Directly employs approximately 4000 £710m MRC health through excellent science staff and supports around 3,000 (ftp site login from molecular science to public researchers and 1400 postgraduate required) health research. students in universities, hospitals and its own institutes. Natural Environment Research Council (NERC) www.nerc.ac.uk Increases knowledge and Directly employs approximately 2500 £440m NERC understanding of the natural staff and supports around 1200 (ftp site login world – including climate researchers and 1000 postgraduate required) change, biodiversity and natural students in universities and its own hazards institutes. Science and Technology Facilities Council (STFC) www.stfc.ac.uk Responsible for the UK research Supports more than 700 current £650m STFC programme in astronomy, space research grants, over 600 students, (ftp site login science, particle and nuclear and UK access to world-wide required) physics, and for investment in facilities. large scientific facilities used across the research base.

Figure 4 Diagram of Government – Research Council hierarchy

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC 3.2 RCUK Priority Themes 3.2.1 The Research Councils are focusing on the following grand challenges (leading council in brackets): • Energy ( EPSRC): a co-ordinated portfolio of energy-related research and training t o address t he outstanding i nternational i ssues of c limate change and security of energy supply. The theme aims to sustain strong research in power generation and supply, and grow research in demand reduction, alternative energy vectors, transport, and security of supply. • Living with environmental change (NERC): a major interdisciplinary research programme to deliver new knowledge and tools to mitigate, adapt to and capitalise on environmental change. • Global Food Security (BBSRC): a multi-agency programme bringing t ogether Research Councils, Executive Agencies and Government Departments to address the challenge of meeting rising food demand in ways that are environmentally, socially and economically sustainable, and in the face of global climate change. • Global uncertainties: security for all in a changing world (ESRC): researching how to better understand, predict, detect and respond to five inter- related global threats to security - Poverty (and Inequality & Injustice), Conflict, Transnational Crime, Environmental Stress and Terrorism. • Lifelong health and wellbeing (MRC): by 2051, 40% of the population will be over 50 and 25% over 65. There are considerable benefits to having an active and healthy older population: this programme supports multi-disciplinary research into the factors that influence healthy ageing and wellbeing in later life. • Digital economy (EPSRC): multidisciplinary, user-focused research to build the national capacity t o harness t he power of ICT t o transform the way business operates, the way that government can deliver, and the way science is undertaken. • Nanoscience th rough engineering to application ( EPSRC): a multidisciplinary programme that aims to evolve nanoscience and promote the responsible development and management of nanotechnology.

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC

4. EPSRC N.B. T his section is descriptive in a general way of EPSRC structures, processes etc. Further contextual information which is specifically relevant to EPSRC investments supporting mathematical sciences is provided in from section 5 onwards.

4.1 Overview 4.1.1 The EPSRC is the UK's main agency for funding research in engineering and the physical sciences investing around £850 m illion a year to promote and enc ourage research and postgraduate training in the areas of chemistry, engineering, information and communications technologies, materials, mathematical sciences and physics. The EPSRC currently supports about 5,400 research, training and public engagement grants worth a total of £3.5 billion. In t he f inancial year from 1 April 2009 t o 31 Mar ch 2010, EPSRC managed the peer review of 1951 research grant applications and provided funding for 659, with a total value of around £898 million. The complete portfolio of awards can be explored at http://gow.epsrc.ac.uk/. 4.1.2 Figure 5 gives an ov erview of EPSRC’s s tructure ( in pl ace s ince 20 08), with indicative commitment levels for 2008-11. ‘Core’ funding commitment levels are agreed by Council following advice received from Advisory Bodies11, while the remainder is ring-fenced for investment through ‘ Mission’ pr ogrammes in t he cross-council pr iority themes l isted i n section 3.2 above (in some areas, e.g. energy, ‘core’ funding supports additional research relevant to priority themes)12. Table 3 outlines the coverage of the current core research programmes and their antecedents.

Figure 5 EPSRC - The Big Picture (2008-11)

4.1.3 EPSRC uses co-funding by programmes and between research councils to support research proposals spanning different programme areas and/or disciplines; the intention is t o t reat s uch proposals equally w ith proposals that fall w ithin a s ingle programme or council remit. EPSRC’s Cross-Disciplinary Interfaces programme, which develops and

11 See section 4.2 12 EPSRC does not have ring-fenced funds to support the ‘Global Food Security’ RCUK Priority Theme; support for this is distributed across the EPSRC portfolio and includes for example research into energy use during food production and water supply and control systems for crops. 15

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC manages opportunities at the interface between disciplines, programmes and organisations, has a remit query service in place to help academics whose research spans two or more of the research councils’ remits. 4.1.4 Unlike several other UK research councils EPSRC does not employ researchers directly, but invests mainly through research and training grants to universities, using independent peer review to identify the highest quality grant applications.

Table 3 EPSRC’s core programmes post April 2008 Current Programme Current Programme Research Areas Include: Information and Communications, computer science, people & interactivity, Communications semiconductor materials for device applications including modelling, Technology (ICT) & electronic & photonic devices Materials, Mechanical Aero & hydrodynamics, control & instrumentation, generic and Medical manufacturing, mechanical engineering, medical engineering, Engineering structural materials engineering including materials modelling at the macro-level, and synthetic biology Mathematical Sciences Pure mathematics, applied mathematics, statistics & applied probability, & mathematical aspects of operational research. Physical Sciences Organic, physical & inorganic chemistry, condensed matter, atomic & molecular physics, plasma physics, laser physics, optics, quantum information processing, surface science, soft condensed matter, & fundamental physical & functional aspects of materials, i.e. synthesis, growth & characterisation of materials & materials modelling at the atomic scale Process, Environment Process engineering, Built environment & civil engineering, Water, and Sustainability waste & coastal engineering, Transport management, Energy impact & adaptation to climate change (responsive mode only), Energy generation & distribution & electrical engineering (responsive mode only), Combustion, Science & heritage, Sustainable urban environment

4.2 Governance 4.2.1 EPSRC’s Council is the senior decision making body responsible for determining overall policy, priorities and strategy; it is also accountable for the stewardship of EPSRC's budget and the extent to which performance objectives and targets have been met. Council is advised by a Societal Issues Panel (SIP), a Technical O pportunities Panel (TOP) and a User Panel (UP). 4.2.2 TOP's main r ole is t o i dentify new r esearch opp ortunities ar ising f rom developments i n EPSRC’s mainstream disciplines and interdisciplinary areas. Members are drawn predominantly from the academic sector. UP represents the ‘user’ community13, advising on r esearch needs and t he value of EPSRC’s r esearch and t raining programmes. UP members are prominent individuals drawn from user sectors, including industry, commerce, government an d ed ucation. SIP aims to help EPSRC take m ore ac count of public thinking when deciding how to invest public funds in research and to help promote a healthier relationship between science and society in general. 4.2.3 EPSRC identifies programme level priorities with the help and support of Strategic Advisory T eams ( SATs) and through di alogue with t he wider research an d s takeholder community. Each programme has a SAT comprised of independent experts from academia and the stakeholder community who serve on average for 3 - 4 years. Specifically, the role of SATs: • alert EPSRC to new and emerging research and training opportunities, paying particular attention to multidisciplinary and international opportunities; • advise on the balance between research and training activities; • advise on areas or issues that need further exploration or investigation; and

13 The user community consists of technology supply-chain users and end-users who could benefit from EPSRC-funded activities, through take-up of research outputs or as potential employers. 16

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC • develop input on specific topics as requested by the Chief Executive. Whilst the SATs are programme-based, there is some cross-representation of disciplines. SAT members are appointed for one year, but a degree of continuity year-on-year is sought.

4.3 Defining programme priorities 4.3.1 EPSRC’s approach to meeting the challenges posed by t he ‘ Science and Innovation Investment Framework 2004-14’ is articulated in the 2006 Strategic Plan14 and the current Delivery Plan (2008-2011)15. These set out that 18% of the budget now supports the national challenges identified as cross-council priority themes, as well as the strategic basis f or m aking a major investment in new centres for doctoral t raining. At t he same time, as noted in section 2.3.2 above, the introduction of fEC reduced in real terms in the volume of research that could be supported during the period 2008-11. All these factors were taken into account by the SATs, TOP, UP and Council in agreeing individual programme funding levels for the period 2008-11. 4.3.2 More recently (May 2010) a new Strategic Plan 2010 has been issued setting out EPSRC’s long-term vision and goals for the next three to five years. EPSRC intends to deliver change through three clear goals over the next three to five years. These are: • Delivering Impact: supporting excellent research and talented people to deliver maximum impact for the health, prosperity and sustainability of the UK; building strong partnerships with organisations that can capitalise on our research and inform our direction; promoting excellence and impact, and ensuring it is visible to all; • Shaping Capability: shaping the research base, stimulating creativity and rewarding ambition to ensure it delivers high quality research for the UK, both now and in the future; the research portfolio will be actively focused on the strategic needs of the nation, such as green technologies and high-value manufacturing, and will retain the capability to tackle future challenges and capitalise on new opportunities; • Developing Leaders: committing greater support to the world-leading individuals delivering the highest quality research and meeting UK and global priorities; creating a career environment that supports them and allows others to benefit from their ability; fostering their ambition and adventure and ensuring they are connected with the best researchers worldwide. Knowledge Transfer Activities 4.3.3 In addition to the core programmes set out i n Table 3 and the mission programmes addressing cross-council priority themes EPSRC invests significant resource to facilitate knowledge transfer. Sector teams network with specific business sectors, learning about their needs, providing information about funding opportunities, and giving access to extensive k nowledge about t he ac ademic c ommunity. T he s ectors currently a ddressed are: Aerospace, Defence and Marine; Creative I ndustries; Electronics, C ommunications and IT; Energy; Infrastructure and Environment; Manufacturing; Medicines and Healthcare; Transport systems and Vehicles; and Cross-Cutting Themes (http://gow.epsrc.ac.uk/Sectors_Def.htm).

4.4 Support for Research Projects

Standard research grant funding process 4.4.1 For standard research grant proposals EPSRC uses a two–stage peer review process: in the first stage, proposals are sent in confidence to at least three reviewers; at least one of whom is chosen from three nominated by the applicant. The applicant does not know the identities of the selected reviewers, who prepare written comments and provide a number

14 See http://www.epsrc.ac.uk/SiteCollectionDocuments/Publications/corporate/EPSRCSP06.pdf 15 See http://www.epsrc.ac.uk/SiteCollectionDocuments/Publications/corporate/DeliveryPlanUpdatedFor2010- 11.pdf 17

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC of ‘ scores’ on v arious aspects of the proposal. These are returned t o EPSRC, and fed back anonymously to the applicant. 4.4.2 Proposals which are strongly supported by at least two reviewers are taken to the second stage in which a pplications are batched by broad programme area and presented for consideration by a prioritisation panel. Panels are programme-specific; membership is not static, but each panel has members with prior experience to ensure a consistent standard is applied. EPSRC convenes several such panels per quarter, with the annual number for any programme dependant on t he v olume of demand. Applicants are i nvited t o identify factual inaccuracies and to respond to questions raised by their reviewers; panels see the proposals, the reviewers’ reports and the applicants’ responses. 4.4.3 Using scientific excellence as the primary criterion, panels rank the proposals in order of priority f or f unding, and following each panel meeting t he head of t he r elevant EPRSC programme decides how much of their budget to commit, or in other words ‘how far down the list’ they can afford to fund. 4.4.4 Guidance on t he peer review pr ocess is available on the E PSRC w ebsite 16 and in t he EPSRC Funding Guide which can be downloaded from the website17. The guide covers all aspects of the application procedures for EPSRC research grants and fellowships, and sets out the standard terms and conditions governing any funds awarded. 4.4.5 The great majority of research proposals are processed as described above. Occasionally however proposals are received which it may not be appropriate to consider using the standard process. EPSRC recommends that such proposals are discussed with programme staff at an early stage so that a suitable peer review measures can be designed and implemented as necessary. Research support modes 4.4.6 EPSRC supports two modes of application: responsive mode (unsolicited research proposals submitted at any time by anyone eligible to apply) and targeted mode (proposals submitted in response to calls, characterised by closing dates and/or eligibility criteria). Responsive Mode 4.4.7 During t he f inancial year 2009-10, EPSRC r eceived 2726 proposals requesting i n total £1013 million in responsive mode; 716 of these, with a value of £272 million were funded, giving an overall funding rate of 26% by num ber and 27% by v alue.18 Of the proposals considered, 37% were for 36 months (53% were for between 30-42 months), and there is also a clear concentration in t he level of support requested in t he region of £250k to £350k (see Figure 6). 4.4.8 Responsive mode funding is very flexible. EPSRC fund projects ranging from small travel grants to multi-million pound research programmes. Academics can apply for funding to cover a wide range of activities, including research projects, feasibility studies, instrument development, equipment, travel and collaboration, and core support to develop or maintain critical mass. Targeted Mode 4.4.9 This mode supports proposals submitted in response to t argeted f unding mechanisms. There is often a preliminary outline stage, with full proposals only being invited from the higher ranked outlines. In general, funding rates in the targeted mode are higher than in responsive mode. Some calls for proposals involve a stage to build managed consortia and involve a number of universities collaborating on one proposal. Targeted mode includes schemes designed to support individual researchers, research groups and/ or research areas19.

16 See http://www.epsrc.ac.uk/funding/apprev/basics/Pages/prprinciples.aspx 17 See http://www.epsrc.ac.uk/funding/apprev/basics/Pages/fundingguide.aspx 18 See section 7.3.1 for comparative data relating specifically to the Mathematical Sciences programme 19 Such schemes include : Portfolio Partnerships, Platform Grants, Research Chairs, Postdoctoral Mobility Grants, Science and Innovation Awards, Follow on Fund and Discipline Hopping Awards 18

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 6 Distribution of requested support duration and value - Responsive mode applications 2009-10

4.4.10 During the f inancial year 2009 -10, EPSRC received 653 proposals requesting in total £411 million in targeted mode; 310 of these, with a values of £188 million were funded, giving an overall funding rate of 47% by number and 46% by value. Demand Management 4.4.11 Since 01 April 2009, EPSRC has implemented a policy of rejecting uninvited resubmissions of proposals. The number of proposals received by research councils had doubled over the previous two decades, placing great pressure on the peer review system and the measure was introduced to help alleviate this. Previous policy had allowed unfunded proposals to be resubmitted after six months, but the majority of such resubmissions were no more successful at their second attempt. 4.4.12 This policy applies to all investigators on grant proposals, including first grant and fellowship applicants, unless there is compelling evidence from peer review when a resubmission may be specifically invited by EPSRC. Repeatedly unsuccessful applicants are constrained to submitting one application only for 12 months and EPSRC expects them to review their submission behaviour. 4.4.13 A repeatedly unsuccessful applicant is defined as anyone who, as a principal investigator, has: • Three or more proposals within a two-year period ranked in the bottom half of a funding prioritisation list or rejected before panel (including administrative rejects), AND • An overall personal success rate of less than 25% over the same two years. The two year period is from the date on the letter the applicant is sent informing them of the outcome of their proposal (i.e. the date the decision is made). The personal success rate i s c alculated b y d ividing t he t otal n umber of funded proposals b y the t otal num ber submitted where decisions have been made in the past two years.

4.5 Sustaining Research Capacity 4.5.1 EPSRC offers substantial long-term grants to support leading groups in the UK in selected research areas. Three key support mechanisms are currently available; in each

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC case applicants are required to discuss their interest with EPSRC staff prior to submitting a proposal: • Platform grants: up to five years long, these provide flexible support designed to allow leading groups20 to retain key staff during funding gaps which may arise between research projects, as well as to carry out feasibility studies, longer-term research and international networking. • Programme grants: up to six years in length, these provide long term funding to support a suite of related research activities focussing on one major theme. Proposals are expected to be interdisciplinary and collaborative, but may also address key research challenges in a single discipline. • Strategic Packages: tailored funding to attract the very best international research leaders (sometimes called ‘star recruits’) by allowing them to maintain the momentum of their research programmes while they become familiar with and embedded in the UK research funding system. 4.5.2 In addition to the above currently available mechanisms EPSRC has over the past several years s upported m athematical s ciences research c apacity through Science and Innovation Awards, Multidisciplinary Critical Mass Centres, and investment in major Centres such as the Isaac Newton Institute (IMI) at Cambridge and the International Centre for the Mathematical Sciences (ICMS) at Edinburgh, all of which are described in more detail in sections 7.4.3 and 7.4.4.

4.6 Support for Doctoral Training 4.6.1 Since 2003/04, RCUK have invested over £ 100M in researcher development, with at least £50M of this being for postgraduate researchers (usually known as “Roberts” funding). In January 2009, the 1994 Group of universities produced a report that said “as a result of this [ Roberts] funding, skills training and related support for early career researchers are now firmly embedded within institutions” and that the amount, range and quality of the training and other support has improved considerably” 21. Research Councils are now seeking to embed support for researcher development within its normal funding mechanisms. There is currently an ongoing Review of Roberts funding which is due to report in October 2010. 4.6.2 In addition to the provision of skills funding v ia “ Roberts” funding, enhanced stipends have been payable to attract students in identified shortage areas including Engineering, ICT, Materials and Mathematics (particularly statistics and operational research). 4.6.3 EPSRC is a significant funder of doctoral training, currently supporting around half of all Research Council funded PhDs in t he UK, an d around 35% of UK research P hDs in Engineering and Physical Sciences. The size of the EPSRC studentship portfolio remained largely constant through the 1990s at approximately 7000; since then it has shown an upward trend and in 2009 was approximately 9,600 with around 25% engaged on collaborative research with industry. 4.6.4 EPSRC does not provide funding to students directly: innovations introduced by EPSRC during the l ast f ew years, such as Doctoral Training Accounts (DTAs) and Centres for Doctoral Training ( CDTs), have changed the way PhDs are supported and the way universities manage and deliver PhDs beyond the scope of EPSRC funding alone. There are currently 4 main r outes t hrough which EPSRC allocates pos tgraduate s tudentships as shown in Table 4:

20 Programme grants in mathematical sciences are focussed at the level of Department rather than research group – see section 8.3.6 21 http://www.1994group.ac.uk/researchenterprise.php 20

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Table 4 Main EPSRC PhD Training Funding Mechanisms Approach Main features Doctoral Training • Flexible 4 year blocks of funding (Doctoral Training Grants Accounts (DTA) (DTGs)); awarded annually to qualifying universities. • Aligned with a university’s research funding. • Allocation and recruitment of students is carried out by the University. • PhDs can be 3 to 4 years in length. • 45 Universities received DTGs in 2010. • Include a 10% target for collaborative training (CASE). • Selected universities are allowed to convert 10% of DTG to support International students via the International Doctorate Scheme. Centres for Doctoral • Funded after a competition. Training (CDT) • Centres funded for 5 years of annual student intakes. • PhDs are 4 years in length and students are trained in cohorts. • Combine PhD research with taught coursework. • EPSRC currently has more than 50 centres22. Industrial CASE • Enable companies to work with academic partners of their choice Studentships and to strongly influence the topic of research. • Allocated directly (or indirectly via agents such as Knowledge Transfer Networks). • >460 companies currently involved. • Companies contribute minimum 1/3rd of the EPSRC support – in cash. • In 2009/10 288 Industrial CASE awards made. Project Students • Currently awarded on research grants. 4.6.5 EPSRC recently invested £280M in new Centres for Doctoral Training and student numbers are projected to grow significantly; it is expected that an additional 1,800 students will be trained by 2013 via t he 4 -year cohort-based training. Evidence f rom earlier funded CDTs indicates that this approach adds real value, producing students who are highly competitive internationally, are more productive, have higher impact in academic circles, and a breadth and depth of knowledge which allows them to be highly mobile and flexible in subsequent careers. CDTs are intended to provide highly innovative an d ex citing training e nvironments, drawing upon the research ex cellence of their host universities. A significant number of CDTs have industrial collaborations, with more than 570 Industrial organisations recorded as supporting them. Around a quarter of the new CDTs are Industrial Doctorate Centres (IDCs), where students spend up to 75% of their time in an industrial environment with potentially much higher [economic] impact to said partner. Some 250 c ompanies ar e i nvolved with I DCs l everaging approximately £100 million of co-funding. 4.6.6 EPSRC s upported P hDs are al most as l ikely t o enter business or t he p ublic s ector as academia on completion. A r ecent r eport on f irst destinations of UK domiciled doctoral graduates indicated that in engineering and physical sciences 42% entered the education sector with significant proportions entering manufacturing ( 25%) and business finance and IT (20%)23.

Masters 4.6.7 EPSRC is a niche provider of Masters-level postgraduate training, having historically supported no more t han 5 -10% of UK Masters-level training c ourses offered w ithin our remit area. In 2006-07 only about 3-4% of qualifying Masters in EPS disciplines received EPSRC support.

22 See also section 7.5.14 23 see ‘What Do PhDs Do? – Trends’ available from http://www.vitae.ac.uk/policy-practice/14772/What-Do- PhDs-Do-Trends.html 21

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC 4.6.8 In 2009 EPSRC decided to focus most of its support for Masters-level training on activities which use it as a means of developing highly skilled doctoral-level researchers, rather than those which view it as an end in itself. Masters-level training is therefore delivered via the CDTs, DTAs and through Taught Course Centres funded by the Mathematical Sciences programme. It is estimated that around £11M per year will be invested in Masters-level training over the next five years, with more than 3,000 students benefitting from this funding in that period.

4.7 Support for People EPSRC operates a number of schemes intended to support academic researchers at different stages of their careers. Fellowships generally provide funding to help researchers spend their time on research rather than administration and teaching, while other mechanisms are designed to help researchers move discipline or establish research groups of their own. Figure 7 shows in a schematic arrangement how the balance of available support is distributed across the career path aggregated across all EPSRC programmes.

Figure 7 People Support across the Career Path - 2009 Portfolio, headline numbers for all Programmes (EPSRC data)

Early Career 4.7.1 The Postdoctoral Research Fellowship (PDRF) is a 3-year award available to researchers who have recently completed a PhD to help them establish an independent research career. It is available only in mathematical sciences, theoretical computer science, cross- disciplinary interfaces ( physical sciences and/or engineering with the life or social sciences), theoretical physics and engineering. 4.7.2 Postdoctoral Mobility follow-on funding is available to encourage the effective transfer into different research disciplines of knowledge gained by post-doctoral Research Assistants (PDRAs) working on existing EPSRC research grants and to enhance the competencies and career opportunities of these researchers. Typically, proposals are sought during the final year of an EPSRC grant for a 12 months ‘follow-on’ grant. An example request might be to transfer a PDRA supported by the Mathematical Sciences programme into a more applied discipline/department. Any proposal that involves transfer of knowledge between two distinctly different disciplines can be considered, however the balance of the research proposed for the follow-on year must be within EPSRC remit. 4.7.3 The Career Acceleration Fellowship (CAF) offers up to f ive years’ funding for talented researchers with 3-10 years of postdoctoral experience at the time of application. Available across the whole of EPSRC’s remit, CAFs are highly sought after with on average 450 applications for around 25 available awards annually. Applicants are 22

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC expected to have a strong research publication record and to be able t o demonstrate independence from their current/previous supervisors. 4.7.4 First Grant Scheme This scheme is restricted to researchers who obtained their PhD (or equivalent pr ofessional qu alification) within the previous 10 years and who are within 3 years of their first academic appointment. Applicants can apply at any time and there are no closing dates, but there is cap on the f EC project cost o f £125k and a maximum duration of 2 years. Mid-Career 4.7.5 The Leadership Fellowship scheme provides up to five years of funding for researchers with the most potential to develop into the UK’s future research leaders. Applicants must have a permanent academic post at a UK higher education institution. As with CAFs, competition is intense, with on average 250 applications for around 20 awards annually. Other 4.7.6 EPSRC supports the most successful researchers through the later stages of their careers through mechanisms such as Programme and Platform Grants which are described in section 4.5.1 above. 4.7.7 The Mathematical Sciences programme is taking part in the ‘Dream Fellowship’ scheme that is being piloted across the majority of the EPSRC remit. These awards will support talented individual researchers with adventurous and potentially transformative research ideas. T he a ward will provide 12 m onths f unding enabling researchers to take time out from t heir normal activities, to g ive t hem t he f reedom t o gai n new knowledge of novel creative problem solving techniques, explore new radical ideas and develop new ambitious research directions.

4.8 Support for Knowledge Transfer & Exchange (KTE). 4.8.1 EPSRC encourages knowledge transfer and exchange in all its funding schemes to improve the uptake and exploitation of research outcomes, and since May 2009 has required all researchers submitting funding applications to include a ‘Pathways to Impact’ statement to encourage applicants to explore, from the outset, who could potentially benefit from their work in the longer term, and consider what could be done to increase the chances of their research reaching those beneficiaries 24. The following bespoke mechanisms are also supported: KTE ‘beyond’ academia 4.8.2 Follow-on fund This offers up to 12 months’ funding for EPSRC grant holders to work on the very early stages of turning research outcomes into a commercial proposition. There is an annual call for proposals issued in the autumn. 4.8.3 Innovation and knowledge centres (IKCs). Centres of excellence with five years' funding to accelerate and promote business exploitation of an emerging research and technology field. They provide an entrepreneurial environment where researchers, potential customers and skilled professionals from academia and business can work side by side to scope applications, business models and routes to market. Four have been funded to date as listed below: • Advanced Manufacturing Technologies for Photonics and Electronics - Exploiting Molecular and Macromolecular Materials at the University of Cambridge. • Ultra Precision and Structured Surfaces at Cranfield University • Centre of Secure Information Technologies at Queen’s University Belfast • Regenerative Therapies and Devices at the University of Leeds 4.8.4 Knowledge Transfer Accounts (KTAs): these are a recent innovation by EPSRC intended to overcome barriers to the exploitation of EPSRC-funded research by fostering an environment in which impact and knowledge t ransfer/exchange are highly valued and encouraged. Twelve KTAs have been funded to support a range of activities including

24 See http://www.epsrc.ac.uk/funding/apprev/preparing/Pages/economicimpact.aspx 23

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC proof-of-concept, ent repreneurship t raining, networking, pe ople exchange (including t he popular ‘Knowledge Transfer Partnerships’ (KTPs)25), business relationship-building and start-up generation. Leveraged use of KTA funding with resources from other agencies is welcomed. To support business-academic interactions at Universities which do not hold a KTA EPSRC has more recently provided ‘Knowledge Transfer Secondment’ (KTS) grants to fund short-term exchanges of researchers to or from organisations that can exploit EPSRC-funded research results. Decisions about t he use of KTS f unding are made by the university holding the grant to ensure maximum flexibility and responsiveness. KTE ‘within’ academia 4.8.5 EPSRC encourages fruitful exchange between academics and stakeholders and to promote movement within the science community. The following schemes help promote such activities: • Overseas travel grants: small grants of short-term funding for visits to non-UK centres to study techniques or develop international collaborations. • Visiting researchers: up to one year’s funding is available to support visits to UK research organisations by scientists or engineers of acknowledged standing. They may be from anywhere in the world (including from within the UK). • Workshops and schools: funding is available for national and international (‘bilateral’) research workshops, summer schools to train postgraduate students, and similar events. • Networks: funding to allow researchers, industry and other groups to develop or enhance collaborations through workshops, visits, travel and part-time coordinators. • Postdoctoral Mobility – see section 4.7.2 above

4.9 Support for Public Engagement. 4.9.1 Engaging with the public enables research to be informed by public values and attitudes. It also ensures that the outputs of research, in particular their implications and applications, are shared with the society that supports t hem. EPSRC encourages is researchers to get involved in public engagement (PE) activities throughout all stages of the research process. 4.9.2 EPSRC’s public engagement aims and strategy complement RCUK’s vision f or publ ic engagement; together with the other Councils EPSRC supports and contributes to RCUK’s public engagement activities and initiatives such as Researchers i n Residence and the Beacons for Public Engagement. 4.9.3 Until 2010, EPSRC ran two funding schemes specifically for public engagement activities: Partnerships f or Public E ngagement ( PPE) a nd Senior Me dia F ellowships ( SMF). The PPE scheme w as des igned t o pr ovide r esearchers w ith t he s upport a nd op portunity t o develop their own public engagement activities related t o t heir research interests. In 2009, the PPE Starter grants were introduced; these were small scale grants (of less than £20k) for researchers with limited public engagement experience. The SMF scheme was established t o enable l eading researchers to devote time to de veloping a higher media profile. 4.9.4 As part of the forthcoming delivery plan, EPSRC has been working towards embedding public en gagement as an integral element of the ac tivity we support. From April 20 11, public engagement will be funded through inclusion of PE activities as Pathways to Impact or as integral work packages in applications for research funds. Universities will be encouraged to take an institution-wide view of public engagement within the ‘delivering impact’ agenda. Within EPSRC, mainstream programmes and themes will take on greater responsibility for the public e ngagement with research as an integral aspect of their business. The UK Research Funders' Concordat for Engaging the Public with

25 A KTP is a three-way project between an academic, a business and a recently qualified person. Projects last between 10 weeks and 36 months, and enable academics to participate in collaborations with businesses that require up-to-date research-based expertise; for further information see http://www.ktponline.org.uk/strategy 24

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Research ( to be launched on 7 th December 2010) will provide context, setting out t he expectations for and responsibilities of universities.

5. Mathematical Sciences Research in the UK

5.1 Introduction

The aim of the following sections is to offer evidence that relates specifically to UK research in the Mathematical Sciences, as well as to the role of EPSRC in supporting this research. The structure of the document is as follows. • Sections 5.3 and 5.4 provide an overview of research in the mathematical sciences, including the context in which research in this area is situated with respect to other research areas. • Section 6 gives an overview of funding for mathematical sciences research in the UK. This section includes information on funding of research in the mathematical sciences from different sources, including the research councils and the different Programmes within EPSRC, as well as data on the distribution of funding for mathematical sciences research (in terms of institutions and departments). • Section 6.5 focuses on the training aspects, and provides an overview of the UK support for training in the mathematical sciences, including the different mechanisms offered by EPSRC. • Section 7 analyzes the support mechanisms that are centred on people. This section includes information on fellowship schemes, as well as support for researchers at all stages of their careers. • Section 7.5 provides demographics data for the mathematical sciences community, including information on gender balance, and age profiles for mathematical sciences researchers in the UK. • Section 8 provides information about the international engagement of the mathematical sciences community. The analysis in this section is based on EPSRC data. • Section 9 gives a short overview of the role of knowledge transfer and public engagement in relation to research in the mathematical sciences. • Section 10 provides an overview of bibliometric data related to UK research in the mathematical sciences.

The sections described above are supplemented by Annexes which can be found at the end of the document and which provide additional data.

For further information or advice about the data please contact EPSRC, Ben Ryan: [email protected]

5.2 Analysing the Evidence As mentioned in the Preface, the data provided in this document has been obtained from a range of different sources. The following points must be considered when drawing conclusions: • Every agency classifies its portfolio in a different way: therefore, where data is provided by different agencies, direct comparison will not be possible. • HESA data on academic research staff and funding is categorised by “Cost Centres” which map broadly to academic departments – there is, however, inevitably some blurring as University structures do vary and it is left to individual universities to decide on the details of which staff and other data to return against each cost centre.26 Throughout this document we have used data for the ‘Mathematics’ cost centre as it is the most closely aligned to mathematical sciences and is likely to represent most ‘mathematical sciences’ departments in universities, but it should be recognised that this will exclude much of the mathematical sciences research being carried out in departments such as physics, engineering and biology.

26 A description of the HESA Cost Centres is available at http://www.hesa.ac.uk/index.php/component/option,com_studrec/task,show_file/Itemid,233/mnl,10051/href, a%5E_%5ECOSTCN.html/ 25

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC • The categories used by the Higher Education Statistics Agency (HESA) for recording student data are different to those used for recording academic research staff and funding data. Student data is categorised according to subject of study: individual institutions are responsible for submitting student data and code the submissions according to the classification of the students’ research activity. • Financial mathematics and mathematical biology are not identified as separate sub-areas in EPSRC data, although they are at least partially included within the remit of EPSRC. These areas are partly covered by mathematical sciences funding and to this extent it is possible we have included them in the data presented here.

5.3 The Character of Mathematical Sciences Research 5.3.1 Research in the mathematical sciences is pervasive. Mathematical methodology underpins all other scientific disciplines, and it is challenging to define the boundaries of research which crosses t he boundaries between disciplines a nd takes pl ace in a wide range of departments and institutions. It is therefore extremely difficult to fully map research in the mathematical sciences, and t he available data provides only a partial representation of the whole landscape. 5.3.2 A significant proportion of mathematical research is funded by universities from sources such as QR funding, scholarships and consulting for industry; a situation more applicable to the mathematical sciences than the experimental sciences since there is little need for extensive infrastructure. Hence research in the mathematical sciences is spread more extensively through the university system. We do not have data on the extent of university funding for research in the mathematical sciences. 5.3.3 The EPSR C Mathematical S ciences programme funds r esearch i n novel mathematics, spanning pure and applied mathematics, statistics, probability and the mathematical foundations of operational research. The Programme does not generally fund the application of existing m athematical knowledge to other disciplines and research areas, which are typically taken up by the application areas themselves. Research councils have a mechanism in place to evaluate research projects where common interests exist.27 5.3.4 In the mathematical sciences, research meetings provide an important method for facilitating research (as well as giving PhD students access to international experts in their area of study). The Isaac Newton Institute ( INI) at Cambridge, t he I nternational Centre f or t he Mathematical S ciences ( ICMS) at Edinburgh, t ogether with t he Durham and Warwick Symposia all have strong reputations for excellence in attracting international experts to the UK to participate in topic-based programmes. EPSRC funds the INI and ICMS as national facilities for the whole U K mathematical sciences community. 5.3.5 Similarly, the mathematical sciences community organises many other forums for bringing researchers together to examine state of the art research, forge new collaborations and open up prospects for new research. These include research networks, workshops, conferences and training events.

5.4 The boundaries of the EPSRC Mathematical Sciences Programme 5.4.1 The above description of mathematical sciences research naturally has an impact when one tries to define the boundaries to t he EPSRC Mathematical Sciences programme. Indeed, when trying to define said boundaries the pervasive nature of research in mathematics becomes evident. In the rest of this section we give a short overview of the Programme, focusing in particular on the relationships between the Programme and other research areas. More detailed information on the Mathematical Sciences programme (including the programme’s strategy and the research landscape funded by the programme) can be found in Annex C. 5.4.2 An i ndication of the p ervasive n ature of the m athematical s ciences c an b e obtained by considering the extent in which researchers funded by the Mathematical Sciences programme also receive funding from other programmes (and other research councils). In the last 4 years the Mathematical Sciences programme has funded more than 650

27 See section 4.1.3. 26

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC different researchers28, and in the same time-frame, almost 45% of them have also received funding from a different programme.

Table 5 Number of common Investigators between the Mathematical Sciences programme and other EPSRC programmes – all fEC grants awarded since 2005 (EPSRC data)29 Investigator % Mathematics Programme number Investigators Cross-Discipline Interface 144 21.8% Materials, Mechanical and Medical Eng 140 21.2% Information & Communication Technology 138 20.9% Process Environment and Sustainability 119 18.0% Physical Sciences 96 14.5% Other 63 9.5% Energy 35 5.3% Infrastructure & International 33 5.0% Digital Economy 24 3.6% Innovative Manufacturing 12 1.8% Nanoscience to Engineering 1 0.2%

5.4.3 Figure 8 gives a pictorial representation of the data presented in Table 5. Each research programme in EPSRC is represented by a rectangle in the diagram; the size of the overlap between each programme’s rectangle and the Mathematical Sciences’ rectangle represents the number of common investigators between the two programmes. The overall size of Mathematical Sciences rectangle is also in proportion with these overlaps; however this is not true f or the s izes of the other programmes’ rectangles, nor for their overlaps with each other.

Figure 8 Overlap between the Mathematical Sciences programme and other EPSRC programmes (EPSRC data)

28 This number includes Principle and Co-Investigators on grants that have received at least partial funding by the Mathematical Sciences Programme. 29 It should be noted that in the table each researcher is counted only once per research programme (independently of the number of grants for which they have received funding by the programme). The same investigator, though, could be counted multiple times if they have received funding from more than one programme. 27

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC

5.4.4 An analogous analysis can be done in relation to the overlap between the EPSRC Mathematical Sciences programme and the other research councils. The available data is shown in Table 6.

Table 6 Number of common investigators between the EPSRC Mathematical Sciences programme and other research councils on fEC grants awarded since 2005 (Research Councils data) Number of Investigators also % of EPSRC Mathematical Research funded by EPSRC Mathematical Sciences programme Council Sciences programme Investigators BBSRC 28 4.3% NERC 15 2.3% STFC 39 6.0%

5.4.5 The data shows that more than 80 researchers that have received funding from the EPSRC Mathematical S ciences pr ogramme (more t han 10% of t he t otal n umber) ha ve also received some funding by at least another research council. The overlap is the greatest with STFC. This does not come as a surprise, since the mathematical physics portfolio of the EPSRC Mathematical Sciences programme spans many areas (such as general relativity, quantum f ield t heory and string t heory) t hat are c lose t o t he remit of STFC. I t should be noted t hat only 2 investigators have received funding both by the Mathematical Sciences programme and by 2 other Councils.

6. Funding for Mathematical Sciences Research in the UK

6.1 Overview 6.1.1 As mentioned in section 2, research in the mathematical sciences is supported by funding from a variety of sources. While data on the value of externally awarded research grants and contracts is readily available at the level of ‘mathematics cost centre’ in HEIs, there is no such disaggregated data on the value of ‘internal’ support provided by the universities themselves from the QR funding allocated to them by the Funding Councils30. Thus it is possible to assess the relative importance of the individual external sources in the context of total external support received, but it is not possible to set these in the context of the total support given to mathematical sciences from ‘ internal’ and external sources. Individual HEIs may be able to provide the panel with further information on this aspect on a case-by-case basis. 6.1.2 The aim here is therefore to give an overview of funding for UK research in the mathematical sciences beyond that provided by individual universities. A s explained in previous sections, a number of caveats (including those listed in section 5.2) should be considered while analysing the data provided. 6.1.3 Aside from the contribution coming from the Funding Councils, research in the mathematical sciences is funded by a variety of different sources, including (but not limited to) the Research Councils. 6.1.4 Table 7 and Table 8, which list the value and source of research income (from contracts and grants) received by all UK Higher Education Institutions (HEIs) in the years 2004/05 to 2008/09 for the Mathematics cost centre and all cost centres respectively, show that the total research funding has increased steadily in the last 5 years. It is however necessary to take into account the following when interpreting this data: • In September 2005 full economic costing (fEC)31 was introduced. All grants from Research Councils announced after April 2006 had an average increase of 45%, though there was considerable variation by subject. Since grants that had

30 See section 2.1.4 for more information about dual support, and QR funding. 31 See Annex A for more information about full economic costing (fEC). 28

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC already been awarded were not affected by the introduction of fEC, the effect of such a change is spread over time (as older grants end and are substituted by new grants with fEC). Steady state in fEC was not expected to be achieved until 2010/11. • The data in Table 7 and Table 8 refer to university income and reflects the level of expenditure by Research Councils. Elsewhere in this document data on EPSRC investment is generally stated in terms of ‘commitment’, i.e. the value of new investments entered into during a particular period.32 • The data in tables Table 7 and Table 8 does not separately indentify funding by the Mathematical Sciences programme. Over the period programme funding has been in decline but the steady increase in total research council funding reflects other investments that involve the mathematical sciences, particularly EPSRC funding for Science & Innovation (S&I) awards.33 • Note also that S&I awards and some other investments were not specific to the mathematical sciences. Nevertheless, their impact (in terms of the relative increase in funding) is more significant for the mathematical sciences because core funding in this area is much lower.34

Table 7 UK university research funding - 'Mathematical Sciences' cost centres (£M) (excludes QR funding) (HESA data) 2004-05 2005-06 2006-07 2007-08 2008-09 5-yr incr. Research Councils 23.4 (62%) 28.7 (63%) 34.3 (68%) 41.7 (70%) 47.2(69%) 102% UK industry35 2.9 (8%) 2.9 (6%) 2.3 (5%) 2.7 (5%) 3.3 (5%) 15% UK govt. & health auth. 4.4 (12%) 3.5 (8%) 4.0 (8%) 4.4 (7%) 4.4 (6%) 0% UK-based charities 2.4 (6%) 2.4 (5%) 2.9 (6%) 3.4 (6%) 3.6 (5%) 55% EU government 3.1 (8%) 4.6 (10%) 4.6 (9%) 4.6 (8%) 4.5 (7%) 43% EU other sources 0.1 (0%) 0.1 (0%) 0.2 (0%) 0.3 (1%) 0.7 (1%) 421% Other overseas sources 1.1 (3%) 2.4 (5%) 1.5 (3%) 1.8 (3%) 3.7 (5%) 249% Other sources 0.5 (1%) 0.8 (2%) 0.6 (1%) 0.7 (1%) 0.8 (1%) 57% Totals 37.8 45.4 50.3 59.6 68.2 80%

Table 8 UK university research funding - all cost centres (£M) (excludes QR funding) (HESA data) 2004-05 2005-06 2006-07 2007-08 2008-09 5-yr incr. Research Councils 919.7 1,064.4 1,140.7 1,349.3 1,521.9 65% UK industry35 240.9 254.5 289.1 295.5 311.8 29% UK govt. & health auth. 558.7 568.7 596.0 629.3 698.4 25% UK-based charities 698.7 720.4 765.4 824.1 894.3 28% EU government 200.8 217.4 256.2 277.5 322.9 61% EU other sources 34.2 40.2 45.8 51.7 66.5 95% Other overseas sources 150.9 171.2 198.2 216.5 255.3 69% Other sources 59.6 55.8 55.6 51.8 49.2 -17% Totals 2,863.5 3,092.5 3,347.1 3,695.7 4,120.3 44%

32 EPSRC accounts for and reports to government in expenditure. Generally, expenditure data is more consistent, in the sense that data from different sources is likely to be based on expenditure and is more readily comparable. However, expenditure in EPSRC terms represents the results of investments made over several past years (because grants spend over an extended period) and so can be difficult to interpret. By contrast, reporting commitment year-on-year (i.e. new investment) shows immediately whether funding is rising or falling. 33 See section 8.4.3 for more information about Science and Innovation awards. 34 In 2005/06, for example, EPSRC funded four S&I awards in the mathematical sciences totalling £14M. This figure equated to some 60% of research funding by the Mathematical Sciences Programme in that year. EPSRC also funded 3 S&I awards in the physical sciences totalling £11M; this equated to less than 10% of research funding by the Physical Sciences Programme that year. 35 UK Industry includes public corporations 29

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC 6.1.5 A rough estimate of the funding of the different research councils to mathematical sciences-related r esearch can be obtained by c onsidering the f igures i n Table 9 which represent the 2008/09 Research Council component of the UK university research funding for the “Mathematical Sciences” cost centre shown in Table 7 above. 6.1.6 While Table 7 shows that research councils are collectively the largest single s ource of research grant and contract income received b y ‘ Mathematical Sciences’ cost centres, funding more than 45% of the total received i n 2008/09, Table 9 shows t hat EPSRC provided almost 70% of the funding from this source.36

Table 9 Research Councils’ contribution to UK university research funding - 'Mathematical Sciences' cost centres (HESA data)37 Funding (£ M) % of total funding AHRC 0 0% BBSRC 2,894 6% EPSRC 32,295 68% ESRC 1,770 4% MRC 708 2% NERC 1,335 3% STFC 5,759 12% Other 2,407 5% Totals 47,168 100%

6.2 EPSRC support for Mathematical Sciences Research 6.2.1 The most significant individual component of EPSRC research funding to the mathematical sciences comes from the Mathematical Sciences programme38. 6.2.2 Table 10 below shows the budgets for EPSRC Programmes in the last 5 years

36 Recall that support drawn from QR funding is not included in this analysis – see section 7.1.1 37 This data is only available for 2009/10. The percentages are not expected to vary significantly in other years. 38 Details on the activities run in the areas of the mathematical sciences are included in Annex C 30

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Table 10 EPSRC research budget by programme: new investment39 (£M, 2005/6 - 2009/10) (EPSRC data). Programme Current Programme 40 2005/06 2006/07 2007/08 2008/09 2009/10 (pre-2008) Life Sciences Cross-Disciplinary Interface 26.9 31.2 22.5 36.8 41 Interface Digital Economy - - - - 46.2 19.9 Energy Energy 26.3 41.2 56 57.9 43.8 ICT 88.8 87.2 83.3 75.6 72 Infrastructure & International - - - - 21.3 8.2 41 42 Mathematical Sciences 16.4 21.5 24.2 15.3 14 Materials, Mechanical & Medical Eng 60.7 63 Engineering 77.7 96.8 94.4 Process Environment & Sustainability 27 30.6 Nanoscience to Engineering - - - - 21.1 5.6 Next Generation Healthcare - - - - 3.7 7 Chemistry 48.2 51.7 42.8 Physical Sciences Materials 47.4 52.6 56.3 100.4 88 Physics 38.2 49.2 33.1 User-Led Research - - - - 82.5 68 6.2.3 Researchers in the mathematical sciences have been successful throughout this period in obtaining funding from other research programmes at E PSRC 43, emphasising the underpinning role and impact that mathematics has across all EPSRC research areas. 6.2.4 Over the last five years, the Mathematical Sciences programme was the lead funder on slightly m ore than 700 projects, and 123 of these received some element of co-funding from another EPSRC programme. These contributions cover more than 15% (in terms of number of grants) and 9% (in terms of value) of the research funded by the programme. Table 11 gives a breakdown of this co-funding by EPSRC programme

Table 11 Contribution (in value and number) of other EPSRC research programmes to new Mathematical Sciences programme grants (years 2005/06-2009/10) (EPSRC data) Total Value of Total Contribution Total Funding Programme Contribution as % of grants Number of (£M) value Grants Cross-Discipline Interface 3.4 19% 28 Energy Research Capacity .2 50% 1 Information & Communication Technology 3.3 8% 34 Materials, Mechanical and Medical Eng 2.2 7% 23 Physical Sciences 2.5 9% 19 Process Environment and Sustainability 1.2 6% 18 Grand Total 12.7 9% 123 6.2.5 Table 7 gave an indication of the extent to which research in the mathematical sciences is supported by a range of bodies beyond EPSRC. This is also confirmed by the data in Table 12, which shows the contribution of other organisations to projects funded by the Mathematical Sciences programme.

39 The figures give the value of new grants funded per year, and do not compare directly with the expenditure data presented in Table 8 or Table 9. See footnote 32 for more information 40 In 2008 there was a restructuring of EPSRC. This column shows which former Programmes correspond to current Programmes; “-“ indicates that the current programme did not exist before restructuring. 41 The 2007/08 figure includes a one-off commitment of £6M for the renewal of the Isaac Newton Institute that was given by council in addition to the mathematical Sciences Programme budget. 42 £2million of this figure has been invested to supporting training. 43 As discussed in section 4.1.3 projects spanning more than one programme area can be funded by two (or more) EPSRC programmes. 31

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Table 12 Co-Funding by other public sector organisations to Mathematical Sciences programme grants awarded between 2005/06 and 2009/10 (EPSRC data)44 Co-funding Total Value Number of Organisation Contribution (£k) Grants BBSRC 378 2 DSTL 337 8 ESRC 49 1 Inst. of Actuaries 143 11 NERC 865 3 Total 1,772 25 6.2.6 Figure 9 and Figure 10 give additional illustration of the support given by EPSRC to research in the mathematical sciences. Figure 9 shows the value of current EPSRC research grants and fellowships awarded by other EPSRC programmes (i.e. not Mathematical Sciences) to mathematics and statistics departments45. The total value of these contributions amounts to almost 50% of the total funding received by these departments from EPSRC. As noted, the data refers to a wide set of departments; Figure 10 is obtained by restricting the data to only those departments focussing on mathematical sciences (including statistics) 46. The value of current EPSRC research grants and fellowships held by this restricted set of departments amounts to approximately £100M, of which 70% comes from the Mathematical Sciences programme. Figure 10 shows the distribution by programme of the remaining 30%.

Figure 9 EPSRC Funding (£M) to Mathematics and Statistics Departments – excluding Mathematical Sciences programme funding (EPSRC data, current grants)

44 Institute of Actuaries contributions resulted from targeted activity on quantitative finance which ran until 2005/06. 45 The classification of departments in the UK is left to individual universities. For this reason, it is difficult to define “Mathematics Departments”. In the data presented in Figure 9 we have considered all departments including an element of mathematics. The list clearly includes all the departments of (pure/applied) mathematics and the departments of statistics, but it also includes other departments such as, for example, the School of Mathematical and Physical Sciences, the Faculty of Computing Info Systems and Maths, the School of Maths, Statistics and Actuarial Science and the School of Computing, Engineering and Maths. This choice was made in order to include as many departments in which research in the mathematical sciences takes place. 46 This restriction implies only considering data from departments of (pure/applied) mathematics and/or statistics. 32

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC

Figure 10 EPSRC Funding (£M) to Mathematics and Statistics Departments (EPSRC data, current grants). Data referring to a restricted set of Departments.

6.3 Distribution of EPSRC Mathematical Sciences Funding 6.3.1 The largest single element of EPSRC funding to mathematics research comes from the Mathematical Sciences programme. Support is given both by means of “ Responsive Mode” and through m anaged calls and activities. During the financial year 2009/10, the programme received 238 proposals in responsive mode requesting in total £43 million; 91 of these, with a combined value of £12 million were funded, giving an overall funding rate of 38% by number and 28% by value. 6.3.2 Figure 11 shows the distribution of Mathematical Sciences programme funding by department type. As expected, the majority of the funding is directed towards mathematics and statistics departments, but a significant part of funding (more than 10%) is also directed towards departments focussing on research in other areas of science.

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 11 Mathematical Sciences programme funding (£M) by department (current grants) (EPSRC data)47

6.3.3 Within the remit of EPSRC, there are also a significant number of research projects whose main focus is within the scope of different research programmes, but which have a significant element of mathematics research included. Table 13 shows the data related to funding that the Mathematical Sciences programme has invested in supporting projects of this type. As for the above data, this information does not give a complete picture, but it constitutes an element of evidence for the pervasiveness of research in the mathematical sciences.

Table 13 Contribution (in value and number) of the Mathematical Sciences programme to new research grants in other areas of the EPSRC remit (years 2005/06-2009/10) (EPSRC data) Value of Maths Contribution Contribution as % of Number Lead Programme (£M) grants value of Grants Cross-Discipline Interface .9 21% 18 ICT 4.2 26% 61 Materials, Mechanical & Medical Eng 1.2 25% 18 Physical Sciences .4 23% 5 Process Environment & Sustainability .5 26% 7 Grand Total 7.2 25% 109 6.3.4 Almost 60 universities throughout the UK currently receive grants from the Mathematical Sciences programme, however more than 70% of the f unding has been awarded to 15 institutions shown in Figure 12. This is analogous to the case for the whole EPSRC portfolio, which is similarly concentrated across a small number of institutions, as shown by Figure 13; such concentration is not a result of a specific policy but is the outcome of peer review consideration of individual research proposals. The institutions which win the largest share of Mathematical Sciences programme funds are not necessarily those with the largest share of EPSRC funding overall.

47 The classification of departments in the UK is left to individual universities: for this reason there is a wide range of different names. In Figure 11 all departments covering the same research areas have been merged under a common “umbrella” title. The heading “other” includes, amongst others, the Isaac Newton Institute, the ‘CoMPLEX’ Doctoral Training Centre at UCL, and other departments which cover a broad range of subjects. 34

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 12 Top 15 institutions by value of funding from the Mathematical Sciences programme (current grants, includes research grants and fellowships; values in £M) (EPSRC data)

Figure 13 Top 15 institutions by value of funding from EPSRC (current grants, includes research grants and fellowships; values in £M) (EPSRC data)

6.4 National and International Facilities and Services 6.4.1 The Isaac Newton Institute (INI) in Cambridge and the International Centre for Mathematical Sciences (ICMS) in Edinburgh provide major centres of interaction for UK mathematics with the broader community. Both centres offer a range of programmes and workshops that span the whole breadth of the mathematical sciences, also covering the overlap with research in other areas. Comparable with these centres is the Institut des

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Hautes Études Scientifique ( Orsay, France), which provides an important platform for collaboration of UK mathematicians with the French community and beyond. 6.4.2 The Isaac Newton Institute for Mathematical Sciences 48 is a national and international visitor research institute situated in Cambridge. It runs programmes on selected themes in the mathematical sciences with applications over a wide range of science and technology. The programmes vary in length from 4 weeks to 6 months, and are typically open to both UK-based and international participants. At any particular time there are typically 2 programmes running, each with up to 20 participants working at the Institute. Shorter workshops, conferences and satellite meetings are organised as part of the programmes. 6.4.3 The International Centre for Mathematical Sciences 49 is situated in Edinburgh and has the development and organisation of international workshops and conferences in all areas of the mathematical sciences as its core activity. In addition to workshops, ICMS is involved in a number of other mathematical activities such as postgraduate training, journal management and outreach. ICMS also holds events organised by other mathematical and related organisations, as well as running Research in Groups activities in all aspects of t he mathematical sciences and interdisciplinary areas with significant mathematical content.

6.5 Support for Mathematical Sciences PhDs 6.5.1 Postgraduate training is recognised in the UK as an important element of research, and this section provides an overview of mathematical sciences training available at the level of postgraduate researchers. Data for this section is drawn from both HESA and EPSRC. It is important to bear in mind the caveats noted in section 5.2 regarding the analysis of the evidence provided. In particular, the HESA data refers to the Mathematics cost centre: while this cost centre is the most aligned with the Mathematical Sciences programme, it does not include research in mathematics done in other departments and, at the same time, the data includes research in other areas which takes place in mathematics departments. 6.5.2 Unless otherwise indicated, all HESA numbers are given as FPE (full person equivalents). The studentship data is based on a ‘restricted population’ extract from the HESA Student Record; it refers only to students in their first year of study to avoid double-counting between years, and covers only the subset of the “ Mathematical and Computer Sciences” students classified as working in the various areas of the mathematical sciences50. 6.5.3 An element of under-reporting also occurs in the HESA data concerning research council supported students: this is a function of the fields used to select research council students ( “major source of funding” and “major source of tuition fees”) as alternatives occur ( for ex ample r esearch c ouncil-funded s tudents whose m ajor s ource of f unding is industrial), so not all students receiving funding from EPSRC will be flagged as “research council students”. 6.5.4 Funding for postgraduate study comes from a range of sources, which vary significantly depending on whether students are UK, EU or other nationals, as well as on whether they are part-time or full-time and on the different subject disciplines in both taught and research programmes. Postgraduate training in the UK is divided into Masters (typically one year) and Doctoral training (three to four years duration). 6.5.5 Masters degrees are typically centered around taught courses, as distinct from PhD training which focuses on research and has a smaller taught element. Funding for masters courses is limited: university scholarships are rare and highly competitive. Consequently, students who choose to f ollow a Masters course often fund themselves through Career Development Loans. It should be noted that a Masters degree is not required in order to undertake doctoral level training.

48 http://www.newton.ac.uk/ 49 http://www.icms.org.uk/ 50 More information on student records and classification can be found on the HESA website: http://www.hesa.ac.uk/index.php?option=com_studrec&task=show_file&Itemid=233&mnl=07051&href=jacs 2.html. 36

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC 6.5.6 Funding for PhD-level training in the UK comes from a variety of different sources. Most bursaries are offered directly by the universities, using funding received from governmental bodies and from ot her sources. Studentships are a lso d irectly funded by UK research charities (such as t he Wellcome Trust), pr ivate sponsors, or by means of career loans. 6.5.7 Figure 14 shows in a schematic ar rangement how t he balance of available support is distributed across the career path aggregated across all EPSRC programmes.

Figure 14 People Support across t he C areer P ath - 2009 Mathematical S ciences P ortfolio (EPSRC data)

EPSRC Support for Training – Masters level 6.5.8 Most of EPSRC’s training budget is directed towards training at a doctoral level ( see below an d section 4.6 for further i nformation), a nd support f or Masters t raining i s very limited due to t he main objective of t he Research Councils being t o support research. However, individual programmes can strategically support Masters training where needed to ensure a high enough number of qualified students to undertake a research career. In this context, the Mathematical Sciences programme allocated £800k to support statistics and operational research Masters students in 2010/2011. EPSRC Support for Training – Doctoral level 6.5.9 The information in this section refers specifically t o EPSRC support for doctoral level training in the mathematical sciences. For a general description of how EPSRC supports doctoral l evel t raining see 4.6. Over t he 5-year period 20 04/05 t o 2008/09, H ESA data shows 3,650 first-year doctoral students in the mathematical sciences; supplementary data provided directly to EPSRC by Universities indicates that EPSRC funding is used for approximately one t hird of t hem. The funding for t he remaining students comes from a variety of different sources, including university scholarships, industrial collaborators and home government funding for international students amongst others. 6.5.10 EPSRC’s support to postgraduate training takes place through block grants to Doctoral Training Accounts (DTAs) held by universities or through individual research grants. The block grants give institutions a high degree of autonomy with regard to the students they fund and enable them to leverage EPSRC support with funding from other sources. There are currently more than 700 DTA-funded studentships in mathematical sciences. The

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC allocation of EPSRC’s DTA budget by research programme over the last 5 years is shown in Figure 15.51

Figure 15 Allocation of DTA budget by EPSRC programme over past 4 years (EPSRC data)

6.5.11 Data52 based on students starting during the four years’ 2005/06 - 08/09 (the period for which the data is most complete) indicates that overall approximately 20% of DTA-funded doctoral students in the mathematical sciences are female; more detailed analysis shows that the ratio varies by discipline area, for example in Statistics the figure is approximately 30%. 6.5.12 Research grants also currently support training by funding the employment of PhD “project students” as research staff. There are currently approximately 70 such PhD students supported by the Mathematical Sciences programme. Figure 17 shows the number of project students starting over the last 5 years by discipline. 6.5.13 The most recent information available indicates that the proportion of mathematics PhD students from overseas was just under 50% in 2008/09, having risen during the preceding five years from just over 40% in 2004/05, as shown in Figure 16.

51 It should be noted that the EPSRC programmes have been restructured during the past 4 years; Engineering was split in 2009 to form the Process, Environment and Sustainability Programme and the Materials, Mechanical and Medical Engineering Programme, the Physical Sciences Programme was formed in 2009 from the Chemistry, Physics and Materials Programmes. 52 Provided directly to EPSRC by HEIs 38

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 16 Recorded domicile of PhD students in mathematical sciences (HESA data)

Figure 17 Number of EPSRC project studentships starting per year, by programme (2005/06 to 2009/10) (EPSRC data) 51

Centres for Doctoral Training 6.5.14 EPSRC has funded 50 Centres for D octoral Training ( CDTs) w ith a total i nvestment i n excess of £280M 53. Each has an intake of approximately ten students per year for 5-year period of the grant. Students will enrol on a 4-year programme in which first year allows time for explorations before deciding on a challenging and original research project. The students also take part in other activities to develop breadth of knowledge and transferable skills. T hree CDTs in the Mathematical S ciences were f unded in 2 009 an d received their first cohort of students in October 2010. The three centres are listed below: • Doctoral Training Centre in Statistics and Operational Research, Lancaster University http://www.stor-i.lancs.ac.uk/ • Cambridge Centre for Analysis, University of Cambridge http://www.maths.cam.ac.uk/postgrad/cca/ • MASDOC: A CDT for the Mathematical Sciences, University of Warwick http://www2.warwick.ac.uk/fac/sci/masdoc/

53 See also section 4.6.5 39

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC In addition to the three CDTs funded by the Mathematical Sciences programme, there are a further three in Complexity Science and five in Life Sciences that relate to the mathematical sciences.54

Other Activities 6.5.15 In addition to funding PhD studentships as described above, EPSRC supports the following other training activities aimed at postgraduate students.

Mathematical Sciences Taught Course Centres55 6.5.16 In response to the International Review of Mathematics in 2004 the Mathematical Sciences programme invested £3 million over 5 years to fund six Taught Course Centres (TCCs). These are: • Scottish Mathematical Sciences Training Centre • TCC for the Mathematical Sciences based at Oxford, Warwick, Imperial, Bath and Bristol • Mathematics Access Grid: Instruction and Collaboration (MAGIC) • National Taught Course Centre in Operational Research (NATCOR) • Academy for PhD Training in Statistics (APTS) • London Taught Course Centre The i nvestment by EPSRC was more t han matched by a £4 million i nvestment over 5 years from universities. About 750 PhD students per annum use the TCCs, the majority (more than 90% ) coming from the mathematical sciences. Further information on the TCCs is available in Annex A56. Mathematical Sciences Short Courses 6.5.17 The EPRSC Mathematical Sciences programme funds additional short courses organised by the London Mathematical Society and the Royal Statistical Society. These courses are complementary to the TCCs, providing greater depth in the subject. Industrial Mathematics Internships Programme 6.5.18 The Industrial Mathematics Internship (IMI) Programme is a scheme run by the Industrial Mathematics Knowledge Transfer Network and supported by the EPSRC Mathematical Sciences programme. This scheme provides a way for companies and university research groups to develop long term working relationships, through engaging a dedicated postgraduate researcher to work on a specific industrial project over a period of 3-6 months. Each internship is jointly funded by EPSRC and the participating company. CASE Awards 6.5.19 CASE awards are intended to provide funds to support the training of a PhD student on a project in collaboration with a UK firm. During their PhD the student is required to spend a period working directly at the firm, while the firm is expected to make a financial contribution to both the student and the project. The Mathematical Sciences programme allocates funds for up to 30 such awards through the DTA budget.

6.6 Destination of UK PhD students 6.6.1 Table 14 shows proportionately the destinations of UK PhD students in the engineering and physical sciences; the data is drawn f rom a survey of UK and EU-domiciled P hD students who qualified or left higher education between 1/Aug/’07 – 31/Jul/’08; the response rates for the survey varied, ranging from almost 80% for full-time UK students to ar ound 50% f or par t-time s tudents. While t he most significant share of Mat hematics students remains in higher education it is evident that mathematical sciences PhD students are valued by employers.

54 http://www.epsrc.ac.uk/funding/students/centres/Pages/default.aspx 55 http://www.epsrc.ac.uk/about/progs/maths/train/Pages/research.aspx 56 A recent review of TCCs has recently taken place, and the report can be found on the EPSRC website: http://www.epsrc.ac.uk/SiteCollectionDocuments/Publications/reports/EPSRCReviewOfTCCs.pd f 40

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Table 14 Destination of UK PhD students by subject area (HESA data) Computer Destination Chemistry Physics Mathematics Science Engineering Materials Total Further study 5% 1% 5% 3% 2% 5% 3% Higher education 35% 39% 45% 46% 32% 26% 37% Research related 24% 16% 8% 7% 9% 11% 13% employment Other employment 24% 33% 30% 30% 48% 45% 36% Education 2% 2% 2% 2% 1% 0% 2% Not employed 7% 7% 8% 7% 6% 13% 7% Not known or not 2% 2% 3% 4% 3% 0% 3% reported

7. Support for People The support of world-class people and projects is an ongoing strategic priority for EPSRC. ‘Supporting Excellence’ was an objective in the 2006 Strategic Plan57 and ‘Developing Leaders’ is one of the three priorities of the 2011 Strategic Plan58. Consequently, significant effort is devoted to activities which focus on supporting the best researchers throughout their whole career path. This section aims to provide information on the support that is available to mathematical sciences researchers in the UK; i t begins with an overview of non-EPSRC s ources of s upport, and t hen describes the mechanisms and schemes offered by EPSRC59.

7.1 Non-EPSRC sources of support for People 7.1.1 Mathematical sciences researchers in the UK are supported not only by Funding and Research Councils, but also by a number of i ndependent c harities a nd t rusts, t ypically through fellowships or research grants. The principal such sources of support in mathematical sciences are described in outline below: 7.1.2 The Royal Society http://royalsociety.org/ The Royal Society runs a number of grants schemes open to researchers in most areas of science (including mathematics). Awards aimed at early-career researchers include: • Dorothy Hodgkin Fellowships: support excellent early-career scientists, who require a flexible working pattern, for up to four years. • JSPS Postdoctoral Fellowships: provides opportunities for early career researchers from the UK to conduct research in Japanese institutions. • Newton International Fellowships: attract the world's best early-career researchers to the UK for two years. • University Research Fellowships: provide outstanding scientists, who have the potential to become leaders in their chosen field, with the opportunity to build an independent research career. Funding is available for up to 8 years. The Royal Society also administers fellowship schemes aimed at senior researchers, including: • Leverhulme Trust Senior Research Fellowships: allow scientists to concentrate on their research by relieving them of teaching and administrative duties for a year. • Royal Society Wolfson Research Merit Awards: provide a five year salary enhancement to attract or to retain outstanding scientists within the UK.

57 More information on the 2006 Strategic Plan can be found here: http://www.epsrc.ac.uk/SiteCollectionDocuments/Publications/corporate/EPSRCSP06.pdf 58 For the 2010 Strategic Plan, please refer to http://www.epsrc.ac.uk/plans/approach/strategicplan/Pages/default.aspx 59 It is acknowledged that the EPSRC schemes represent only a subset of the support available to researchers, but it is a subset on which we can provide detailed data. 41

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC • Royal Society Research Professorships: provide ten years' funding for internationally recognized scientists. The Society also runs a number of award schemes to support research capacity, infrastructure and innovation. 7.1.3 The Leverhulme Trust http://www.leverhulme.ac.uk/ The Leverhulme Trust has an annual budget of approximately £50M, and provides research funding across most academic disciplines (including the mathematical sciences). The Trust’s awards are typically focussed on people support, at all career stages. Awards include: • Early Career Fellowships: approximately 70 awards per year, 2-3 years duration, covering 50% of the Fellows’ salary, with a contribution towards other research expenses. • Research Fellowships: aimed at experienced researchers, up to 2 years duration. • Study Abroad Fellowships: designed to support a period overseas in a stimulating academic environment, up to 1 year duration. • Emeritus Fellowships: aimed at senior established researchers to assist them in completing a research project, up to 2 years duration. The Trust also provides support for international networks and for visiting professorships, as well as running Research Project and Research Programme Grants schemes. 7.1.4 The Daphne Jackson Trust http://www.daphnejackson.org/ The Daphne Jackson Trust is an independent charity dedicated to returning talented scientists, engineers and technologists to careers after a break of two years or more. Their Fellowship scheme is open to graduates from any of the STEM subjects (including the mathematical sciences) who have taken a career break of two or more years’ duration. Each year, the Trust awards approximately 20 new fellowships.

7.2 EPSRC support for People 7.2.1 A general analysis can be done on t he distribution of funding of EPSRC f ellowships. Figure 18 shows how fellowships funded by the Mathematical Sciences programme are distributed across different institutions, while Figure 19 shows the distribution of all fellowships f unded by a ll EPSRC pr ogrammes. In c omparison w ith Figure 13 it c an be seen that the concentration of funding is even more pronounced in the case of fellowships. Some noticeable differences are also apparent when comparing the universities which appear in Figure 18 with those in Figure 13.

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 18 Number of current Fellowships funded by EPSRC Mathematical Science Programme

7.2.2 Figure 19 shows the distribution of all current fellowships funded by all EPSRC programmes and may likewise be compared with Figure 13.

Figure 19 Number of current Fellowships funded all EPSRC Programmes

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Support for Early Career Researchers 7.2.3 EPSRC provides targeted funding to address demographic concerns by assisting the best early-career stage academics to establish research careers (see also section 4.7). Specific support mechanisms are available to researchers in the mathematical sciences, depending on their experience level and on whether or not they already hold a permanent academic post. Postdoctoral Research Assistants 7.2.4 Postdoctoral Research Assistants (PDRAs) are typically PhD-qualified junior researchers employed on fixed-term contracts by universities, occasionally with teaching responsibilities; their employment conditions have been evolving over recent years 60. They are often funded through research grants and supervised by more senior academics. Figure 20 shows the number of PDRAs in time for different EPSRC programmes. 7.2.5 The relatively small number of PDRAs in the mathematical sciences compared to other EPSRC programme areas is due to the relatively small research budget of the Mathematical Sciences programme. In practice, the ratio bet ween PDRA numbers and programme budget is very similar across the EPSRC remit. 7.2.6 It is also interesting to consider the role that PDRAs have within the different areas of the mathematical sciences. In Figure 21 below, data is presented on the number of PDRAs on grants funded by the Mathematical Sciences programme in the last 5 years. It should be noted that the split into research topics is done according to the internal EPSRC classification, and as with any classification is at least partially arbitrary.

Figure 20 Number of EPSRC-funded PDRAs by programme (2005/06 to 2009/10)

60 see http://www.vitae.ac.uk/1315/Rights-and-responsibilities.html 44

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 21 Total number of PDRAs on grants funded by the Mathematical Sciences programme by area (from 2005 to 2010) (EPSRC data)61

7.2.7 Figure 21 shows that the largest proportion of PDRAs funded by the Mathematical Sciences pr ogramme work in the areas of a lgebra, g eometry, t opology, and/or number theory (AGNT). Comparing the number of PDRAs with the number of grants funded in the same period it is found that PDRAs are least common in AGTN research projects, with less than 45% of successful grant applications having a PDRA (the average for the whole Mathematical Sciences programme is approximately 55%). Conversely, PDRAs are common in the areas of numerical analysis (more than 80% of funded grants) and analysis (more than 65% of funded grants). Postdoctoral Research Fellowships 7.2.8 Postdoctoral Research Fellowships (PDRFs) enable the most talented new researchers with 0 -3 years of postdoctoral experience t o establish an independent research career shortly after completing a PhD. The awards are for a period of up to t hree years and cover the salary costs and travel expenses of t he Fellow. PDRFs are not available across the breadth of the EPSRC remit; in 2010 only fellowships in theoretical physics, mathematical sciences and cross-disciplinary interfaces were offered. 7.2.9 Postdoctoral Research Fellowships (PDRFs) are extremely popular within the mathematical sciences community. The Mathematical Sciences programme typically receives between 70 and 90 applications per year, and awards on average ten fellowships. While applications are open to all areas of the mathematical sciences, most applications are either in pure mathematics (including algebra, geometry, topology, number theory, logic and combinatorics) or in mathematical physics (together these two areas account for more t han 70% of t he a pplications r eceived b y the Programme, with pure mathematics accounting for approximately 35% of the applications). 7.2.10 In order to better manage demand for these PDRFs, a cap of seven applications per institution was introduced in 2008. In 2010/11 the cap was lowered to 5 applications[1].”

61 The category “other” represents PDRAs co-funded by the Mathematical Sciences Programme and other programmes (therefore they will have research topics from other programmes). The numbers are given in single-student-equivalents (if a PDRA is on a grant coded as 50% logic and 50% ICT, then there will be 0.5 PDRA in logic and 0.5 students in other). [1] This cap is intended to be variable as a function of the quality of the applications submitted by each institution. 45

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 22 Total number of PDRFs funded by the Mathematical Sciences programme by area (from 2005 to 2009) (EPSRC data)

7.2.11 Figure 22 shows the distribution of PDRFs funded by the Mathematical Sciences programme since 2005. As can be seen, the largest numbers of PDRFs are awarded in the ar eas of pure m athematics and m athematical ph ysics ( which is c onsistent with t he fact that the Programme receives the majority of the applications in these areas), while no PDRFs have b een awarded in statistics and applied probability or in the mathematical aspects of operational research (OR). Career Acceleration Fellowships 7.2.12 Career Acceleration Fellowships (CAFs) provide up to five years funding to outstanding researchers with three to ten years postdoctoral experience who have not held a permanent academic pos t. The expectation is t hat the f ellows will h ave es tablished an independent career of international standing by the end of the award. The scheme is run by EPSRC c entrally, an d t here is an outline stage before a small n umber of applicants are invited to submit a full proposal and to the interview stage. Table 15 below shows the number of Career Acceleration Fellowship applications received, invited to full proposal, and awarded per year (comparison between the whole of EPSRC and the Mathematical Sciences programme). As with the PDRFs, typically the majority of applications fall within the areas of pure mathematics (20-25%) and mathematical physics (30-40%).

Table 15 Number of Career Acceleration Fellowships per year (2007/8 – 2009/10), all EPSRC vs. Mathematical Sciences programme (EPSRC data) 2007/08 2008/09 2009/10 Outlines 437 50 415 53 449 61 Invited 61 4 70 9 72 11 Awarded 22 1 24 4 30 5 7.2.13 Of the 10 awarded fellowships, 2 are in mathematical physics, 1 is in numerical analysis, 2 are in analysis and the remaining 5 are in the area of AGTN. First Grants 7.2.14 First Grants are open to academic staff within three years of their first academic appointment; applications are reviewed in competition against each other rather than against work of more experienced researchers. From January 2009 the funding was capped at £125,000 with a maximum duration of two years. 7.2.15 Table 16 below shows the number of First Grants for EPSRC and the Mathematical Sciences programme by year.

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Table 16 Number of First Grant applications per year (2005/6 – 2009/10), all EPSRC vs. Mathematical Sciences programme (EPSRC data) 2005/06 2006/07 2007/08 2008/09 2009/10 Successful 130 14 133 18 123 11 116 14 114 17 Unsuccessful 196 15 185 27 152 21 290 30 188 26

7.2.16 Most of the applications (more than 50% on average) within the remit of the Mathematical Sciences programme fall in the areas of AGTN, analysis and statistics and applied probability, while most of the awarded grants are either in AGTN ( more than 30% on average) or analysis (more than 20% on average).

7.3 EPSRC Support for Established Researchers 7.3.1 EPSRC aims at supporting researchers at all stages of their careers. The schemes considered in this section in some sense complement those described in 7.2 in that they target researchers at more advanced s tages of their c areer. The s chemes des cribed in the following section are all in addition to funding of standard research proposals (which are processed in responsive mode and are available to all UK academics with a permanent academic post). Leadership Fellowships 7.3.2 Leadership F ellowships ( LFs) pr ovide up to f ive years support for talented r esearchers with the most potential to develop into international research leaders with the ability to set and drive new agendas, by the end of the award. As with the Career Acceleration Fellowships, applicants initially submit o utline pr oposals which ar e then e valuated by a Panel. As a second stage, applicants are invited to submit a full proposal and to interview. LFs ar e up t o 5 years long, and their value is typically around £1M ( in t he mathematical sciences). 7.3.3 Table 17 shows the number of Leadership Fellowship applications received, invited to full proposal, and awarded per year (comparison between the whole of EPSRC and the Mathematical Sciences programme). As with the PDRFs, typically the largest proportion of applications falls within the area of pure mathematics (~ 25%).

Table 17 Number of Leadership Fellowships per year (all EPSRC vs. Mathematical Sciences programme) (EPSRC data) 2007/08 2008/09 2009/10 Outlines 296 47 253 52 206 37 Invited 59 11 49 7 38 10 Awarded 23 7 17 2 16 2 7.3.4 Of t he 11 awarded leadership fellowships, 5 are in AGTN and 2 in analysis. One has been awarded to each of the areas of mathematical physics, numerical analysis and statistics, and one fellowship focuses on research which is at the boundary between AGTN and non-linear systems mechanics. 7.3.5 It should be noted that there is a single budget for Career Acceleration and Leadership Fellowships, the best candidates from across the two schemes are awarded fellowships. Consequently, the numbers of LFs awarded are not constant throughout the years. Programme Grants 7.3.6 Programme grants are a flexible mechanism to provide funding to world-leading research groups to address significant major research challenges. They are intended to support a suite of related research activities focussing on one strategic research theme. The Mathematical Sciences programme has funded one programme grant in Applied Derived Categories with a value of £1.2M lead by Professor Richard Thomas at Imperial College London.62

62 http://gow.epsrc.ac.uk/ViewGrant.aspx?GrantRef=EP/G06170X/1 47

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC 7.3.7 Applicants must discuss their suitability for programme grants with EPSRC before submitting an outline; the Mathematical Sciences programme is currently in discussions with two other institutions.

7.4 Previous Schemes

Advanced Research Fellowships 7.4.1 Advanced Research Fellowships were offered annually until 2007 to outstanding researchers within 10 years of completing their PhD ( whether they held a permanent academic post or not). The awards were for up to five years and fellows were expected to have established themselves as an independent researcher of international standing by the end of t he a ward. During t he years that this s cheme was run 69 f ellowships were awarded in the mathematical sciences, 20 of these are still current. Senior Fellowships 7.4.2 Senior F ellowships were a warded t o out standing ac ademic r esearchers of i nternational standing to enable them to devote themselves full-time to personal research for up to five years. Up to three fellowships were awarded per year. During the years that this scheme was run ten fellowships were awarded in the mathematical sciences, three of these are still current. Science and Innovation Awards 7.4.3 A number of Science and Innovation A wards have been made in recent years to build new activity in areas of national strategic importance, with a particular focus on supporting new research leaders and groups. Awards were typically £3-5 m illion over 5 years and required a commitment from the host research organisation to continue support after the end of the grant. Out of a total of 37 S&I awards made, 8 were in areas of mathematical science with total funding in the region of £29 million.

Table 18 Science & Innovation Awards since 2005 in areas of Mathematical Science (EPSRC Data) Value Title Date Location (£ million) Edinburgh, Numerical Algorithms and Intelligent Software for the Aug-09 Heriot-Watt, 4.5 Evolving HPC Platform Strathclyde Lancaster, The LANCS Initiative in Foundational Operational Nottingham, Sep-08 5.4 Research: Building Theory for Practice Cardiff, Southampton Analysis of Nonlinear Partial Differential Equations Oct-07 Oxford 2.8 Centre for Analysis and Nonlinear Partial Differential Edinburgh, Aug-07 2.8 Equations Heriot-Watt Cambridge Statistics Initiative (CSI) Mar-07 Cambridge 2.3 The Centre for Discrete Mathematics and its Mar-07 Warwick 3.8 Applications (DIMAP) SuSTaIn - Statistics underpinning Science, Aug-06 Bristol 3.5 Technology and Industry The Centre for Research in Statistical Methodology Oct-05 Warwick 4.1 (CRISM) Total 29.2 Multidisciplinary Critical Mass Centres 7.4.4 Multidisciplinary Critical Mass Centres connect mathematics (including statistics) to other disciplines in the remit of EPSRC ( engineering, materials, IT and computing science, physics, chemistry and life sciences interface).

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Table 19 Multidisciplinary Critical Mass Centres funded bythe EPSRC Mathematical Sciences programme Value (£ Title Date Location million) The Bristol Centre for Applied Non-linear Oct-02 1.07 Bristol Mathematics Renewed Oct-07 1.77 Interdisciplinary Programme for Cellular Regulation: Oct-03 Warwick 1.26 Mathematical Architecture of Biological Regulation Bath Centre for Complex Systems Nov-04 Bath 1.01 Oct-05 Kent, 1.10 National Centre for Statistical Ecology Renewed May- Cambridge, 1.04 10 St. Andrews Multidisciplinary Critical Mass in Computational Sep-05 St. Andrews 1.09 Algebra and Applications New Frontiers in the Mathematics of Solids Oct-06 Oxford 1.16 Total 9.49

7.5 Demographics 7.5.1 The most recent (2008/09 academic year) HESA data records 2,885 academic staff working in the Mathematics cost centre at universities in the UK. They are defined as academic professionals responsible for planning, directing and undertaking academic teaching and research within higher education institutions. 7.5.2 There are proportionately more women training as mathematical sciences researchers than being employed as such. As noted previously ( see 6.5.11), male doctoral student outnumber females by approximately 4:1. A similar ration of 4:1 is typically present at the level of ‘Researcher’ an d ‘ Lecturer’ grade s taff, but by the time they have moved up to ‘Senior Lecturer’ grade and above, males outnumber females nearly 6:1.

Figure 23 Distribution of staff number analysed by grade and gender between 2004/05 and 2008/09 (HESA data)

1200

1000 2008/09F 2008/09M 800 2007/08F 2007/08M 2006/07F 600 2006/07M 2005/06F 400 2005/06M 2004/05F 2004/05M 200

Lecturers Professors Researchers Other Grades

Senior Lecturers & Researchers

7.5.3 Figure 24 shows the distribution of staff numbers by age between the years 2004/05 and 2008/09. As may be expected there is s ome c orrelation bet ween the data i n Figure 23 and Figure 24.

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 24 Distribution of staff numbers analysed by age between 2004/05 and 2008/09 (HESA data)

7.5.4 Figure 25 presents the distribution of staff numbers by salary in the last five years.

Figure 25 Distribution of staff numbers analysed by salary between 2004/05 and 2008/09

7.6 EPSRC Mathematical Sciences Programme Demographics 7.6.1 Figure 26 shows the age distribution of principal and co-investigators (PIs and Co-Is) currently funded by the EPSRC Mathematical Sciences programme in relation to all EPSRC currently funded PIs and Co-Is. The distribution peaks with the 46-50 age groups; this is consistent with the HESA data in Figure 24 as this includes ‘Researcher’ grade staff at an early stage in their careers compared to EPSRC principal and c o- investigators who by definition will have gained some experience already.

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 26 Age and gender of current EPSRC mathematical sciences portfolio principal and co-investigators, by proportion (July 2010) (EPSRC data)

20%

18%

16%

14%

12% Maths Unknow n Maths Female 10% Maths Male 8% A ll EPSRC

2% Proportion of Principal and CoInvestigators and Principal of Proportion

>65 <=25 25-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65 Unknown Investigator Age

7.6.2 Figure 27 shows the number of EPSRC Mathematical Sciences grants awarded between 2005 and 2010, divided by age group. The increase in the number of announced grants for the 25-35 age groups in the last t wo years c an b e explained by t he c ap in value of First Grants (which has increased the overall number of funded First Grants).

Figure 27 Number of EPSRC Mathematical Sciences grants announced, by age group from 2005 to 2010 (EPSRC data)

Number of EPSRC Maths grants announced by age group (2005 - 2010)

200

180

160

140 Unknown >65 120 61-65 56-60 51-55 100 46-50 41-45 80 36-40 31-35 60 25-30 Number of Announced Grants Announced of Number 40

0 2005/2006 2006/2007 2007/2008 2008/2009 2009/2010

PI Age Range

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC

8. International Engagement

8.1 Overview 8.1.1 Numerous sources of funding are available to support the flow of people through travel grants and visiting fellowships; these serve both to enhance the research capabilities of individual r esearchers and to develop collaborative links. Support is available for U K- based researchers wanting to travel abroad, and for overseas researchers who want to come to the UK. Schemes to support international collaborations are run by most major organisations such as , for example, the learned societies, the Royal Society, the Leverhulme T rust, t he Wellcome T rust, t he Royal Society of Edinburgh and the B ritish Council. 8.1.2 It is not easy to quantify the extent of the international engagement of the UK mathematical sciences community, however the landscape documents and the departmental s ubmissions should provide qualitative evidence. This s ection f ocuses on the analysis of that part of the community that is funded by EPSRC since it is the set for which we can access the most reliable data. Even in this case it should be clear that the figures will probably greatly underestimate the extent of international collaborations.

8.2 International collaboration with EPSRC-funded Mathematical Sciences Researchers 8.2.1 As said above, while the perception is that most of the research that is being done by the mathematical sciences community benefits by some form of international collaboration, it is extremely difficult to find data that can fully reflect this perception. In the following, data is presented which can in part represent the breadth of international collaborations of the community. It should be noted that the data is restricted to EPSRC-funded research. 8.2.2 Figure 28 shows the origin of visiting researchers on projects funded by the Mathematical Sciences programme. While this parameter certainly gives an indication of the extent in which international collaborations are part of research in mathematics, it should only be taken as very partial information. In particular, it should be noted the data presented does not include information on visits abroad by UK-based researchers, nor does it include all visitors to the UK (but only those who were identified in the grant application).

Figure 28 Number and origin of Visiting Researchers on Mathematical Sciences projects (years 2005/06-2009/10) (EPSRC data)

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC 8.2.3 As may be expected, the majority of collaborations take place with scientists from the EU and US. Figure 28 also shows that UK mathematical science researchers collaborate with researchers throughout the world. 8.2.4 Another parameter that does reflect the extent of collaborations in the UK mathematical sciences community can be obtained by considering whether publications have been written in collaboration with overseas scientists. Analysis of final reports from all EPSRC projects completed during the past three years shows that the proportion of mathematical sciences projects with at least one internationally co-authored publication, at above 60%, is much higher than the average for EPSRC (just below 45%); the overall proportion of publications reported on mathematical sciences grants with international c o-authors is 37%, which again is much higher than the average across all EPSRC areas (22%). 8.2.5 EPSRC also funds the “Representation Theory Across the Channel” network, in collaboration with CNRS63. The network aims at strengthening research in representation theory both in the UK and in France encouraging collaborations between the two countries. A number of activities are supported by the network to achieve this, including conferences, visit schemes, invited lecture series and annual meetings.

9. Impact 9.1.1 In this Section, we give an overview of different activities that can be classified under the general heading of “Impact”. This includes information on Knowledge Transfer/Exchange (KTE) and Public Engagement (PE) activities undertaken by the UK mathematical sciences community. The information in this section is supplemented by the case studies that will be presented to the Panel during the review week.

9.2 Knowledge Transfer 9.2.1 The exploitation of academic r esearch and its economic impact i s subject to i ncreased attention within the UK (see ‘Support for Science and Innovation in the UK’ section 2). As a consequence KT is a central element of the review (sections F and G of the evidence framework). Obtaining r eliable qu antitative evidence on KT ef fectiveness and economic impact which can be directly linked to specific funding interventions is notoriously difficult. The ‘Warry’ report about the economic impacts of the Research Councils' work underlines 64 these difficulties and provides a detailed analysis of the key issues. 9.2.2 Measuring knowledge transfer in the Mathematical Sciences is arguably even harder than for other research areas, since a significant part of the research in this field is speculative and might require many years before it reaches the application stage. It is therefore extremely difficult to find reliable data on these collaborations. 9.2.3 Despite this, much fruitful collaboration exists between researchers in the mathematical sciences and users in a variety of sectors. An important part in this is played by the Industrial Mathematics KTN65 which is funded by a number of sources (including EPSRC, Government departments and the TSB) and whose aim is to facilitate the involvement of the mathematical sciences community with users. Additionally, universities holding a KTA or KTS grant from EPSRC are able to allocate resource from these to support knowledge transfer/exchange connected with the mathematical sciences.

9.3 Public Engagement 9.3.1 Public Engagement with Research ( PER) provides considerable opportunities for the mathematical sciences community to improve knowledge and understanding of the value of mathematics to society. The current emphasis among funders on the need for researchers to demonstrate the impact of research includes scope for better engagement with the general public. Impact is defined broadly to include economic, policy, quality of life, cultural an d societal benefits, s o that an element of publ ic eng agement can be an integral aspect of an impact plan.

63 http://www.maths.abdn.ac.uk/~geck/REPNET/ukrepnet.html 64 The ‘Warry’ report is available to panel members on the review ftp site. 65 http://www.industrialmaths.net 53

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC 9.3.2 While there are a small number of enthusiasts for PER within the mathematical sciences community e.g. Marcus Du Sautoy, Chris Budd, David Abrahams, it is not evident that the community at large recognises the value of PER as an integral aspect of research. Nevertheless, it is encouraging that more mathematicians are including PER activities in the ‘pathways t o i mpact’ statements which are now a required element in research proposals, albeit that many of these limit their audience to schoolchildren. Inspiring young people is important, but research councils define PER much more broadly to include dialogue with t he general public on societal and ethical issues connected with research t hrough t o dissemination, explanation and understanding of the outcomes of research to a wide p ublic audience. The adoption of this def inition offers much wider opportunities for involvement in PER. 9.3.3 The mathematics learned societies actively support PER and have collaborated with EPSRC in a number of i nitiatives. There is scope for f urther c ollaboration as EPSRC moves away from direct funding for PER projects to embedding PER within the academic community as a natural feature of research. Because of this change in emphasis the landscape will undergo some change in the short term. Of particular note is the work of Professor Marcus Du Sautoy (Oxford), holder of an EPSRC Senior Media Fellowship. Professor Du Sautoy has used the fellowship to raise the popular profile of mathematics through books, television programmes and popular lectures, together with involvement in a number of PPE projects.

10. Bibliometric Evidence 10.1.1 Although bibliometric indicators must be used with caution66 they are commonly used as a source of information when seeking to compare national research quality in fields such as mathematical sciences where the journal coverage is good and where publication in journals is the usual mode of disseminating outputs. Providing evidence from such indicators may therefore assist the panel in benchmarking UK performance relative to the rest of t he w orld. It i s expected t hat much of t he ev idence will be gathered during the institution visits to the Panel. 10.1.2 The details of t he bi bliometric analyses are provided in Annex G. The key points (Jan 2000-Apr 2010) are as follows: • Globally the UK is ranked 5th behind the USA, France, China and Germany in terms of the total numbers of citations; • the UK is ranked 5th in the number of papers globally; • the UK is ranked 3rd, behind the USA and Australia in terms of Citations per paper67; • the UK hosts 5 of the top 100 institutions based on number of papers and 4 of the top 100 in terms of number of citations; • the UK has 6 of the top 100 institutions based on citations per paper. 10.1.3 The relative citation impact of UK papers (and some of the UK’s closest competitors) in ’Mathematics’ using f ive-year rolling averages to 2008 is shown in Figure 17 (relative citation impact is the number of citations per paper for Mathematics in each country, divided by the worldwide average for the discipline). This shows that the UK is on a slight downward trend (possibly now levelling off) after a blip in 2000-4/2001-5. This is against a rising trend amongst selected other developed and developing countries (notably China).

66 For more information, and a detailed statistical analysis, of issues related to bibliometric data and their use, please refer to the “Citation Statistics Report” by the International Mathematical Union: http://www.mathunion.org/fileadmin/IMU/Report/CitationStatistics.pdf 67 Limited to countries publishing at least 500 papers a year 54

International Review of Mathematical Sciences 2010 PART I Information for the Panel Evidence Prepared by EPSRC Figure 29 Citation impact of Mathematics papers68 (Thomson Reuters ‘InCites’)

68 Papers here refers to journal articles and review articles 55

Making sense of research funding in UK higher education September 2010

Research Information Network factsheet www.rin.ac.uk

How is research funded in the higher education sector? Dual support and the quest for sustainability

Research in the higher education sector is funded to choose which research to support themselves, primarily by the Government, with additional at arm’s length from any political control. It has support from charities (particularly for biomedical evolved into a system that provides multiple points research), international sources and the private of decision-making about what research should sector. Public funding comes from a range of be supported and where resources should be Government Departments with research budgets concentrated. at their disposal, but the bulk comes from the • Higher Education funding bodies provide core Department for Business, Innovation and Science funding as block grants to HEIs for research (BIS), which funds research from its science and infrastructure and to support their strategic higher education budgets. research priorities. The bulk of their recurrent Most of the funding BIS provides for research quality-related (QR) funding is allocated through in higher education institutions (HEIs) is delivered a formula that takes account both of institutions’ either through non-departmental public bodies volume of research activity and an assessment such as Research Councils, or through the Higher of the quality of their research over the previous Education Funding Council for England (HEFCE). period. The devolved administrations support equivalent • Research Councils provide grants to specific funding bodies in other parts of the UK (see research projects and programmes of research. boxes 1 and 2). Awards are made, after expert peer review of applications submitted by individual researchers The dual support system and teams, on the basis of research excellence This division of labour between funding bodies and potential, and on the importance of the and Research Councils constitutes an organising proposed topic. principle for research funding known as ‘dual • Each Research Council seeks to achieve a support.’ With its origins in the 19th Century, it rests balance between ‘responsive mode’ awards made on an idea (known as the ‘Haldane Principle’) that in response to proposals submitted by researchers Research Councils and universities should be able in any area they choose, and awards for proposals

Annex A - 1 that relate to strategic priorities for which • directly allocated costs: related to resources councils may earmark funds. Councils’ priorities that are used by a project but shared by other are set out in their strategic plans, which are activities, such as accommodation and the developed through extensive consultation with contributions of principal investigators; the academic community and other stakeholders. • indirect costs: non-specific costs charged across all projects, including administrative Ensuring the full costs of research are funded and library costs. The dual support system came under strain in the 1980s and 1990s as core funding declined Directly-allocated and indirect costs are markedly in relation to the amount provided as calculated using standard rates established by project grants. The two funding streams were not the university through the Transparent Approach meeting the full costs of the research and the UK to Costing (TRAC) methodology that underpins research base was running a large recurrent deficit. the FEC regime. Universities then use the results In 2002 the Government reviewed the dual support of their calculation of project costs under all three model and concluded that changes were required headings to set the price for the research projects to address under-investment in infrastructure. In its 10-year Science and Innovation Investment Box 2: Research councils Framework (2004), the Government reiterated its commitment to a dual support system, but The UK’s Research Council system is supported signalled that HEIs would be asked to recover by the Government’s Science Budget. BIS has more of the full economic costs (FEC) of research statutory control of the councils, supported by and to manage their research portfolios in the Director General of Science and Innovation. a more sustainable way (see box 3). The FEC There are currently seven Research Councils: regime attributes costs to projects under the • Arts and Humanities Research Council three headings: (AHRC) • directly incurred costs: explicitly-identifiable • Biotechnology and Biological Sciences costs arising from the conduct of a project, Research Council (BBSRC) including equipment, staff, travel and subsistence; • Engineering and Physical Sciences Research Council (EPSRC) Box 1: Funding bodies • Economic and Social Research Council There is a devolved system of funding for (ESRC) higher education. The funding councils for • Medical Research Council (MRC) England, Scotland and Wales are the: • Natural Environment Research Council • Higher Education Funding Council for (NERC) England (HEFCE) • Science and Technology Facilities Council • Scottish Funding Council (SFC) (STFC) • Higher Education Funding Council for They are collectively represented by an Wales (HEFCW) umbrella organisation, Research Councils BIS supports HEFCE, while the Scottish and UK (RCUK). Welsh equivalents are supported by the Scottish The Research Councils’ expenditure for 2008/09 Government and Welsh Assembly Government totalled nearly £3.4bn (ranging from the AHRC’s respectively. In Northern Ireland, funding comes £122m to the EPSRC’s £796m), almost all of directly from the Department for Employment which comes from the Science Budget. A large and Learning (DELNI). part of that money is distributed to universities, Over £2.2bn was allocated in 2008/09 by but significant sums also go to institutions the four funding bodies in recurrent and run by the Research Councils, to large-scale capital funding; the largest share of this, international collaborations and to independent £1.9bn, came from HEFCE. research organisations in the UK.

Annex A - 2 UK higher education institutions’ income from research grants & contracts and funding council grants*

HM TREASURY

Dual support system

BIS Scottish Welsh DELNI Government Executive Assembly Departments Government other than BIS, (£3.1bn from local authorities, Science Budget) health & hospital authorities EPSRC BBSRC NERC HEFCE SFC HEFCW

STFC ESRC AHRC MRC

Research councils Funding bodies £706m £1,892m £2,266m

UK higher education institutions

UK-based UK industry EU and other Other charities commerce & overseas sources £51m £896m public £648m corporations £312m

* Wherever possible, the figures in the diagram are for the latest available year, 2008–09; figures from earlier years (2006–07 or 2007–08) have been used otherwise. Information about sources for this financial data can be found at www.rin.ac.uk/making-sense-funding

they wish to undertake. The proportion of the Box 3: TRAC sustainable management FEC that universities recover will depend on the The guidance document An overview of TRAC price that they set. Research Councils currently pay defines sustainable management as follows: 80% of the FEC of the research projects they fund, which is intended to rise to 100% in the next few ‘An institution is being managed on a years. Government Departments, however, are sustainable basis if, taking one year with now expected to pay 100% of the FEC. another, it is recovering its full economic costs across its activities as a whole, and is investing There is a broad consensus in the sector that the in its infrastructure (physical, human and move towards the FEC approach is beneficial, intellectual) at a rate adequate to maintain but some concerns persist as to whether Research its future productive capacity appropriate to Councils’ increased contributions are causing them the needs of its strategic plan and students, to fund less research, about the use of funding body sponsors and other customers’ requirements.’ money to meet the balance of support, and about how money is dispensed to departments through university administrations.

Annex A - 3 From RAE to REF Acronyms QR funding is another area undergoing major reform. The purpose of the Research Assessment BIS Department for Business, Innovation Exercise (RAE), first conducted in 1989, was to and Science enable funding bodies to derive a formula through FEC Full economic costs which to allocate funds to institutions selectively and for extensive periods to promote stability. HEFCE Higher Education Funding Council The system’s focus on rewarding performance is for England claimed to have had a positive effect on the volume HEIs Higher education institutions and quality of research outputs. However, so profound has been the influence of RAE scores on QR Quality-related [funding] universities’ reputation and ranking that immense RAE Research Assessment Exercise time and effort has been put into maximizing REF Research Excellence Framework scores. Concerns about the intensiveness of the system also played into fierce debates around the funding bodies’ policies, including the degree of and develop the approach to impact evaluation; selectivity in funding (how much is awarded to these will conclude in the autumn of 2010. On top research universities as opposed to others). that basis, it is anticipated that the REF will be In 2006 the Government announced its intention introduced in 2014. that the RAE should be replaced after 2008 with a new assessment system to be termed Research Excellence Framework (REF). This was intended About the Research Information Network to reduce the administrative burden on HEIs, The Research Information Network has been make less use of academic panels to judge established by the Higher Education funding research quality and more use of statistics, bodies, the Research Councils and the National including the number of times research is cited Libraries in the UK. We investigate how by other researchers (known as bibliometrics) efficient and effective the information services and departments’ external research income. provided for the UK research community are, Following two rounds of public consultation how they are changing and how they might in 2007 and 2009, and a series of pilot projects, be improved for the future. We help to ensure HEFCE significantly modified the proposal: the that researchers in the UK benefit from world- use of bibliometric indicators will feature less leading information services so that they can prominently than initially suggested; instead, the sustain their position among the most successful REF will be essentially a process of expert review, and productive researchers in the world. informed by indicators where appropriate; it will We provide policy, guidance and support, thus bear much similarity to the RAE. However, focussing on the current environment in one important innovation will be the incorporation information research and looking at future into the new assessment system of measures aimed trends. All our publications are available at evaluating the impact of research on the UK on our website at www.rin.ac.uk economy, society, public policy, culture and quality of life. Further pilot projects are being run to test

Useful resources • Science & innovation investment framework • Research Excellence Framework website 2004 – 2014 (July 2004) www.hm-treasury.gov. www.hefce.ac.uk/Research/ref uk/spending_review/spend_sr04/associated_ • Research Councils UK www.rcuk.ac.uk documents/spending_sr04_science.cfm • The financial information in this document • An overview of TRAC (June 2005) www.jcpsg. is derived from a variety of sources, which ac.uk/guidance/downloads/Overview.pdf are detailed at www.rin.ac.uk/making- • Research Assessment Exercise website sense-funding www.rae.ac.uk

Annex A - 4 International Review of Mathematical Sciences 2010 PART I - Annex B Information for the Panel Previous Review Recommendations

Annex B Main Recommendations to the IRM 2003 and the Review of Operational Research and Report on Subsequent Actions

Introduction EPSRC has invested more than £30M in direct response to recommendations made in the above reviews, mainly through Science & Innovation awards, a Council-wide scheme that operated until 2009/10 aimed at increasing research capacity in threatened areas. In addition, some other investment reflecting current EPSRC policy aims has been directed to areas highlighted in the reviews. A summary of actions is set out below.

Research Training “The [PhD] programme is of shorter duration and more narrowly focused than those in most other countries. As a result, PhD graduates have difficulty competing for research fellowships and academic posts…”

In 2005 Research Councils increased funding levels to support an average PhD length of 3.5 years.

Working with the community, EPSRC has funded 6 Taught Course Centres (TCC) that provide a range of short instructional courses across the breadth of the mathematical sciences. These provide mainly first year PhD students with opportunities to widen their mathematical knowledge by choosing to study a number of topics outside the area of their PhD. Four TCCs based on regional groupings of universities provide courses in pure and applied mathematics and some statistics, while 2 TCCs provide a national facility in statistics and operational research. Pump-priming funding by EPSRC continues until autumn 2011.

In preparation for consideration of further support, EPSRC has funded a management review of the TCCs to test whether they are meeting the objectives of the initiative and that organisation and management arrangements are fit for purpose. This review has confirmed that the centres are meeting the objectives and are generally working well. There is scope to improve the sharing of information between centres and to adopt best practice. A short extract from the management review is included in this briefing document.

Research Career Development Although not a direct response to the reviews, current EPSRC policy puts significant emphasis on the need to nurture and develop the best researchers all through their careers. One of EPSRC’s strategic objectives for the next funding period is explicitly concerned with the development and support of research leaders. EPSRC provides various schemes across the career path: • Postdoctoral fellowships (up to 3 years post-PhD experience) • Career acceleration fellowships (3-10 years experience) • First grants (first time applicant to EPSRC within 3 years of permanent appointment) • Leadership fellowships ( to develop or consolidate world-leading research status) • Programme grants (major, potentially transformative research programmes led by world- leaders in field) More consolidated funding arrangements may be expected to be developed.

Research opportunities – Statistics Despite the high quality of research in statistics, the review identified serious shortages in producing and attracting people to fill academic posts, exacerbated by strong demand for statisticians from industry and Government. EPSRC has funded 3 Science & Innovation awards in statistics at Bristol, Cambridge and Warwick Universities (total ~£10M), aimed at increasing research capacity by funding new academic posts. The Mathematical Sciences programme also funded a small number of Statistics Mobility fellowships to draw researchers from other disciplines into statistics.

Subsequently, the programme has funded 2 Centres for Doctoral Training (CDT) at Lancaster (statistics and OR) and Warwick (statistics, applied maths and probability) to improve the supply of PhD graduates. The programme has also recently renewed funding for the National Centre for

Annex B - 1 International Review of Mathematical Sciences 2010 PART I - Annex B Information for the Panel Previous Review Recommendations Statistical Ecology (St Andrews and Kent Universities leading a consortium) for which NERC has agreed 50% of the funding.

Research Opportunities – Analysis and PDEs The IRM found that UK research in analysis and PDEs was sparse and sporadic, perhaps because the short duration of PhD training discouraged students from opting for this area. EPSRC has funded 2 S&I awards at Oxford and Edinburgh Universities, together with, in 2009, a CDT in analysis at Cambridge University.

Research Opportunities – Discrete algorithms and computational complexity; numerical analysis and computational science The IRM considered that the UK was under-represented in these fields and failing to take advantage of research opportunities. EPSRC has funded an S&I award at Warwick University (£4M) in discrete algorithms and another at Edinburgh (£5M) in numerical analysis, with emphasis on developing new algorithms for novel architecture computing.

Research Opportunities – Operational Research One S&I award to the LANCS consortium (Lancaster, Nottingham, Cardiff and Southampton Universities) (~£5M) to increase capacity and focus support at leading centres, working together; and a CDT at Lancaster in statistics and operational research to improve the supply of new PhDs.

Research Opportunities – Mathematical biology and its new dimension The Mathematical Sciences programme has been working on development of an initiative in ‘new maths for biology’ together with EPSRC’s Cross-disciplinary programme. A one-week sandpit was held in 2009 on the topic of evolutionary processes, from which research projects of some £2M, mostly to mathematicians, resulted.

Annex B - 2 International Review of Mathematical Sciences 2010 PART I - Annex C Information for the Panel EPSRC Mathematical Sciences Programme Overview 2010

Annex C EPSRC Mathematical Sciences Programme Overview 2010

Scope of the Overview This section describes the scope and remit of the Mathematical Sciences programme together with its context, strategy and funding over recent years and the makeup and highlights of its activities.

Scope and Remit The EPSRC Mathematical Sciences programme funds research and training concerned with new mathematical knowledge and techniques. Its remit spans pure and applied mathematics, statistics, probability and the mathematical foundations of operational research. Generally, it does not fund the application of existing mathematical knowledge to other subjects and disciplines. However, boundaries are not fixed and some degree of joint funding between the programme and other programmes within EPSRC and or other Research Councils occurs. For example, the Isaac Newton Institute at Cambridge is funded two thirds by the Mathematical Sciences programme and one third by a consortium of other programmes – physical sciences, engineering and ICT, in recognition of the relevance of many of the programmes run at the INI to application areas.

Context Within the current organisation of EPSRC, the Mathematical Sciences programme is one of seven programmes in the Research Base Directorate. These seven programmes represent the areas and disciplines that are at the heart of EPSRC’s broad remit. They are augmented by theme- based programmes (such as Energy) aimed at addressing today’s significant societal challenges. The Research Base programmes provide the fundamental capability in research to enable the UK to respond to advances and developments in any area of science and technology and to tackle new societal challenges as they emerge.

The Mathematical Sciences are underpinning disciplines that provide knowledge and techniques that may be applicable across the whole scope of science, including areas within the remits of other Research Councils. Funding the mathematical sciences represents an investment in long term, in the expectation that advances in knowledge will provide the foundation for new applications, albeit that the timing and particular area of impact cannot be predicted.

Some of the more applied aspects of research in the mathematical sciences have more immediate potential impact and in these cases, such as operational research, industrial mathematics and financial mathematics, greater emphasis is placed on connections with application areas to ensure that the results of research are taken up in the most effective manner.

Strategy The programme strategy has been formulated on the basis of contributing to the broader strategic objectives set by EPSRC within successive Delivery Plans. In the current Delivery Plan period, ending March 2011, the strategy has been: • To encourage the development of transformative research among leading researchers to increase the impact of mathematics research on a world scale; • To promote better and more extensive cross-disciplinary connections between the mathematical sciences and application areas, especially in engaging with the challenges of the societal themes; • To maintain and develop a fundamental capability for research across the whole scope of the mathematical sciences.

Programme activities Funds have been allocated over the last three years to support transformative research through ‘Programme grants’. The community has been slow to respond to this initiative and just one programme grant (Professor RM Thomas, Imperial College) has been awarded so far. Two other applications (also in pure maths) are presently under review.

The community has needed much encouragement to develop better cross-disciplinary links and to engage with the societal themes. For two years, the programme ‘sign-posted’ opportunities to obtain earmarked funding for research in these areas, but received very few proposals. In the current year, a more active approach has been taken through a ‘call for proposals’ for research

Annex C - 1 International Review of Mathematical Sciences 2010 PART I - Annex C Information for the Panel EPSRC Mathematical Sciences Programme Overview 2010 underpinning Energy and/or the Digital Economy, which has resulted in a satisfactorily large response. £5M has been invested as a result of this call from the Mathematical Sciences, Energy and Digital Economy programmes.

Support for the final objective has been made available through responsive mode grants, for which the community has a strong affinity, together with emphasis of maintaining the flow of people into research, through the doctoral training and fellowships programmes.

Other significant programme activities over recent years include: • Funding for research meetings and symposia through the Isaac Newton Institute (INI), the International Centre for Mathematical Sciences (ICMS), the Durham and Warwick Symposia; • Taught Course Centres and specialist course provision; • Doctoral Training Account and Centres for Doctoral Training; • Platform Grants • Larger, longer grants • Knowledge Transfer Network Internships; • International links

Research meetings and symposia The Mathematical Sciences programme has been the principal funder of the Isaac Newton Institute (INI) since its foundation in 1991. Current funding is £9.6M over 6 years to 2014 (by far the single largest element of programme funding). An interim review of this grant, coupled with an application for funding by other Research Councils is currently underway. The INI, hosted by Cambridge University, has a world-wide reputation for organising high-quality programmes in the mathematical sciences and its application areas, and it provides a resource for the whole UK community.

The programme also funds the International Centre for Mathematical Sciences (ICMS), hosted by Edinburgh/Heriot-Watt Universities, (£2M over 4 years to 2012) which provides similar programmes to the INI but generally of shorter duration. It too has an international reputation and serves the whole UK community.

The programme also funds the Durham and Warwick Symposia. It has recently funded the former on a multi-year basis and is considering a similar funding arrangement for Warwick.

Taught Course Centres and specialist course provision Six ‘Taught Course Centres’ (TCCs) have been established in response to the recommendation in the 2003 International Review that UK doctoral training was narrow in focus compared to international standards, reducing the competitiveness of doctoral graduates in the jobs market. EPSRC has recently undertaken a management review of the centres in preparation for consideration of further funding when existing support expires in late 2011. Further information is given in section 3.5.2. In addition, the programme funds the provision of specialist courses for doctoral students managed by the London Mathematical Society and the Royal Statistical Society.

Doctoral Training Account and Centres for Doctoral Training The programme invests about £11M a year in doctoral training, allocating funds through Doctoral Training Accounts. It uses a peer review process to determine allocations at departmental level based on published criteria. The allocations include some provision for CASE awards to encourage collaborative research with industry. EPSRC has largely withdrawn from masters funding but the Mathematical Sciences programme continues to fund masters in statistics and operational research in order to encourage graduates to continue to research training.

EPSRC has funded a major initiative to establish more than 50 Centres for Doctoral Training (CDT) aimed at improving the quality of doctoral training through integrated 4 year training programmes for groups or cohorts of students. Three CDTs in the mathematical sciences have been funded at Lancaster (statistics and operational research), Warwick (applied maths, probability and statistics) and Cambridge (analysis). This funding is additional to the programme’s doctoral training budget.

Platform Grants The programme has recently funded three Platform grants, aimed at providing infrastructural funding for the whole range of research activity within a department, to help ensure that the major suppliers to EPSRC are secure in a potentially austere financial climate. The expectation is that

Annex C - 2 International Review of Mathematical Sciences 2010 PART I - Annex C Information for the Panel EPSRC Mathematical Sciences Programme Overview 2010 holders of Platform grants will align their research strategies more closely to those of EPSRC and be in a better position to respond to initiatives by the Mathematical Sciences programme. Grants have been awarded to Oxford, Warwick and Manchester Universities and a fourth is in preparation. There is scope for additional Platform grants, funds permitting, and a variation based on topic- based infrastructural support spanning a number of universities working collaboratively.

Larger, Longer grants In addition to Programme grants (for transformative research) EPSRC is encouraging a general increase in the number of larger grants it funds since these give more scope for extended research projects and programmes and allow for greater adventure and risk. In the Mathematical Sciences programme, average grant values in responsive mode are just ~£0.2M. Nevertheless, the programme has funded a number of ‘critical mass’ grants in recent years, two of which have subsequently been renewed and others are likely to seek further funding over the next few years. Note that ‘large’ in relation to the Mathematical Sciences programme is about £1.5M (compared to, say, Physical Sciences £5-8M) in view of the limited funding available to the programme.

Knowledge Transfer Network Internships The Knowledge Transfer Network (KTN) in industrial mathematics (one of a series funded by the Technology Strategy Board) has developed an Internships scheme to enable doctoral students in mathematics to be seconded to a company for up to 6 months to work on a practical industrial project. The expectation is that a successful project will be of economic value to the company as well as giving the student an insight into working in industry and be of relevance to his/her PhD work. It is a model of knowledge transfer through people, a demonstration of the practical use and value of mathematics and a means of broadening the training of PhD students. The scheme is funded jointly by the Mathematical Sciences programme and the TSB.

International Links The mathematical sciences community is good at forging international links without the framework of specific collaboration initiatives and grants are available, subject to competition, to support travel, visits and collaborations. Nevertheless, the programme has supported small-scale initiatives, such as the European Science Foundation’s international networks programme to promote international collaboration. It has also funded a network in representation theory through a bilateral agreement with CNRS in France. Following the opening of RCUK offices in India and China, consideration is being given to initiatives with both these countries, which seem to offer good prospects for more formal connections. There is also scope for more explicit agreements within Europe, either through the EU or direct with national agencies. The programmes of the INI and ICMS also bring many international visitors to the UK and provide opportunities for UK academics to develop new collaborations. A recent trial of international peer review involving a mathematics proposal from groups in the USA, UK and Germany had mixed results, suggesting that better understanding is needed between national agencies before a common procedure could be established.

Funding Funding for the Mathematical Sciences programme has reduced over the current Delivery Plan period as a consequence of EPSRC’s decision to allocate funds to thematic programmes such as Energy and the Digital Economy. Research funding (i.e. new investments) over the last five years has been:

2006/7 2007/8 2008/9 2009/10 2010/11 £M 21.6 18 16 14 14

In addition, the programme has ~£11M a year for postgraduate training and ~£9M a year for fellowships, thus a total budget of ~£34M. This is less than 4% of EPSRC’s total budget. The programme has benefited from some funding that is additional to programme allocations - £10M for 3 CDTs and some £25M over the period 2005-10 for Science and Innovation awards (see section 3.4.1).

Programme organisation and management The programme team comprises a Head of Programme and 3 Portfolio Managers. EPSRC does not appoint subject specialists to its programmes, so none of the current staff have qualifications in the mathematical sciences. Specialist advice to support decisions on scientific matters is obtained from experts, in a process generally known as peer review. The programme also has a Strategic

Annex C - 3 International Review of Mathematical Sciences 2010 PART I - Annex C Information for the Panel EPSRC Mathematical Sciences Programme Overview 2010 Advisory Team (SAT) comprising a dozen members of the community, to act as a sounding board for the Head of Programme on strategy and planning, to provide feedback from the community and to provide intelligence on emerging research issues. Membership is rotated periodically. The programme team has regular liaison meetings with the relevant learned societies to discuss issues of common interest including an annual meeting attended by the EPSRC Chief Executive. The team also visit universities frequently, individually or collectively, to promote knowledge of the EPSRC programme and funding opportunities and to obtain feedback from the community on the programme and other relevant matters.

Looking Forward The UK Government is currently undertaking a Comprehensive Spending Review which will establish funding for Government Departments and their agencies (including Research Councils) for 4 years from April 2011. This is being conducted in a climate of severe financial constraint, so it is likely that funding will be lower than current levels. The outcome, at Departmental level, will be announced in October, but it may be as late as December before individual Research Council budgets (and thereafter, programme allocations) are known. Further information, if available, will be provided at the briefing meeting. Programme-level planning will not be undertaken until budgets are available; however, EPSRC’s strategic priorities for the next period are: • Shaping Capability • Developing Leaders • Maximising Impact

These strategic objectives set the framework for individual programmes to decide how their activities can best be organised to contribute to the achievement of EPSRC’s aims. In particular, ‘shaping capability’ implies, in a constrained funding environment, a need to make choices on what can be funded, or, perhaps, how much activity must be supported to maintain the capability to address future challenges across the EPSRC remit.

Other EPSRC Funding for the Mathematical Sciences Aside from the Mathematical Sciences programme itself and any joint funding with other programmes, some additional funding is also available from the Cross-Disciplinary programme. This programme exists to promote research and training at the interfaces between disciplines, which is a strong theme within the whole EPSRC programme. The Cross-Disciplinary programme (C-Dip) takes the lead on the broad topic of complexity which has a significant mathematical element. It has, for example, funded several CDTs in complexity. It has also worked jointly with the Mathematical Sciences programme in developing an initiative in ‘new maths for biology’. In 2009 a one week sandpit on the topic of evolutionary processes was held, from which research totalling some £2M was funded wholly by C-Dip. Most of the recipients were from the mathematical sciences.

The Mathematical Sciences Programme Research Landscape EPSRC programme teams periodically review the portfolio of research funded by programmes to assess the balance of funding across research topics and to analyse trends. The most recent review was in 2009. This programme landscape is quite separate from the landscape reviews set out elsewhere in this evidence document. The latter are written by experts within the community with the aim of informing the International Review Panel and represent an authoritative review of current status and trends in a particular area. By contrast the programme landscape is a review of what is funded by the EPSRC programme and the perceptions of programme staff of trends. To avoid duplication and possible confusion, a brief resume only of the latest programme landscape is set out below.

Algebra and geometry is the largest topic within the programme by value of funding and researchers in this area have been consistently successful in applying for and obtaining support in research and fellowships. The topic encompasses a variety of high-quality groups in specialised areas, so that the research supported by EPSRC is varied and dynamic. It is apparent that small groups can thrive in niche areas in this research area, as exemplified by small teams at Aberdeen and East Anglia Universities, as well as larger groups at Oxford, Imperial College, Bristol and Warwick.

By contrast, logic and combinatorics is one of the smallest topics. Research quality is generally high but there have been relatively few applicants for grants. The area also appears, from an

Annex C - 4 International Review of Mathematical Sciences 2010 PART I - Annex C Information for the Panel EPSRC Mathematical Sciences Programme Overview 2010 EPSRC perspective, to be shrinking, since the number of grant holders is less than half that recorded in 2005. There are some indications that activity in this area is migrating towards ICT.

Analysis is a wide-ranging and varied area within mathematics, with many departments having relatively small but good quality research groups in the topic. Funding from EPSRC has been stable in recent years at about 16% of the total by value. The award of two S&I awards in 2007 should help to focus research in analysis; more recently the award of a CDT at Cambridge could reinvigorate research. There is some concern at the low number of early career researchers obtaining research funding in this area.

Three topics covering continuum mechanics, non linear systems and numerical analysis are usually combined under the umbrella heading applied mathematics. This represents the second largest element within the portfolio and is even more diverse than algebra and geometry. Success rates among researchers have fallen in recent years. While there are individual high spots – CICADA at Manchester and applied non-linear mathematics at Bristol, there is generally a dearth of larger projects. The community has been slow to take advantage of the opportunities offered through engagement with EPSRC mission programmes.

While the RAE results show a large and high-quality community in statistics and probability, the topic continues to struggle for funding within the EPSRC programme. This may mean that researchers in this area have other sources of support and have less dependency on EPSRC funding than other topics. However, our impression is that this community is unduly critical of research proposals to the extent that proposals find it hard to get funding. Fellowships are scarce, as there is high demand for statistics graduates and PhDs from industry and Government. EPSRC funded three S&I awards in statistics to increase capacity and more recently a CDT at Lancaster covering statistics and OR.

The topic ‘mathematical aspects of operational research’ is also a relatively small within the programme. Our impression is that OR applications work is highly-rated and reasonably well- funded by other EPSRC programmes and by industry. There have been few underpinning projects funded through the Mathematical Sciences programme, but support has been boosted by the award of an S&I grant to the LANCS consortium and a CDT at Lancaster in statistics and OR. There may be scope, across EPSRC, to focus research in OR more explicitly on engagement with the mission programmes.

Mathematical physics represents about 10% of the programme by value. Research is of high quality and is spread among several prominent groups. There is some overlap in this area with STFC support for particle physics and astrophysics and remit boundary issues can be a problem for researchers and programme staff alike. The area may struggle to remain attractive to funders if financial constraints are severe.

Annex C - 5

International Review of Mathematical Sciences 2010 PART I - Annex D Information for the Panel EPSRC Mathematical Sciences Programme Overview 2010

Annex D Additional Funding Data from Other Research Councils

D.1 BBSRC support for Mathematical Biology

BBSRC classifies mathematical biology as research that involves the use of mathematical sciences to contribute to an understanding of biological systems. Mathematical, computational and statistical techniques are used to analyse experimental data, make predictions from data, discriminate between models and pose questions, which can guide future research efforts. To be classified as mathematical biology the project may include one or more of the following elements:

• Mathematical modelling; dynamic, deterministic, stochastic and spatial modelling; simulation; Markov, Monte Carlo and Bayesian methods. • Studies are often fully integrated within interdisciplinary / multidisciplinary projects, for example through systems biology approaches. • Spatio-temporal heterogeneity – systems that possess properties, which are distributed heterogeneously on all scales of interest within the spatial and/or temporal domain. • Stochasticity – systems that behave in an inherently probabilistic manner, or determining systems, which are subject to stochastic perturbation. • Scaling – systems where di fferent processes are i mportant at di fferent scales. For example, using i ndividual based models to develop population level models, development of models used on one time/spatial scale to study processes occurring on different time/spatial scales etc. • Data Analysis – new methods for analysis of complex information. Examples are novel approaches to bioinformatic analysis techniques or for combined analysis of different data types.

Overview of funding

Since financial year 2005/06 to financial year 2008/09 BBSRC’s support for mathematical biology has increased from £24.5M to £55.0M per annum. An analysis of the data from 2008/09 demonstrates that BBSRC provides support for mathematical biology by a number of funding routes which comprise: • Responsive Mode (£17.4M including £0.7M within the BBSRC Sponsored Institutes). • Managed Mode Initiative (£27.2M including £1.7M within the BBSRC Sponsored Institutes). • Institute Strategic Programme Grants (£9.1M awarded to the BBSRC Sponsored Institutes). • Research Fellowships (£1.3M).

In addition each year BBSRC supports circa 110 places on Masters courses and circa 2000 PhD studentships. Of these a significant proportion will incorporate mathematical biology. For more information on BBSRC funding mechanisms please see the Introduction to BBSRC document.

D.2 Overview of ESRC funding for Mathematical Sciences ESRC’s support for research in the mathematical sciences occurs through its support of quantitative research. Quantitative research is used throughout the social sciences and the objective is to develop and employ mathematical models, theories and/or hypotheses pertaining to phenomena. It is often contrasted with qualitative research which is the examination, analysis and interpretation of observations in a manner that does not involve mathematical models. Research methods used to gather quantitative data emphasise on objective measurements and numerical analysis of data collected through polls, questionnaires or surveys, and methods used to gather qualitative data focus on understanding social phenomena through interviews, personal comments etc.

The data included within this document are from ESRC funded grants which have been highlighted as having a quantitative element to their research. This includes grants funded through responsive mode and managed calls from all ESRC Research Programmes. Examples of the type of research techniques this would include are:

• Psychometrics - tasks aimed at assessing personality or certain managerial qualities, e.g. leadership ability.

Annex D - 1 International Review of Mathematical Sciences 2010 PART I - Annex D Information for the Panel EPSRC Mathematical Sciences Programme Overview 2010 • Experimental Design - methods to test hypotheses in positivist approaches. • Statistics and data analysis - study of the relationship between numbers representing social and economic activities Statistical Theory - hypotheses that are topically and internally consistent about the relationships between members representing social and economic activities. • Statistical Modelling Techniques - ways of representing by numbers the relationship between different social and economic activities. • Datasets and Services - numbers that represent economic and social activities and ways to make them available for secondary analysis. • Statistical Computing and Methodology - ways of using software to model concepts and data.

Many of ESRC’s long term infrastructure investments have a focus on quantitative research. For example, the National Centre for Research Methods (NCRM) was established in 2004 to promote a step change in the quality and range of methodological skills and techniques used by the UK social science community. This represents a £17.6M investment and its research has a strong focus on quantitative methodological development. ESRC also provide an enhanced stipend to students who use advanced quantitative methods as part of their PhD research, which demonstrates ESRC’s commitment to building capacity in these methods and their application to social sciences.

A different type of support for mathematical sciences by ESRC is the £3M joint Targeted Initiative on Science and Mathematics Education in partnership with organisations including the Department for Children, Skills and Families (DCSF), the Institute of Physics, the Association for Science Education, the Gatsby Foundation, and the council for Mathematical Sciences. This was launched in 2009 and the focus of the initiative is to support research which is capable of leading to significant increases in participation in science/mathematics in schools and colleges and significant enhancements in learners’ achievement in science/mathematics by children and young people in the UK up to, and including A level and transitions into higher education or the workplace.

D.3 Overview of MRC funding for Mathematical Sciences Overview commentary is available at the end of Part I. D.4 Overview of NERC funding for Mathematical Sciences

NERC funds research and monitoring conducted by the UK environmental sciences community, delivered by Higher Education Institutions and through NERC Centres. Environmental research is inherently multi-disciplinary, with mathematics embedded within it, underpinning major elements, such as modelling and the use of super-computers. NERC does not manage its investments, nor does it classify them, along disciplinary lines. This has limited the availability of hard data for NERC.

The only disciplinary classification available to us is the departmental affiliation of investigators on research grants. Searching for principal investigators affiliated to departments with mathematics and mathematics-associated names identifies 89 NERC grants over the period 2005/06 to 2009/10. This is the base data for the NERC submission.

An alternative analysis was attempted using modelling related science topics, but was an insufficiently selective proxy to give meaningful data. Likewise, no classification of Centre research projects could be used as a proxy to isolate mathematics.

Skills reviews have highlighted a requirement for environmental scientists with stronger maths and statistical skills. Statistical analysis, and maths and applied maths are two of the 33 priority knowledge/skills areas categorised as both being needed to address top priority challenges and currently being in shortage.

Annex D - 2 International Review of Mathematical Sciences 2010 PART I - Annex D Information for the Panel EPSRC Mathematical Sciences Programme Overview 2010 Value of NERC grants held by PIs based in mathematics departments

3.5 3.08 3.0

2.5 2.18 1.93 2.0 1.6 1.5

spend (£m) 1.0 0.64 0.5

0.0 2005/06 2006/07 2007/08 2008/09 2009/10

D.5 Overview of STFC funding for Mathematical Sciences STFC Astronomy and Space Science grants provide support for novel techniques, theory and modelling and data analysis covering the range of science from using our knowledge of space to understand fundamental physic of matter, particles and gravity to the origins of the Universe and our role within it. All of these areas require generation of theories, mathematical modelling and formulation and extensive use of data handling and analysis techniques, many of these being adapted for specific tasks. They also require access to high performance computing, most often provided by in-house systems, adapted to specific requirements of the research. Astronomers are highly numerate researchers many of which go on after training to careers in finance, where there continues to be a demand for their analytical and modelling skills. Grants include 5 year rolling grants (reviewed every 3 years), standard grants (e.g. responsive RA awards, typically 2/3 years) and Visiting Researcher grants. The percentage of any grant that can be identified as ‘maths’ is a rough estimate and should be used with caution. Higher percentages are given for pure theory awards, which by nature require the development of new mathematical techniques or adaptation of existing ones (an example might be the work of Stephen Hawking at Cambridge). However percentages are given to grants that mix modelling of complex, 3-dimensional systems with observational tests of theory (and example might be Seb Oliver’s work on cosmology at Sussex or Bob Nichol’s work on galaxy evolution at Portsmouth.

STFC Theoretical Particle Physics Grants

STFC Theoretical Particle Physics grants cover research in the following areas: • Formal Theory (e.g. supersymmetry, string theory)

• Phenomenology (application of theory to the particle physics experimental programme, e.g. LHC exploitation)

• Lattice QCD (a non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons, formulated on a grid or lattice of points in space and time). Types of theoretical particle physics grants include 5 year rolling grants (reviewed every 3 years), standard grants (e.g. responsive RA awards, typically 2/3 years) and Visiting Researcher grants. In addition, a ten-year grant, with critical review points built in, was awarded to support the Institute for Particle Physics Phenomenology in Durham in 2008. As these grants can cover various theoretical particle physics topics it has not been possible to identify the percentage of maths in the grants. Please note that the rolling grants are awarded for 5 years but reviewed after 3 years at which point a new grant is usually issued; therefore the commitment figure for the rolling grants is higher than the total spend figures.

Annex D - 3

International Review of Mathematical Sciences 2010 PART I - Annex E Information for the Panel Research Assessment Exercise (RAE)

Annex E Research Assessment Exercise (RAE)

This annex provides an introduction to the RAE, the 2008 RAE results and an analysis of trends in previous RAE results and submissions.

Introduction

The purpose of the Research Assessment Exercise is to rate the quality of research conducted in universities and higher education colleges in the UK. The quality ratings, together with data on the number of research active staff, are used to inform the allocation of over £1 billion per year for unspecified research by the Funding Councils.

The last RAE was in 2008, and further assessments were carried out in 2001, 1996 and 1992. The results which are relevant to mathematical sciences are presented here.

The RAE is a process of peer review by experts of high standing covering all subjects. All research assessed is allocated to one of 67 discipline-based ‘units of assessment’ (UoA), each with a panel of experts to agree ratings using their professional skills, expertise and experience: it is not a mechanistic process. Assessment is by voluntary submission of evidence to a panel; submissions consist of information about the academic unit being assessed, with details of up to four publications and other research outputs for each member of research-active staff.

RAE 2008 Results

The 2008 RAE results are presented below; for a detailed explanation of how the UK Funding Councils translated these results into QR allocations please refer to the Appendix of the LMS briefing note on Funding. Assessed research outputs were allocated a quality rating as defined in Table E-1 (with any submitted outputs below national quality or not recognised as research being unclassified). Quality judgements were then aggregated to create an overall ‘quality profile’ to describe each submission received. The quality profiles for submissions to the ‘Pure Mathematics’, ‘Applied Mathematics’ and ‘Statistics and Operational Research’ sub panels are given in Tables E- 2, E-3 and E-4 respectively. The submissions are ranked by the ‘Quality Index’, a weighted score proposed by HEFCE to help determine future funding allocations. The formula used is: QI = {(4*% x 7) + (3*% x 3) + (2*% x 1)} 7 Research rated 1* or unclassified is omitted. In the tables ‘†’ denotes the universities being visited/met with by the Mathematical sciences International Review panel.

Table E-1 Definitions of 2008 RAE ratings

Rating Description 4* Quality that is world-leading in terms of originality, significance and rigour. Quality that is internationally excellent in terms of originality, significance and 3* rigour but which nonetheless falls short of the highest standards of excellence. Quality that is recognised internationally in terms of originality, significance 2* and rigour. Quality that is recognised nationally in terms of originality, significance and 1* rigour. Quality that falls below the standard of nationally recognised work. Or work Unclassified which does not meet the published definition of research for the purposes of this assessment.

Annex E - 1 International Review of Mathematical Sciences 2010 PART I - Annex E Information for the Panel Research Assessment Exercise (RAE)

Table E-2 Pure Mathematics unit of assessment 2008 RAE ratings

Overall quality profile (percentage of research activity at each quality level) FTE Category A Quality staff submitted 4* 3* 2* 1* unclassified Index Imperial College London† 21.8 40 45 15 0 0 61.4 University of Warwick† 32 35 45 20 0 0 57.1 University of Oxford† 55.16 35 40 25 0 0 55.7 University of Cambridge† 55 30 45 25 0 0 52.9 University of Bristol† 34.53 30 40 25 5 0 50.7 Joint submission: Universities of Edinburgh† and Heriot-Watt† 41 25 45 30 0 0 48.6 University of Bath† 10 25 35 40 0 0 45.7 King's College London† 13 20 50 25 0 5 45.0 University of Aberdeen† 14 20 45 35 0 0 44.3 University College London† 15.25 20 40 35 5 0 42.1 University of Durham† 15 20 40 35 0 5 42.1 University of Manchester† 27 20 40 35 5 0 42.1 University of East Anglia† 7 15 45 35 5 0 39.3 University of Sheffield† 17.25 15 40 45 0 0 38.6 Queen Mary, University of London† 20.2 10 50 40 0 0 37.1 University of Birmingham† 18 15 40 35 5 5 37.1 University of Glasgow† 16.32 15 40 35 10 0 37.1 University of Nottingham† 15 15 35 45 5 0 36.4 Loughborough University† 11.4 10 45 45 0 0 35.7 University of Exeter† 5 10 45 40 5 0 35.0 University of Leeds† 23.2 10 45 40 5 0 35.0 University of Leicester† 10 10 40 50 0 0 34.3 Lancaster University† 10 10 40 35 15 0 32.1 University of York† 12.34 10 35 50 5 0 32.1 University of Liverpool† 15 10 35 45 10 0 31.4 University of Southampton† 15.75 5 45 40 10 0 30.0 London School of Economics and Political Science 12.5 5 40 50 5 0 29.3 Queen's University Belfast† 8.2 5 40 50 5 0 29.3 London Metropolitan University 4 10 25 50 15 0 27.9 Swansea University 20.5 5 35 50 10 0 27.1 Aberystwyth University 8.3 5 35 45 15 0 26.4 Cardiff University 30.45 5 35 45 15 0 26.4 University of Newcastle upon Tyne† 10 5 30 60 5 0 26.4 University of St Andrews† 12 5 30 55 10 0 25.7 University of Kent† 6 0 35 55 10 0 22.9 Open University 16.5 5 25 40 30 0 21.4 Royal Holloway, University of London 26.6 0 25 35 20 20 15.7

*WIMCS (Wales Institute of Mathematical and Computer Science) will represent the Welsh institutions during the review week.

Annex E - 2 International Review of Mathematical Sciences 2010 PART I - Annex E Information for the Panel Research Assessment Exercise (RAE)

Table E-3 Applied Mathematics unit of assessment 2008 RAE ratings

Overall quality profile (percentage of research activity at each quality level) FTE Category A Quality staff submitted 4* 3* 2* 1* unclassified Index University of Cambridge† 80.3 30 45 25 0 0 52.9 University of Oxford† 54.25 30 45 25 0 0 52.9 University of Bristol† 38 25 45 30 0 0 48.6 University of Warwick† 29.25 30 30 35 5 0 47.9 University of St Andrews† 15 25 45 25 5 0 47.9 University of Bath† 19.8 20 50 30 0 0 45.7 University of Manchester† 28.8 25 35 40 0 0 45.7 Imperial College London† 37.2 20 45 35 0 0 44.3 University of Portsmouth† 16 15 60 25 0 0 44.3 University of Durham† 26 15 60 20 5 0 43.6 University of Nottingham† 34.4 20 45 30 5 0 43.6 University of Southampton† 17 15 55 30 0 0 42.9 University of Surrey† 17.42 15 55 25 5 0 42.1 Joint submission: Universities of Edinburgh† and Heriot-Watt† 35.46 15 50 30 5 0 40.7 King's College London† 18 15 50 25 10 0 40.0 Keele University† 9 15 45 35 5 0 39.3 University of Liverpool† 23.33 15 45 35 5 0 39.3 University of Newcastle upon Tyne† 11 15 45 35 5 0 39.3 University of Leeds† 24.9 15 40 40 5 0 37.9 University of Exeter† 18 10 50 40 0 0 37.1 University of York† 22 10 40 45 5 0 33.6 University of Strathclyde† 24.33 10 40 45 5 0 33.6 Brunel University† 19 5 55 30 10 0 32.9 University of Kent† 6 10 35 55 0 0 32.9 Loughborough University† 20.4 10 40 40 10 0 32.9 Queen Mary, University of London† 15 10 40 40 10 0 32.9 University of Sussex 13 10 40 40 10 0 32.9 University College London† 16.5 10 40 40 10 0 32.9 University of Glasgow† 19 10 40 40 10 0 32.9 University of Sheffield† 17 10 35 50 5 0 32.1 University of Dundee† 10.7 10 35 45 10 0 31.4 University of Birmingham† 25 10 35 35 20 0 30.0 University of Leicester† 12 5 40 50 5 0 29.3 University of East Anglia† 7 5 40 45 10 0 28.6 University of Reading† 21.3 5 35 45 15 0 26.4 City University, London 8.2 5 15 65 15 0 20.7 Coventry University 7 0 20 70 10 0 18.6 University of Plymouth 4 0 25 55 20 0 18.6 University of Glamorgan 4 0 20 45 30 5 15.0 University of Brighton 3 0 15 45 35 5 12.9 University of Chester 5.8 0 15 45 35 5 12.9 Oxford Brookes University 5.71 0 10 55 35 0 12.1 University of the West of England, Bristol 9 0 10 50 40 0 11.4 Glasgow Caledonian University 1 5 5 30 60 0 11.4 Staffordshire University 1 0 0 0 95 5 0.0

Annex E - 3 International Review of Mathematical Sciences 2010 PART I - Annex E Information for the Panel Research Assessment Exercise (RAE) Table E-4 Statistics and Operational Research unit of assessment 2008 RAE ratings

Overall quality profile (percentage of research activity at each quality level) FTE Category A Quality staff submitted 4* 3* 2* 1* unclassified Index University of Oxford† 24.5 40 50 10 0 0 62.9 University of Cambridge† 16 30 45 25 0 0 52.9 Imperial College London† 13.9 25 50 25 0 0 50.0 University of Bristol† 23 25 45 30 0 0 48.6 University of Warwick† 24 25 45 30 0 0 48.6 University of Leeds† 11 25 40 30 5 0 46.4 University of Nottingham† 9 20 50 30 0 0 45.7 University of Kent† 12 20 45 30 5 0 43.6 University of Bath† 15 20 40 35 5 0 42.1 University of Southampton† 28 15 50 30 5 0 40.7 Lancaster University† 21.65 15 45 35 5 0 39.3 University of Manchester† 10.9 20 35 30 15 0 39.3 London School of Economics and Political Science 13 15 40 35 5 5 37.1 University of St Andrews† 7 10 50 35 5 0 36.4 Brunel University† 10 15 35 40 10 0 35.7 University of Sheffield† 10.7 10 50 30 10 0 35.7 University of Glasgow† 13 15 35 40 10 0 35.7 University of Newcastle upon Tyne† 13 10 45 40 5 0 35.0 Open University 7 10 40 45 5 0 33.6 University College London† 13.5 10 40 40 10 0 32.9 Joint submission: Universities of Edinburgh† and Heriot-Watt† 30 10 35 45 10 0 31.4 University of Durham† 11.6 5 45 45 5 0 30.7 Queen Mary, University of London† 8.2 10 30 45 15 0 29.3 University of Strathclyde† 10.33 10 30 45 15 0 29.3 University of Reading† 7.7 5 30 55 10 0 25.7 University of Greenwich 2 0 40 40 20 0 22.9 University of Salford 9.8 0 35 55 10 0 22.9 University of Liverpool† 5 0 35 50 15 0 22.1 London Metropolitan University 4 5 20 40 35 0 19.3 University of Plymouth 4 0 30 45 25 0 19.3

RAE 2008 Sub-panel Subject Overviews

Each of the 67 RAE2008 sub-panels produced an overview report of their discipline, based on the evidence provided in submissions. The Pure Mathematics, Applied Mathematics and, Statistics and Operational Research units were assessed by the same expert panel. All three overview documents discuss similar topics, including research strengths, the research environment, and the interactions with industry and other stakeholders. These overviews are attached at the end of this annex.

Previous RAE Results

Figure E-1 shows that there has been a steady improvement in the selected group of institutions. However it is important to note that after RAEs it is usual for several lowly-ranked departments to close, hence the total numbers of institutions entering the assessment have gradually decreased since 1992. Key examples of these include the closures of the Mathematics departments of Hull University after the 2001 RAE, when it achieved a rating of 4, and Bangor in 2006, after it had

Annex E - 4 International Review of Mathematical Sciences 2010 PART I - Annex E Information for the Panel Research Assessment Exercise (RAE) achieved a rating of 3a. The number of staff submitted has gradually increased in Pure and Applied Maths, but has fallen in Statistics and OR.

The data below was derived by using only those institutions which were present in the 1992, 1996 and 2001 RAEs, in order to gain a direct comparison (in 2008 a different rating system was used). In total there were 39 institutions in the Pure Mathematics UoA, 49 in the Applied Mathematics UoA, and 41 in the Statistics and Operational Research UoA, and these are listed in Table E-5.The total number of institutions entered for each year, and the total number of FTE category A/A* staff are listed in Table E-6

Figure E-1 RAE ratings for ‘Pure Mathematics’, ‘Applied Mathematics’ and, ‘Statistics and Operational Research’ (1992, 1996, 2001)

Pure Mathematics Increasing Quality

2001 3b 3a 4 5 5*

1996 3b 3a 4 5 5*

1992 2 3b/3a 4 5/5*

0% 20% 40% 60% 80% 100%

Applied Mathematics Increasing Quality

2001 3b 3a 4 5 5*

1996 2 3b 3a 4 5 5*

1992 1 2 3b/3a 4 5/5*

0% 20% 40% 60% 80% 100%

Statistics and Operational Research Increasing Quality

2001 3a 4 5 5*

1996 2 3b 3a 4 5 5*

1992 2 3b/3a 4 5/5*

0% 20% 40% 60% 80% 100%

Annex E - 5 International Review of Mathematical Sciences 2010 PART I - Annex E Information for the Panel Research Assessment Exercise (RAE) Table E-5 Institutions included in RAE data

Institutions included in Pure Mathematics RAE Data Goldsmith’s College 1 University of Bath University of Liverpool Imperial College London University of Birmingham University of Manchester King’s College London University of Bristol University of Newcastle Open University University of Cambridge University of Nottingham Queen Mary and Westfield College 2 University of Durham University of Oxford Queen’s University of Belfast University of East Anglia University of Reading Royal Holloway and Bedford College 3 University of Edinburgh University of Sheffield UMIST 4 University of Exeter University of Southampton University College London University of Glasgow University of St. Andrews University College of North Wales 5 University of Hull University of Sussex University College Swansea 6 University of Lancaster University of Wales College Cardiff 8 University College of Wales 7 University of Leeds University of Warwick University of Aberdeen University of Leicester University of York Institutions included in Applied Mathematics RAE Data Brunel University University College of Wales 7 University of Manchester Chester College 9 University of Bath University of Newcastle London Guildhall University 10 University of Birmingham University of Nottingham City University University of Bristol University of Plymouth Coventry University University of Cambridge University of Portsmouth De Montfort University University of Dundee University of Reading Glasgow Polytechnic 11 University of Durham University of Sheffield Heriot-Watt University University of East Anglia University of Southampton Imperial College London University of Edinburgh University of St. Andrews King’s College London University of Exeter University of Strathclyde Loughborough University University of Glasgow University of Surrey Nottingham Trent University University of Hull University of Sussex Oxford Brookes University University of Keele University of Teesside Queen Mary and Westfield College 2 University of Kent University of the West of England Staffordshire University University of Leeds University of York UMIST 4 University of Leicester University College London University of Liverpool Institutions included in Statistics and Operational Research RAE Data Brunel University University of Bath University of Manchester City University University of Birmingham University of Newcastle Coventry University University of Bristol University of North London 10 Goldsmith’s College 1 University of Cambridge University of Nottingham Heriot-Watt University University of Durham University of Oxford Imperial College London University of Edinburgh University of Reading London School of Economics and University of Exeter University of Salford Political Science University of Glasgow University of Sheffield Nottingham Trent University University of Greenwich University of Southampton 12 Open University University of Keele University of St. Andrews Queen Mary and Westfield College 2 University of Kent University of Strathclyde UMIST4 University of Lancaster University of Surrey University College London University of Leeds University of Sussex University of Aberdeen University of Liverpool University of Warwick

1 Now Goldsmiths, University of London 9 Now University of Chester 2 Now Queen Mary, University of London 10 London Guildhall, City of London Polytechnic and 3 Now Royal Holloway, University of London University of North London are now merged as 4 ‘’University of Manchester Institute of science and London Metropolitan University Technology’ now merged with the University of 11 Now Glasgow Caledonian University Manchester 12 In 1996 Southampton entered two departments 5 Now Bangor University (‘Statistics’ and ‘Operational Research’) for the 6 Now Swansea University Statistics and OR UoA. The highest scoring 7 Now Aberystwyth University (Operational Research) is used in Figure G-1. In 8 Now Cardiff University 2001 both departments were awarded the same

Annex E - 6 International Review of Mathematical Sciences 2010 PART I - Annex E Information for the Panel Research Assessment Exercise (RAE) grade, so this is used (still only counting one entry for the University of Southampton)

Table E-6 Number of Institutions and Staff submitted to the RAE (1992, 1996, 2001, 2008)

RAE Number of Institutions Number of Category A/A* Staff Submitted year Pure Applied Stats/OR Pure Applied Stats/OR 2008 38 46 31 685.25 850.05 388.78 2001 47 58 45 509.5 734.71 386.64 1996 45 65 54 468 721.5 423.7 1992 44 67 50 506.4 770.6 407

Annex E - 7

International Review of Mathematical Sciences 2010 PART I - Annex F Information for the Panel Additional Bibliometric Evidence

Annex F Additional Bibliometric Evidence

Introduction

Bibliometric data relating to ‘Mathematics’ publications are included below to provide the Panel with information on the relative performance of the UK internationally. The information was sourced from Thomson Reuters’ (formerly ISI, the Institute of Scientific information) Web of Knowledge ‘Essential Science Indicators’ (ESI) in July 2010.

Some Caveats

When using bibliometric methods to analyse research performance a number of limitations need to be borne in mind. The underpinning data is often incomplete and can be subject to human error. Varying degrees of bias can be introduced by changes in the relative popularity of research fields and behavioural differences depending on the research field and country. In addition, international coverage is problematic: although Thomson Reuters index journals from over 60 countries, and have increased their coverage of LOTE (languages other than English) journals, the coverage is still uneven, and there are still few non-English language publications included, with very few from the less-developed countries. Citations are limited to Journal articles and review articles, i.e. not including books, book chapters, conference papers etc. for either publication or citation counts. Lastly, citations are not always an indication of quality - not all work is cited because of its excellence. Taken together, these factors mean that a simple count of citations may be a poor proxy for the real, positive “impact” of an investigator’s work within the research community.

Despite these cautions citation rates are internationally recognized as a useful indicator of relative research quality in research fields where the journal coverage is good and where publication in journals is the usual mode of disseminating research outputs. It is possible to exclude self citations, and studies have shown strong correlation between citations and quality assessed by other means, such as peer review.

Global Citation Rates

Table F1 presents global citation rates for mathematical sciences in five year intervals since 2002.

Table F1 Mathematics 5 year interval citation rates (ESI)

10.1.4 Five 10.1.5 2002- 10.1.6 2003- 10.1.7 2004- 10.1.8 2005- 10.1.9 2006- years intervals 2006 2007 2008 2009 2010

10.1.10 Number 110,541 115,585 124,552 131,286 134,498 of papers

10.1.11 Times 139,045 149,428 171,822 206,398 222,662 cited

10.1.12 Citations 1.26 1.29 1.38 1.57 1.66 per paper

Annex F - 1 International Review of Mathematical Sciences 2010 PART I - Annex F Information for the Panel Additional Bibliometric Evidence

Table F2 presents the top 30 countries ranked by the total number of citations over the 10 years and four months to 30th April 2010 in the ESI field of Mathematics. Data for the UK is collated from data for the constituent countries of England, Wales, Scotland and Northern Ireland, which may result in the double counting of collaborative papers. The overall position for the UK is 5th.

Table F2 Country rankings for Mathematical sciences field (top 30 ESI)

Ranking Country Papers Citations Citations/paper 1 USA 66,764 305,546 4.58 2 France 23,071 87,984 3.81 3 China 27,458 78,992 2.88 4 Germany 18,795 71,813 3.82 5 UK 14,876 63,653 4.28 6 Italy 13,287 44,910 3.38 7 Canada 11,994 43,312 3.61 8 Spain 11,038 36,252 3.28 9 Japan 12,644 32,918 2.60 10 Australia 5,632 24,379 4.33 11 Russia 12,556 19,523 1.55 12 Israel 4,919 17,986 3.66 13 The Netherlands 3,498 14,869 4.25 14 Belgium 3,283 13,166 4.01 15 Poland 5,533 13,045 2.36 16 South Korea 5,338 12,862 2.41 17 Brazil 4,220 12,478 2.96 18 Switzerland 2,793 12,109 4.34 19 Sweden 2,856 11,844 4.15 20 Austria 2,553 10,502 4.11 21 Taiwan 3,492 10,163 2.91 22 India 4,745 8,935 1.88 23 Czech Republic 2,507 7,488 2.99 24 Denmark 1,382 6,691 4.84 25 Hungary 2,636 6,685 2.54 26 Greece 2,062 6,347 3.08 27 Finland 1,597 6,286 3.94 28 Romania 2,464 5,967 2.42 29 Singapore 1,376 5,860 4.26 30 Norway 1,351 5,845 4.33

Annex F - 2 International Review of Mathematical Sciences 2010 PART I - Annex F Information for the Panel Additional Bibliometric Evidence

Global Institution Rankings

ESI lists a total of 194 academic/industry-based research institutions in Mathematics globally, of which there are 9 from the UK. Based on cites per paper the ranking of UK institutions ranges from 19th to 143rd.The UK has 4 out of the top 100 institutions based on number of citations; and 5 out of the top 100 based on number of papers. 6 of the top 100 institutions in terms of citations per papers are from the UK; these are Imperial (22nd), Oxford (32nd), Bristol (51st), Cambridge (52nd), Warwick (92nd) and Nottingham (100th). Table F3 presents the rank order of the 9 UK universities ordered by citations.

Table F3 World ranking of UK universities Mathematical sciences citation rates (number of citations (ESI))

10.1.19 10.1.17 10.1.18 World 10.1.16 World World ranking 10.1.14 10.1.15 Citations ranking ranking (citations 69 10.1.13 Institution papers Citations per paper (papers) (citations) per paper) Oxford 1,276 7,437 5.83 14th 14th 29th Imperial 750 4,888 6.52 63rd 32nd 19th Cambridge 689 3,546 5.15 85th 57th 49th Warwick 697 3,183 4.57 83rd 74th 89th Manchester 641 2,532 3.79 98th 106th 136th Bristol 442 2,283 5.17 165th 115th 48th Nottingham 486 2,172 4.47 142nd 125th 97th Leeds 531 1,973 3.72 128th 151st 143rd Bath 398 1,757 4.41 176th 187th 101st

Citation Ratings of EPSRC Researchers

Analysis has shown that approximately 1760, or 36%, of researchers supported by EPSRC during the five years 2001-2005 inclusive were awarded 80% of the total research funding. To assess the performance of this group relative to others in the UK and the rest of the world we commissioned Evidence Ltd. in 2006 to carry out a study of their citation impact profiles. Sections of the report relevant to mathematical sciences are available upon request. A more recent study is yet to be commissioned.

Their report showed that the majority of UK output in mathematical sciences-related fields70 is above world average in terms of citations. It also appears that the top EPSRC researchers generally have better citation profiles than other UK researchers, with fewer uncited papers, and a higher proportion above world average for the field and year.

69 For institutions producing at least 20 papers per year 70 The report contains entries for “Mathematics” and “Engineering Mathematics”

Annex F - 3

International Review of Mathematical Sciences 2010 PART I - Annex G Information for the Panel Evidence Framework

Annex G Evidence Framework

The use of a standard framework ensures structured coverage of all relevant strategic issues; the public consultation exercise in particular sought evidence in response to the framework questions, which ar e provided her e for r eference by the pa nel during their visits and particularly when formulating their conclusions and recommendations. The framework is not intended to restrict the panel; additional issues should be addressed by the panel as they arise.

A. What is the standing on a global scale of the UK Mathematical Sciences research community both in terms of research quality and the profile of researchers? • Is the UK internationally leading in Mathematical Sciences research? In which areas? What contributes to the UK strength and what are the recommendations for continued strength? • What are the opportunities/threats for the future? • Where are the gaps in the UK research base? • In which areas is the UK weak and what are the recommendations for improvement? • What are the trends in terms of the standing of UK research and the profile of UK researchers?

B. What evidence is there to indicate the existence of creativity and adventure in UK Mathematical Sciences research? • What is the current volume of high-risk, high-impact research and is this appropriate? • What are the barriers to more adventurous research and how can they be overcome? • To what extent do the Research Councils’ funding policies support/enable adventurous research?

C. To what extent are the best UK-based researchers in the Mathematical Sciences engaged in collaborations with world-leading researchers based in other countries? • Does international collaboration give rise to particular difficulties in the Mathematical Sciences research area? What could be done to improve international interactions? • What is the nature and extent of engagement between the UK and Europe, USA, China, India and Japan71, and how effective is this engagement? • How does this compare with the engagement between the UK and the rest of the world?

D. Is the UK Mathematical Sciences community actively engaging in new research opportunities to address key technological/societal challenges? • What are the key technological/societal challenges on which Mathematical Sciences research has a bearing? To what extent is the UK Mathematical Sciences research community contributing to these? Are there fields where UK research activity does not match the potential significance of the area? Are there areas where the UK has particular strengths? • Are there any areas which are under-supported in relation to the situation overseas? If so, what are the reasons underlying this situation and how can it be remedied? • Does the structure of the UK’s mathematical science research community hamper its ability to address current and emerging technological/societal challenges? If so, what improvements could be implemented? • Are there a sufficient number of research leaders of international stature in the Mathematical Sciences in the UK? If not, which areas are currently deficient?

E. Is the Mathematical Sciences research base interacting with other disciplines and participating in multidisciplinary research? • Is there sufficient research connecting mathematical scientists with investigators from a broad range of disciplines including life sciences, materials, the physical sciences, finance and engineering? What is the evidence?

71 These countries have been identified as strategically important international research partners for the UK

Annex G - 1 International Review of Mathematical Sciences 2010 PART I - Annex G Information for the Panel Evidence Framework • Where does the leadership of multidisciplinary research involving mathematical sciences originate? In which other disciplines are the mathematical sciences contributing to major advances? • Are there appropriate levels of knowledge exchange between the Mathematical Sciences community and other disciplines? What are the main barriers to effective knowledge and information flow, and how can they be overcome? • Have funding programmes been effective in encouraging multidisciplinary research? What is the evidence?

F. What is the level of interaction between the research base and industry? • What is the flow of trained people between industry and the research base and vice versa? Is this sufficient and how does it compare with international norms? • How robust are the relationships between UK academia and industry both nationally and internationally, and how can these be improved? • To what extent does the Mathematical Sciences community take advantage of opportunities, including research council schemes, to foster and support this knowledge exchange? Is there more that could be done to encourage knowledge transfer? • Nationally and internationally what is the scale of Mathematical Sciences R&D undertaken directly by users? What are the trends? Are there implications for the UK Mathematical Sciences research community, and how well positioned is it to respond? Is there any way that its position could be improved?

G. How is the UK Mathematical Sciences research activity benefitting the UK economy and global competitiveness? • What are the current and emerging major advances in the Mathematical Sciences area which are benefiting the UK? Which of these include a significant contribution from UK research? • How successful has the UK Mathematical Sciences community (academic and user-based) been at wealth creation (e.g. spin-out companies, licences etc.)? Does the community make the most of opportunities for new commercial activity? What are the barriers to successful innovation based on advances in the Mathematical Sciences in the UK, and how can these be overcome?

H. How successful is the UK in attracting and developing talented Mathematical Sciences researchers? How well are they nurtured and supported at each stage of their career? • Are the numbers of graduates (at first and higher degree level) sufficient to maintain the UK Mathematical Sciences research base? Is there sufficient demand from undergraduates to become engaged in Mathematical Sciences research? How does this compare with the experience in other countries? • Is the UK producing a steady-stream of researchers in the required areas or are there areas of weakness in which the number of researchers should be actively managed to reflect the research climate. What adjustments should be made? • How effective are UK funding mechanisms at providing resources to support the development and retention of talented individuals in the mathematical sciences? • How does the career structure for researchers in the Mathematical Sciences in the UK compare internationally? • Is the UK able to attract international researchers in the Mathematical Sciences to work the UK? Is there evidence of ongoing engagement either through retention within the UK research community or through international linkages? • Are early career researchers suitably prepared and supported to embark on research careers? • Is the balance between deep subject knowledge and ability to work at subject interfaces/boundaries appropriate? • How is the UK community responding to the changing trends in the UK employment market? • How diverse is the UK mathematical sciences research community in terms of gender and ethnicity and how does this compare with other countries?

Annex G - 2 International Review of Mathematical Sciences 2010 PART I - Annex H Information for the Panel Evidence Framework

Annex H List of Acronyms

AGTN Algebra, Geometry, Topology and Number Theory AHRC Arts and Humanities Research Council BBSRC Biotechnology and B iological S ciences R esearch Council BERR Department f or Business, E nterprise a nd Regulatory Created 2007 Reform Merged with DIUS to form BIS BIS Business, Innovation and Skills Department Established 2009 Formerly DIUS and BERR CAF Career Acceleration Fellowship An EPSRC scheme CCLRC Council for the Central Laboratory of the Research Merged with PPARC to become Councils STFC C-DIP Cross-Disciplinary Interface Programme An EPSRC core programme CIF Capital Investment Framework Co-I Co-investigator CSA Chief Scientific Advisor CSR Comprehensive Spending Review DE Digital Economy An EPSRC mission programme DELNI Department of E ducation an d Lear ning i n N orthern Ireland DGSI Director General of Science and Innovation DIUS Department of Innovation Universities and Skills Created 2007 Merged with BERR to form BIS DTA/DTG Doctoral Training Account / Grant DTC Doctoral Training Centre DTI Department of Trade and Industry Abolished 2007, replaced by DIUS and BERR EPSRC Engineering and Physical Sciences Research Council ERC European Research Council ESRC Economic and Social Research Council EU European Union fEC Full Economic Costing FTE Full-Time Equivalent HE / HEI Higher Education / Higher Education Institution HEFCE Higher Education Funding Council for England HEFCW Higher Education Funding Council for Wales HESA Higher Education Statistics Agency ICT Information & Communications Technology An EPSRC core programme IP/IPR Intellectual Property / Intellectual Property Rights KTE Knowledge Transfer and Exchange KTN Knowledge Transfer Network KTP Knowledge Transfer Partnership M3E Materials, Mechanical and Medical Engineering An EPSRC core programme MRC Medical Research Council NERC Natural Environment Research Council NSF National S cience F oundation (US c ounterpart t o U K Research Councils) OSI Office of Science and Innovation Formerly known as OST (Office of Science and Technology) OST Office of Science and Technology existed within the DTI PDRA Post-doctoral Research Assistant PDRF Post-doctoral Research Fellowship An EPSRC shceme PES Process, Environment & Sustainability An EPSRC core programme

Annex H - 1 International Review of Mathematical Sciences 2010 PART I - Annex H Information for the Panel Evidence Framework PI Principal Investigator PPARC Particle Physics and Astronomy Research Council Merged with CCLRC to form STFC PS Physical Sciences Programme An EPSRC core programme QR Quality Related recurrent grant for research Administered by Funding Councils RA Research Assistant RAE Research Assessment Exercise RCUK Research C ouncils U nited Kingdom or R esearch Councils UK RDA Regional Development Agency REF Research Excellence Framework SET Science, Engineering and Technology SFC Scottish Funding Council SME Small / Medium sized Enterprise SRIF Science Research Investment Fund STFC Science and Technology Facilities Council Created 2007 by merger of PPARC and CCLRC TSB Technology Strategy Board UOA Unit of Assessment

Annex H - 2

MRC funding for mathematical sciences

The MRC supports research programmes which are driven by mathematical science, in areas such as biomedical informatics, statistical genetics, computational biomedicine and clinical trials. We also support a wide portfolio of research which is informed by mathematical science or where it makes a major underpinning contribution, such as statistics, modelling and imaging and in areas such epidemiology and experimental medicine.

MRC funding for programmes with a substantial component of mathematical science as defined broadly by EPSRC comprises around £45m per year.

The breakdown by area is as follows:

Grants: £7.5m Training (Fellowships): £1.9m MRC Unit programmes: £21m Partnerships and Contributions (Large-scale strategic awards): £14.4m

Grants are held at the university level but very little of the current funding relates to Maths departments (only £46k in 2008/09 with a similarly low amount to statistics and biostatistics). Mathematics plays an important role in underpinning many areas of MRC- funded research including trials, epidemiological studies, biostatistics and modelling.

INTERNATIONAL REVIEW OF MATHEMATICAL SCIENCES IN THE UNITED KINGDOM

INFORMATION for the PANEL

PART 2

Landscape Documents Prepared by the members of the Research Community

1 Algebra 2 Geometry & Topology 3 Number Theory 4 Analysis 5a Logic 5b Combinatorics 6 Numerical Analysis 7 Statistics 8 Probability 9 Mathematical Aspects of Operations Research 10 Mathematical Physics 11 Non-linear Dynamical Systems & Complexity 12 Theoretical Mechanics & Materials Science 13 Mathematics in Biology & Medicine 14 Industrial Mathematics 15 Financial Mathematics

INTERNATIONAL REVIEW OF MATHEMATICS ALGEBRA LANDSCAPE

Lead Author: Iain Gordon (Edinburgh) Contributors/Consulted: Dave Benson (Aberdeen), Martin Bridson (Oxford), Kenny Brown (Glasgow), Martin Liebeck (Imperial), Toby Stafford (Manchester).

1.Statistical Overview: The work of 707 individuals working in 38 institutions was submitted to the Pure Mathematics Panel in RAE 2008. According to this panel’s subject overview report, 25% of the outputs in Pure Mathematics were in Algebra. Three years after the RAE 2008 census date, the webpages of mathematics departments across the UK suggest there are 173 permanent faculty and post-doctoral researchers in algebra working in 30 institutions. Thus algebra is a very important component of UK research in pure mathematics, and it is spread throughout UK Pure Mathematics Departments.

2.Subject breakdown: The following data was extracted from departmental webpages. The division into subject areas is somewhat artificial as there are large overlaps between these topics and because many researchers’ work belongs to 2 or more areas.

Number of % of Total Research Area Researchers (IRM2003 total) Infinite Group Theory 42 (43) 24.3% Finite Group Theory 31 (18) 17.9% Semigroup Theory 10 (5) 5.8% Representation Theory 60 (21) 34.7% Noncommutative Algebra 23 (23) 13.3% Other 7 (13) 4%

3. Discussion of research areas:

Infinite Group Theory: The focus of UK research in infinite group theory has generally been shifting over the past few years from the study of classes of groups, such as nilpotent or polycyclic groups, towards geometric group theory. This mirrors a worldwide increase in the prominence and applications of geometric group theory. There have been strong recent appointments, including Drutu and Papasoglu (both Oxford), Nikolov (Imperial), as well as an expansion of the group in Southampton with appointments of Leary, Kassabov and several more junior hires. There is a world-class grouping in the South featuring Oxford, Southampton and Warwick, a strong axis in the North with Glasgow, Heriot-Watt and Newcastle, all complemented by international class researchers in several other departments throughout the UK. Bridson (Oxford) and Lackenby (an Oxford topologist with very strong geometric group theory interests) gave invited lectures at the 2006 and 2010 ICMs respectively. One disappointment has been the dispersion of a first-rate group in Bristol; now only individual excellence remains. Here are some research details, divided into three general areas.

1. Geometric Group Theory. This is an active and highly-regarded international field, which by definition has extremely close links with topology and geometry (at the 2006 ICM there were speakers on geometric group theory in the algebra, geometry and topology sections). The UK has come to occupy a leading role in the world in this field in the last few years, with large groups in Oxford and Southampton, complemented by high quality work in a few other places in the UK. There are flourishing links between the group theorists and the topologists and geometers, illustrated well by the low dimensional topology group in Warwick. There is

3 UK research on mapping class groups (Glasgow, Oxford, Southampton, Warwick); automorphisms of free groups, mapping class groups and arithmetic groups (Glasgow, Oxford); limit groups and residually free groups (Heriot-Watt, Oxford); (relatively) hyperbolic groups (Oxford, Southampton); classifying spaces for proper group actions (Glasgow, Southampton); group cohomology (Glasgow, Southampton); amenable groups and relations with functional analysis (Glasgow, Newcastle, Southampton); actions of groups on spaces of non-positive curvature (Oxford, Southampton); discrete groups of Lie groups (Durham); and actions of groups on buildings (Newcastle). Recent research highlights include the description of finitely presented subgroups of direct products of limit groups by Bridson- Howie-Miller-Short, [4]; Bowditch’s work on the geometry of the curve complex, including [3]; Lackenby’s work on 3-dimensional manifolds based on properties that may be satisfied by their fundamental groups, [13]. The UK is a world leader in this field.

2. Combinatorial Group Theory. Although this has now been largely absorbed into geometric group theory, there are still a number of researchers focusing particularly on this area and its extensions to monoids and semigroups (Glasgow, Heriot-Watt, Newcastle, St Andrews, Warwick). There is interest in automatic groups (Heriot-Watt, Warwick), and connections with theoretical computer science (Glasgow, Newcastle, St Andrews).

3. Pro-p and Profinite Groups. This area is the one closest to the historic strengths of infinite group theory in the UK. There are groups working directly in this field at Imperial, Oxford, Queen Mary, Royal Holloway and Southampton. Research connects to number theory through zeta functions counting subgroups or irreducible representations, through the Langlands’ programme, and through noncommutative Iwasawa theory; to Lie theory via Kac- Moody groups; and of course to finite group theory. Research highlights include Nikolov and Segal’s confirmation of Serre’s conjecture that that in a finitely generated profinite group all subgroups of finite index are open, [16]. The UK is a world leader in this area.

Finite Group Theory: Historically, the UK has been a major centre for finite group theory, particularly concerning the classification of the finite simple groups. Although this lives on, finite group theory in the UK has expanded, both in outlook and in the number of researchers. There are significant groups in Aberdeen, Birmingham, Imperial, Queen Mary, St Andrews, as well as high quality individuals throughout the UK. For the purposes of this document, research in this field may be divided into four subtopics.

1. Structure, geometry and applications of finite simple groups. The major programme of developing a unified proof of the Classification Finite Simple Groups is continuing, with strong UK contributions (Birmingham, Manchester, Warwick). The theory of fusion systems, categories which capture the fusion properties of Sylow p-subgroups, impacts on parts of this programme and also on representation theory and homotopy theory, and has been the focus of substantial recent developments (Aberdeen, Birmingham, Oxford). The theory of maximal subgroups of finite simple groups continues to be a rich topic (Cambridge, Imperial), for example with the recent Liebeck-Seitz classification of maximal subgroups of exceptional groups, [15]; this theory gives strong information about primitive permutation groups, and through this has diverse applications, for example in model theory, Galois theory and combinatorics. Geometries for the simple groups and their p-local subgroups are a major area of research (Birmingham, Queen Mary, Imperial, Southampton). Work of note includes that of Gramlich-Hoffman-Muehlherr-Shpectorov on a general theory of Curtis-Tits and Phan presentations for all Coxeter diagrams and similarly for Kac-Moody groups. The UK leads in this topic.

2. Computational group theory. The main recent advances in this area have been the development of theory and algorithms for the recognition of matrix groups generated by certain sets of matrices (Queen Mary, Warwick, St Andrews). Results include constructive

4 recognition algorithms for classical groups, algorithms for producing maximal subgroups of classical groups, and effective new involution-centralizer methods.

3. Expansion in finite groups. Interest in this topic goes back several decades to the theory of expander graphs, highly connected graphs in which every subset of vertices has boundary of size proportional to the size of the subset. A rich source of examples lies in Cayley graphs of finite simple groups, and recently Kassabov-Lubotzky-Nikolov confirmed the conjecture that (essentially) every family of finite simple groups gives rise to expanders, [11]. Helfgott proved a long-conjectured result that arbitrary generating sets in SL_2(p) expand, bringing to bear a range of techniques from additive combinatorics, [10]. In turn, this has had an impact in the theory of expander graphs and in analytic number theory. Recently, Green-Breuillard- Tao (and independently Pyber-Szabo) have extended this to all families of simple groups, and this work is in the process of being placed in the general setting of “approximate groups", a notion inspired by work of Gowers. The UK is a leader in this area.

4. Asymptotic group theory. This covers a wide range of topics, including those of Paragraph 3 above and overlaps significantly with the work in pro-p and profinite groups included in the infinite groups section. Examples include "width problems" which were at the heart of the Nikolov-Segal proof of Serre's conjecture that in every finitely generated profinite group, every subgroup of finite index is open, [16]. Other major topics are representation growth and subgroup growth, in which there have been serious recent advances involving the use of finite group theory, for example [1]. Again the UK is internationally very strong here.

Representation Theory and Noncommutative Algebra: Representation Theory and Noncommutative Algebra are deeply intertwined in the UK, and beyond algebra they overlap significantly with algebraic geometry, algebraic topology, combinatorics, integrable systems and number theory. Since the last IRM, there has been a substantial rise in the numbers of researchers in these fields, and there have been many new appointments of very high quality: an entire group in Aberdeen including Benson, Bondal, Geck and Linckelmann; Rouquier in Oxford (via Leeds); Smoktunowicz in Edinburgh; and Stafford in Manchester. There are sizeable groups of international calibre in Aberdeen, Edinburgh, Leeds, Manchester, Oxford and York, and world-class individuals working in other departments. There were invited lectures at the 2006 ICM from Crawley-Boevey (Leeds), Grojnowski (Cambridge), Rouquier (Oxford), Smoktunowicz (Edinburgh), and at the 2010 ICM from Benson (Aberdeen) and Gordon (Edinburgh); Smoktunowicz won a European Mathematical Society Prize in 2008. These fields have developed from a strong basis in the UK a decade ago, adding key individuals and a number of points of international excellence.

1. Representation Theory of Groups. This is a well-established field in the UK, with active senior figures based here for some time including Erdmann, Rickard and Robinson. However, there has been an expansion of numbers recently, both at a senior and junior level. There are particularly active links to algebraic topology and to homological algebra. Strong activity in the UK includes the cohomology of finite groups (Aberdeen, Manchester), fusion systems and connections to p-local groups and the classification of finite simple groups (Aberdeen, Birmingham, Oxford), modular representaton theory around Alperin’s weight conjecture and Broue’s abelian defect conjecture (Aberdeen, Bristol, City, Oxford), triangulated categories (Aberdeen, Bristol, City, Oxford). Research highlights include Symond’s proof that the Castelnuovo-Mumford regularity (suitably interpreted) of the cohomology ring of a finite group is zero, [18]; Benson’s work with Iyengar and Krause on the classification of localising subcategories of the stable module category of a finite group, [2]; Chuang and Rouquier’s confirmation of Broue’s abelian defect conjecture for symmetric groups, [6], leading on to Rouquier’s categorification of Kac-Moody Lie algebras.

2. Lie-theoretic Representation Theory. This is a field that is growing quite rapidly in the UK, although there were already a few international leaders in algebraic Lie theory at the time of

5 the last IRM, including Donkin and Premet. There has been a reasonable expansion along these existing lines, but also the emergence of a number of researchers with more geometric interests. There is activity in the representation theory of semisimple Lie algebras and algebraic groups (Birmingham, Manchester, Warwick, York), in the algebraic and homological representation theory of Hecke algebras and other associative algebras (Aberdeen, City, East Anglia, Leeds, Oxford), and through to geometric representations attached to quiver varieties, various flag manifolds and resolutions of singularities (Cambridge, Edinburgh, Leeds, Oxford). Highlights include Premet's solution (in the negative) of one of the basic open problems in Lie theory, the Gelfand-Kirillov conjecture, [17]; Fishel-Grojnowski-Teleman’s work on the failure of the Hodge decomposition for the loop Grassmannian, [7]; Geck’s proof of the cellularity of Iwahori-Hecke algebras of finite Coxeter groups, [8]; Hausel’s confirmation of a conjecture of Kac on the number of absolutely indecomposable quiver representations, [9]. This is a blue riband subject in mathematics, and the UK has great strength here.

3. Noncommutative Algebra. In line with worldwide developments, the number of algebraists who apply ideas and techniques from Artin and noncommutative algebras has grown and the UK has been at the forefront of many of these developments. There are several strands to noncommutative algebra in the UK. These include the study of cluster algebras, cluster categories and cluster combinatorics (Leeds), interactions between noncommutative algebras and algebraic geometry, particularly noncommutative algebraic geometry, aspects of birational geometry, derived categories, Hall algebras and in the construction of moduli spaces (Bath, Edinburgh, Glasgow, Manchester, Oxford), quantum groups and algebras and ring theory (Edinburgh, Glasgow, Kent, Sheffield, Swansea), A-infinity algebras and related parts of homotopical algebra (City, Leicester). Work here includes Smoktunowicz’s construction with Lenagan of infinite dimensional nil algebras of finite Gelfand-Kirillov dimension, [14], thereby significantly strengthening the Golod-Shafarevich example; Marsh and his coworkers’ introduction of cluster categories, bringing together tilting theory for Artin algebras and cluster combinatorics, [5]; the creation of naive blow-ups and the resulting classification of large classes on noncommutative surfaces by Stafford, Sierra and others, for instance [12]. The UK is a world leader in this topic.

Others: There is an active community of semigroup theorists in several UK universities with an international profile. The small commutative algebra community in the UK has now reduced further, although there are researchers in algebraic geometry, algebraic topology and representation theory who have serious interests in this field. There is strong international work on quadratic forms and related questions in K-theory and number theory, but again not a large number of researchers with this as a main interest.

4. Discussion of research community: Since the last IRM there have been a number of new permanent appointments in algebra, at various levels of seniority, and held by British and international algebraists alike. Several of the senior appointments have been mentioned above. Strong algebra groups have grown at a number of universities, including for instance Aberdeen, East Anglia, Kent and Newcastle, and young UK-trained mathematicians with recent faculty positions include Ardakov (Nottingham), Burness (Southampton), Craw (Glasgow), Fayers (Queen Mary), Goodwin (Birmingham), Levy (Lancaster), Lyle (East Anglia), and Paget (Kent), Parker (Leeds), Roney-Dogal (St Andrews), and Turner (Aberdeen). The number of female algebraists has increased - of the 173 algebraists counted, 25 are women - although most, but by no means all, are not yet at a senior level.

An estimate from departmental websites indicates there are 23 post-doctoral researchers in algebra. Of these, however, just over half did not receive their PhD training in the UK. This is perhaps a reflection of the international strength of algebra in the UK, and also an indication of the difficulty UK postgraduates have in a competitive market. Moreover, even without

6 precise figures, it does seem that the number of young UK algebraists going abroad for reasonably long or permanent positions is not high.

It is difficult to measure PhD student numbers in algebra accurately. There seem to be no unified data sets. From email contact with some colleagues and a study of departmental webpages it appears there are currently around 177 algebra (interpreted broadly) PhD students. This suggests an annual intake of around 45 students, a large number of whom come from abroad and bring their own funding. Given the breadth and depth of internationally recognised researchers in algebra, this is some distance from capacity.

5. Cross-disciplinary/Outreach activities: In the UK algebra interacts with many topics, including algebraic geometry, algebraic topology, category theory, combinatorics, geometry, integrable systems and number theory. The links with algebraic topology and combinatorics are well-established through group theory, and they have been bolstered with recent work on fusion systems, on Lie theoretic and cluster combinatorics, and on expanders; links between algebraic geometry and noncommutative algebra were being vigorously developed at the time of the last IRM and are now firmly in place in the UK, with connections particularly to the minimal model programme and homological mirror symmetry (broadening out even to symplectic geometry); interactions between algebra and number theory continue, particularly through the Langlands’ programme, quadratic forms, and Iwasawa theory; integrable systems and algebra connect in the UK through cluster algebras and Lie theory. Beyond mathematics there are links with theoretic computer science, perhaps not as developed as they could be, and to mathematical physics, for instance through string theory and topological quantum field theory. Expander graphs are of interest to electrical engineers, but this has not yet been fully pursued in the UK.

Appendix 1: Main Research Groups in Algebra

Aberdeen: Group and Representation Theory: Benson, Geck, Iancu, Kessar, Le, Linckelmann, Park, Robinson, Sevastyanov, Turner. Birkbeck: Group Theory: Bowler, Hart. Bath: Noncommutative Algebra: King, Su, Traustason. Belfast: Noncommutative Algebra: Hazrat, Iyudu. Birmingham: Group Theory: Brown, Curtis, Flavell, Goodwin, Gramlich, Hoffmann, Magaard, Parker, Shpectorov. Bristol: Group Theory: Gill, Helfgott, Rickard. Cambridge: Group Theory and Noncommutative Algebra: Brookes, Button, Camina, Duncan, Grojnowski, Lawther, , Martin, Saxl, Wadsley. Durham: Geometric Group Theory: Belolipelsky, Guyl, Parker. East Anglia: Representation Theory: Damiano, Guillhot, Lyle, Matringe, Miemietz, Siemens, Stevens. Edinburgh: Representation Theory and Noncommutative Algebra: Chlouveraki, Gordon, Griffeth, Lenagan, Smoktunowicz, Thomas, Wemyss. Glasgow: Group Theory and Noncommutative Algebra: Brendle, Brown, Craw, Kraehmer, Kropholler, Pride. Heriot-Watt: Group Theory: Gilbert, Howie, Lawson, Prince. Imperial: Group Theory: Ivanov, Liebeck, McGerty, Nikolov. Kent: Representation Theory: Fleischmann, Launois, Paget, Rosenkranz, Shank, Woodcock Kings: Representation Theory: Pressley, Rietsch. Lancaster: Representation Theory: Levy, Mazza, Towers. Leeds: Representation Theory: Crawley-Boevey, Hubery, Marsh, Martin, Palu, Parker. Leicester: Representation Theory: Baranov, Mudrov, Schroll, Snashall. London City: Representation Theory: Chuang, Cox, de Visscher. Manchester: Group Theory and Noncommutative Algebra: Bazlov, Borovik, Eaton, Kambites, Premet, Prest, Rowley, Symonds, Stafford, Stohr.

7 Newcastle: Group Theory and Noncommutative Algebra: Boklandt, Duncan, Jorgensen, King, Rees, Vdovina. Nottingham: Noncommutative Algebra: Ardakov, Edjvet, Hoffmann, Pumplin. Oxford: Group Theory and Representation Theory: Bridson, Collins, Craven, Danz, Darpo, Erdmann, Grabowski, Hollings, Johnson, Kremnizer, Rouquier, Segal, Towers, Williamson, Wilson. Queen Mary: Group Theory: Bray, Fayers, Leedham-Green, McKay, Muller, Soicher, Wehrfritz, Wilson. Royal Holloway: Group Theory: Barnea, Klopsch St Andrews: Group Theory: Mitchell, Neunhoffer, Peresse, Quick, Roney-Dogal, Ruskuc. Sheffield: Noncommutative Algebra: Bavula, Jordan, Katzman. Southampton: Group Theory: Anderson, Burness, Kar, Mannan, Martino, Minasyan, Nuckinkis, Renshaw, Stasinski, Voll. Swansea: Noncommutative Algebra: Brzezinski, Garkusha. UCL: Group Theory and Noncommutative Algebra: Johnson, Lopez-Pena. Warwick: Group Theory and Noncommutative Algebra: Capdebosq, Holt, Kraamer, Rumynin, Westbury. York: Group Theory and Noncommutative Algebra: Bate, Donkin, Everitt, Fountain, Gould, Nazarov, Tange.

Appendix 2: Some Recent Papers

[1] M. Belolipetsky, A. Lubotzky, Finite groups and hyperbolic manifolds, Invent. Math. 162 (2005), no. 3, 459--472. [2] D.Benson, S.B.Iyengar and H.Krause, Stratifying modular representations of finite groups, arXiv:0810.1339. [3] B.H.Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008), 281–300. [4] M.R.Bridson, J.Howie, C.F.Miller III and H.Short, Subgroups of direct products of limit groups, Ann. Math. 170 (2009), 1447-1467. [5] A.B.Buan, R.Marsh, M. Reineke, I. Reiten and G.Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572-618. [6] J.Chuang and R.Rouquier, Derived equivalences for symmetric groups and sl_2- categorification, Ann. Math. 167 (2008), 245-298. [7] S.Fishel, I.Grojnowski and C.Teleman, The strong Macdonald conjecture and Hodge theory on the loop Grassmannian, Ann. Math. 168 (2008), 175-220. [8] M.Geck, Hecke algebras of finite type are cellular, Invent. Math. 169 (2007), 501–517. [9] T.Hausel, Kac conjecture from Nakajima quiver varieties, Invent.Math. 181 (2010), 21- 37. [10] H.A.Helfgott, Growth and generation in SL_2(Z/pZ), Ann. Math. 167 (2008), 601-623. [11] M.Kassabov, A.Lubotzky and N.Nikolov, Finite simple groups as expanders, Proc. Natl. Acad. Sci. USA 103 (2006), 6116-6119. [12] D. S. Keeler, D. Rogalski and J.T.Stafford, Naïve noncommutative blowing up, Duke Math. J. 126 (2005), 491-546. [13] M.Lackenby, Heegaard splittings, the virtually Haken conjecture and property tau, Invent. Math. 164 (2006), no. 2, 317–359. [14] T.H.Lenagan and A.Smoktunowicz, An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension, J. Amer. Math. Soc. 20 (2007), 989-1001. [15] M.W.Liebeck, G.M.Seitz, The maximal subgroups of positive dimension in exceptional algebraic groups, Mem. Amer. Math. Soc. 169 (2004), no. 802, vi+227 pp [16] N.Nikolov and D.Segal, On finitely generated profinite groups. I&II, Ann. Math. 165 (2007), 171-238 & 239-273. [17] A.Premet, Modular Lie algebras and the Gelfand–Kirillov conjecture, Inv. Math. 181 (2010), 395-420. [18] P.Symonds, On the Castelnuovo-Mumford regularity of the cohomology ring of a group, J. Amer. Math. Soc. 23 (2010), 1159-1173.

8 International Review of Mathematics Geometry and Topology Landscape

Coordinating Author: J.P.C.Greenlees Supporting Author: U.Tillmann Contributing Authors include: T.Bridgeland, S.Donaldson, M.Haskins, N.Hitchin, F.Kirwan, M. Reid, R.Thomas. 1. Statistical Overview: Total headcount of researchers submitted to UoA F20 (Pure Mathematics) in RAE 2008: 728. Total number attributed to Geometry and Topology: 178 (25%). The percentage in Geometry and Topology is very similar to that in RAE 2001. The proportion of Early Career Researchers’ Outputs submitted to RAE2008 was 23%. 2. Subject breakdown: The table indicates the number of active researchers in designated sub-areas (data extracted from submissions to RAE 2008)

Research Area Number of researchers % of Total in G&T Algebraic Geometry (AG) 44 25 Algebraic Topology (AT) 48 27 Geometric Topology (GT) 36 20 Diﬀerential Geometry (DG) 50 28

3. Discussion of research areas: By subject area, please provide a frank assessment (both positive and negative) of past and current international standing of the field, supported by reference to highlights as appropriate. The period since the last IRM has been one of exceptional activity and progress in the area of Geometry and Topology worldwide, with the solution of the Poincaréconjecture, finite generation of the canonical ring, the Mumford conjecture and the Kervaire invariant conjecture as particular highlights. The UK has world- leading research groups and has been fully involved in international developments as we describe below. Although a number of high-profile researchers have left the country, the general pattern has been of vigour and development. In what follows, particular centres and particular researchers are named as principal points of reference for the Review Panel. These lists are not exhaustive and it is inevitable that important names and places have been omitted. Generally, we have described areas of activity rather than particular results, leaving the Panel to find this more detailed information during their local visits. Algebraic geometry Centres: Bath, Cambridge, Imperial, Liverpool, Oxford, Warwick. Headlines: (a) Major UK contributions to classification and mirror symmetry; thinner coverage of other important areas. (b) Invited ICM talks: T.Bridgeland (ICM 2006), R.Thomas (ICM 2010). The main areas of algebraic geometry internationally can be described as: minimal model theory; moduli spaces; mirror symmetry and other developments inspired by string theory; analytic methods; algebraic cycles; and combinatorial and computational algebraic geometry. The applications of algebraic geometry to other subjects such as number theory, commutative algebra, and representation theory should be discussed in other Landscape documents.

9 Minimal model theory. The major event is the milestone in the minimal model programme represented by the proof of finite generation for the canonical ring by C.Birkar (Cambridge), P.Cascini (Imperial), Hacon, and McKernan, which serves as a foundation for all kinds of further work on classifying algebraic varieties. The UK is strong in minimal model theory: M.Reid (Warwick) and A.Corti (Imperial) played major roles in the developments leading to finite generation. The remaining open problems of the minimal model programme are being hotly pursued and may well be solved in the next five years. Other significant UK geometers in this area include A.Pukhlikov (Liverpool), V.Lazic (Imperial) and I.Cheltsov (Edin- burgh). Reid’s students G.Brown (Kent), A.Craw (Glasgow), and N.Shepherd-Barron (Cambridge) have spread across algebraic geometry. Moduli spaces. The UK is fairly well represented in this area. M.Reid (Warwick) is studying the moduli of surfaces of general type, and G.Sankaran (Bath) the moduli of K3 surfaces and hyperkäler varieties. F.Kirwan (Oxford) has done fundamental work on geometric invariant theory and moduli spaces. Shepherd-Barron is an expert on the moduli space of abelian varieties. Activity on vector bundles over curves continues through Liverpool and the VBAC group. Mirror symmetry. The UK contribution to mirror symmetry is world-leading. T.Bridgeland’s definition of stability conditions, a powerful mathematical idea inspired by physics, has driven great progress in the past 5 years. D.Joyce (Oxford) initiated the study of wall-crossing formulae for Donaldson-Thomas invariants, and with T.Bridgeland (Sheffield, moving to Oxford) and R.Thomas (Imperial) he has made major contributions to enumerative invariants inspired by physics. A.Bondal (Aberdeen) and T.Coates (Imperial) are also significant contributors to this area. Analytic methods. Analysis has always been a powerful tool in algebraic geometry. UK analysts have unfortunately been distant from algebraic geometry in recent years. The main exception is S.Donaldson (Imperial) and his school, who relate extremal metrics to algebro-geometric stability. This exciting work has inspired a large body of research, inside the UK and beyond. Other analytic methods in algebraic geometry, typified by Siu’s nonvanishing theorems and Demailly’s work on hyperbolic complex manifolds, are not represented in the UK. Algebraic cycles. The theory of algebraic cycles, typified by the Hodge conjecture, is an important area of algebraic geometry. The theory embraces Hodge theory, motivic homotopy theory, and applications to quadratic forms and algebraic groups. B.Totaro (Cambridge) and A.Vishik (Nottingham) have made significant contributions to algebraic cycles, but this area of algebraic geometry is underdeveloped in the UK. Combinatorial and computational algebraic geometry. Algebraic geometry has a strong combinatorial side which has developed internationally in recent decades, including subjects like algebraic combinatorics and Schubert calculus, Gröbnerbases, and tropical geometry. D.Maclagan (Warwick) is a welcome recent hire in tropical geometry, but these areas are still underdeveloped in the UK. Many mathematicians and users of mathematics rely on computer algebra programs. The algorithms for computational algebraic geometry have mostly been developed outside the UK, although Reid’s students such as G.Brown (Kent) have begun to contribute. There is potential here for productive interaction between algebraic geometry and other subjects such as statistics, as people like Sturmfels in the US have shown. Algebraic topology Centres: Aberdeen, Glasgow, Leicester, Manchester, Oxford, Sheffield. Headline: Solid contributions in mature areas; opportunities in new applications. The main areas of algebraic topology internationally can be described as: classical algebraic topology, stable homotopy including chromatic methods; applications to moduli spaces and TQFTs; and applications to algebra. Classical algebraic topology. Groundbreaking research in classical algebraic topology has been slowing internationally in recent decades. Instead the machinery of algebraic topology (notably Quillen model categories) is being used to excellent effect in other areas of mathematics (commutative algebra, representation theory, K-theory, combinatorics, and algebraic geometry). Further examples are its use in: higher category theory in Glasgow and Sheffield (E.Cheng (Sheffield), T.Leinster (Glasgow), I.Moerdijk (Sheffield)); toric

10 topology in Manchester (V.Buchstaber, N.Ray); applications to aperiodic tilings J. Hunton (Leicester); configurations spaces M.Farber (Durham). Applications to moduli spaces and TQFT: The great recent achievement is the proof by Madsen and M.Weiss (Aberdeen) of the Mumford conjecture which gives the stable cohomology of the moduli space of Riemann surfaces. Since the last review Madsen, Galatius, U.Tillmann (Oxford) and M.Weiss (Aberdeen) were able to give a much simplified proof by computing the homotopy type of cobordism categories. This result inspired Galatius’s work on the automorphism group of free groups as well as Lurie and Hopkins’s work on the Baez-Dolan cobordism conjecture. Significant work in this area was also done by Tillmann’s students including J.Giansiracusa (Bath). Closely related is the research area of string topology which is represented in the UK by one of its early pioneers, J.Jones (Warwick). G.Segal (Oxford) and M.Atiyah (Edinburgh), the inventors of the cobordism category approach to field theory, continue to contribute at the maths/physics interface of the subject as well as to its foundations (twisted K-theory). The departures of Costello and Telemann are losses to the UK in this area. Stable homotopy theory. Activities in the UK under this heading fall under three headings: chromatic, equivariant and highly structured, with both the stable spaces and the cohomological points of views being important. The main developments in chromatic homotopy theory, have been led from the US, with UK researchers including N.Strickland (Sheffield) and S.Whitehouse (Sheffield). The recent spectacular proof of the Kervaire invariant conjecture, has heavily used equivariant methods developed by J.Greenlees (Sheffield) and others. The global study of derived categories and homotopy categories has recently become accessible. One theme is the classification of thick subcategories (D.Benson (Aberdeen), Iyengar, Krause), and complete algebraic models of categories of rational spectra (Greenlees). Categories of spectra with good smash product have made it possible to study categories of cohomology theories as module categories. The idea of ring spectra has influenced many areas of mathematics: computations of K-theory by Hesselholt and Madsen, derived categories (Fukaya, Seidel), and so on. Formulating notions of commutative algebra in a homotopy invariant fashion and applying them to ring spectra has been very fruitful (Benson, Dwyer, Greenlees, Iyengar). Applications to algebra. The technology of ring spectra has been a powerful method in the study of the representation theory of finite groups. Gorenstein duality for ring spectra (as above) led D.Benson (Aberdeen) to make his regularity conjecture, recently proved by P.Symonds (Manchester) by Quillen’s topological methods. This gives the first explicit bounds for generators of the cohomology ring. The bounds are strong enough to be used effectively by computer algebra programs. The local homotopy theory of classifying spaces (homotopical group theory) has made significant progress, including the classification of p-compact groups. R.Levi (as part of the leading Broto-Levi-Oliver team) and his group in Aberdeen have played a central role in the study of p-local finite groups. This is intimately related to the representation theory of finite groups, also strongly represented at Aberdeen. Geometric topology (including geometric group theory) Centres: Oxford, Southampton, Warwick. Headlines: (a) With a string of new appointments, the UK has ensured that it will continue to be well-represented in this area. (b) Invited ICM talks: M.Bridson (ICM 2006), M.Lackenby (ICM 2010). The main areas of geometric topology can be described as low-dimensional geometry/topology, high- dimensional topology and geometric group theory. Internationally, the field is highly-regarded and very active, contributing 5 Fields medalists over the past 30 years. In the UK, the subject underwent a period of decline about 10 years ago, but has been revived by recent hires. In particular, the UK has moved effectively into the new area of geometric group theory. Low-dimensional geometry/topology. A high point in the last decade was Perelman’s proof of the Geometrization Conjecture. Other impressive advances have been made, for example the solution to the Ending Lamination Conjecture and the Tameness Conjecture. Very recently, remarkable progress has been

11 made by Kahn and Markovic (Warwick), who have shown that every hyperbolic 3-manifold group contains a surface group. In the UK, M.Lackenby (Oxford) has remained very active, working on the interface between topology and geometry in dimension 3. Other strong low-dimensional topologists recently hired include Friedl (Warwick, though now moving to Köln), J.Rasmussen (Cambridge), and Schleimer (Warwick). High dimensional topology. The study of high-dimensional manifolds has been shrinking since its high point in the 1960s and 1970s. Many of the main problems were solved. The most active areas of the subject now are problems such as the Novikov conjecture, where the key methods come from geometric group theory. Homotopy theoretic approaches to K-theory of rings and spaces are not well represented in the UK. The UK has a strong tradition in high-dimensional manifold topology or surgery theory, including A.Ranicki (Edinburgh) and M.Weiss (Aberdeen). Weiss was able to use surgery theory in a new way as part of his proof with Madsen of the Mumford conjecture. Geometric group theory. This subject has grown tremendously in the past 20 years. It can now be considered a major part of geometric topology, closely related to hyperbolic geometry and low-dimensional topology. The UK is world-class in geometric group theory. Oxford has the strong team of M.Bridson, C.Drutu, and P.Papasoglou, as well as M.Lackenby in related aspects of 3-manifold theory. B.Bowditch (Warwick) and J.Howie (Heriot-Watt) remain active in this area. There is a significant group at Southampton using analytic methods in geometric group theory (G.Niblo, J.Brodzki, with I.Leary returning in 2011). This is arguably the one area of geometric group theory that could use further development. Geometric group theory includes hyperbolic geometry and Kleinian groups as an important special case. Significant UK researchers in this area include J.Parker (Durham) and C.Series (Warwick). Differential geometry Centres: Bath, Cambridge, Durham, Loughborough, Imperial, Oxford, Warwick. Headline: (a) Donaldson’s prizes (King Faisal (2006), Nemmers (2008), Shaw (2009)). (b) Growth in geometric analysis, but more needed. The main areas of differential geometry internationally can be described as: classical differential geometry; geometric analysis; symplectic and contact geometry. There are close links with dynamical systems as well as twistor theory and mathematical physics which we leave for other Landscapes. Classical differential geometry. N.Hitchin (Oxford) is one of the world’s most influential differential geometers. His UK-based students include S.Donaldson (Imperial), D.Calderbank (Bath), A.Dancer (Oxford) and T.Hausel (Oxford). The UK is a world-leader in the construction of special metrics (Einstein metrics, Einstein-Weyl metrics, self-dual metrics, constant scalar curvature or extremal Kählermetrics, hyperkählermetrics and other metrics with special holonomy). A distinctive feature of the UK is the variety of tools used: twistor theory, hyperkählerreduction, gauge-theoretic moduli spaces, algebraically completely integrable systems and also methods from geometric analysis. In many of these cases the pioneering work in these areas was done by UK mathematicians. Significant contributions include those made by S.Donaldson (Imperial), A.Dancer, T.Hausel N.Hitchin, Joyce and L.Mason from Oxford, R.Bielawski (Leeds), D.Calderbank (Bath), G.Gibbons and A.Kovalev from Cambridge and M.Singer (Edinburgh). The UK has a significant number of researchers working at the interface of geometry and integrable systems. The main research groups in this area include Bath (led by F.Burstall, D.Calderbank), Lough- borough (led by J.Ferapantov, S.Veselov) and Oxford (especially A.Dancer, N.Hitchin, L.Mason) along with smaller groups elsewhere. The hiring of a number of Russian-trained mathematicians has considerably enriched the area and has led to a significantly broader coverage of the area going beyond traditional UK strengths (such as twistor theory). Other UK strengths include the link between integrable systems, loop groups and harmonic maps to highly symmetric targets (notably F.Burstall, N.Hitchin, Segal-Wilson). J.Wood (Leeds) has also made major contributions to harmonic maps and morphisms. More recent strengths are in special Frobenius manifolds (I. Strachan (Glasgow), S.Veselov and collaborators) and the development of generalised geometry by Hitchin and his recent students.

12 Geometric analysis. Geometric analysis has emerged as one of the most powerful tools in differential geometry. A model example is Perelman’s proof of the Poincaréconjecture. The UK has some leading figures in the field but generally the subject has suffered from the UK’s traditional weakness in PDE theory. S.Donaldson (Imperial) is one of the world’s leading geometers, with particular strength in geometric analysis. His students include A.King (Bath), Thaddeus, Joyce (Oxford), A.Kovalev (Cambridge), A.Macioca (Edinburgh), J.Ross (Cambridge), Seidel, R.Thomas (Imperial), and I.Smith (Cambridge), but only Joyce and Kovalev have gone on to work in geometric analysis. Joyce has long secured his position as a leading geometric analyst through his work on manifolds with special holonomy and the theory of special Lagrangian submanifolds in Calabi-Yau manifolds, though at present his work is closer to algebraic geometry. The UK remains a leader in the areas of special and exceptional holonomy and calibrated submanifolds (Haskins (Imperial)), and these areas likely to remain important in future. There have been strong appointments in this area across the UK: P.Topping (Warwick) and most recently A.Neves (Imperial) work on Geometric evolution equations (such as Ricci and mean curvature flow), P.Blue (Edinburgh) and M.Dafermos (Cambridge) on general relativity, N.Wickramasekera (Cam- bridge) on minimal hypersurfaces and R.Moser (Bath). Internationally, geometric analysis has become a more important subject in recent years. The UK’s strength in this area is rather vulnerable, since few of the young generation have deep roots here; it is important to expand and consolidate this area. To do this we need to ensure young geometers have the background in analysis to make geometric analysis an option. Symplectic and contact geometry. This has become a central area of geometry in the past 25 years. The work of Taubes, Kronheimer, Mrowka and others provides tight links between 3-manifolds and contact structures, and between 4-manifolds and symplectic structures. Mirror symmetry helped to inspire the rich and still incompletely understood notion of Fukaya categories in symplectic geometry. Hamiltonian mechanics describes a vast array of dynamical systems in physics. Donaldson has made fundamental contributions to symplectic geometry. I.Smith (Cambridge) collaborates with many of the leading figures in symplectic geometry in relation to low-dimensional topology. G.Paternain (Cambridge) has made a long series of contributions to Hamiltonian dynamics. In Sheffield and Manchester there is an active group in Poisson geometry and higher order structures. Given the scale of the subject internationally, the UK needs to grow further in symplectic geometry.

4. Discussion of research community in geometry and topology Although a number of high-proﬁle researchers have left the country, the general pattern has been of vigour and development. Amongst the active researchers, the numbers obtaining a PhD in 5-year blocks 1965–2004 are 12, 13, 8, 12, 16, 21, 30, 32, which supports the subjective impression that there was a low point around 1980 and that there has been a resurgence of interest over the period. The corresponding ﬁgures for female researchers only is 1, 0, 1, 1, 1, 0, 5, 3. Altogether, about 10 % of the active researchers in G&T are female. Amongst those who received a PhD from 1990 onwards (i.e., including 2005-09), about 16 % are female, but this does not allow for the number of people who leave mathematics soon after a PhD. Year of award PhDs in G&T Female PhDs in G&T 1965-69 12 1 1970-74 13 0 1975-79 8 1 1980-84 12 1 1985-89 16 1 1990-94 21 0 1995-99 30 5 2000-04 32 3 For those outside the main centres, a major factor in maintaining the research community are the LMS- supported Scheme 3 seminars. These include the COW (Cambridge-Oxford-Warwick) algebraic geometry seminar, the Bristol-Oxford-Southampton geometric group theory seminar, and the Transpennine Topology

13 Triangle (Leicester-Manchester-Sheﬃeld).

5. Cross-disciplinary/Outreach activities There are numerous interactions between geometry, topology and other areas of mathematics. Commu- tative algebra interacts with algebraic geometry, with Warwick a UK centre. Connections between algebraic geometry and representation theory are strong (R.Bocklandt, T.Bridgeland, A.Craw, I.Grojnowski, A.King), as are those between algebraic topology and representation theory (R.Rouquier, and K.Kremnitzer in Oxford), and should be discussed in the Algebra landscape document. The ties between algebraic geometry and number theory are fundamental, for instance at Cambridge, Imperial and Oxford. Geometric group theory is on the border between geometry and algebra. There are close relations between geometric topology and dynamical systems, most visible at Warwick. The department at Aberdeen is centred around the intimate relationship between representation theory of finite groups and homotopy theory, and this is also evident in Sheffield. There are also connections between commutative algebra and topology through the theory of ring spectra. Mirror symmetry in algebraic geometry and symplectic geometry has begun to pay its debts to physics, contibuting a series of new ideas to string theory. Hitchin and his school has had a world-wide impact on the area with influence in apparently far-flung parts of mathematics. Most recently, Hitchin’s work on Higgs bundles and the geometry of the Hitchin fibration in particular played a significant part in Ngo’s proof of the Fundamental Lemma in the Langlands Programme, for which Ngo was awarded the Fields Medal in 2010. Moreover, Higgs bundles look set to play an increasingly important role in the Geometric Langlands programme especially since the work of Kapustin-Witten. J.Hunton (Leicester) has applied algebraic topology to the study of aperiodic tilings (presented at the Royal Society Summer Science Exhibition 2008), and D.Buck (Imperial) used three-manifold theory to illuminate structural and mechanical features of DNA-protein interactions. M.Farber’s group at Durham has an active interest in robotics, including projects with engineers. Carlsson’s initiative in making applications to statistics and data mining is reflected in the recent project from J.Brodzki (Southampton) making similar use of coarse topology.

Number theory landscape

Statistics

The number of academic staff with permanent appointments or long-term fellowships working mainly in number theory is 68, based on data submitted in for the 2008 Research Assessment Exercise. The corresponding figure in the previous exercise (2001) was 61. In common with other subject areas, there is no clear division of Number Theory into subdisciplines. For example, many of the UK researchers in the subject work on some aspect of the Langlands Programme (although they might not naturally describe themselves as such) and could be variously described as working in Arith- metic Algebraic Geometry or Representation Theory/Automorphic Forms. With this caveat, the following table gives a rough distributions of research interests (because of overlaps, the column totals are significantly more than 100%):

Subarea 2008 % 2001 % Algebraic Number Theory 24 34 26 40 Arithmetic Algebraic Geometry 23 32 14 23 Analytic number theory 15 21 12 20 Representation theory 9 13 3 5 Diophantine approximation 7 8 8 13 Computational number theory 8 8 8 13

Community

Number theory in the UK has a long and unbroken tradition, going back to the school of Hardy and Littlewood in Cambridge in the early 1900s. Since then the number of centres of research in number theory has grown substantially, particularly since the explosion of worldwide interest in arithmetic algebraic geometry in the 1970s and 1980s, itself very much catalysed by the conjectures of Birch and Swinnerton-Dyer. Currently there are at over 10 departments with 3 or more permanent staﬀ whose interests lie largely in number theory, and in which 75% of the total researchers are based. The remainder are in smaller groups or are standalone researchers in their departments. Growth has not been uniform, and some large research groups have been created almost from scratch in recent years. Notable are the successful groups in Warwick, which until relatively recently had no number theory at all; and at Bristol, where the formation of the Heilbronn Institute (see below) has led to a major expansion in number theory, particularly in analytic number theory. Likewise (although it is

15 rather less recent history) number theory in London has become very strong, after a marked decline in the 70s and 80s. As well as the broad geographical base there is also a broad spectrum of activity in most aspects of the subject, which reflects international trends well. As the table indicates, the largest growth area in UK number theory has been in arithmetic algebraic geometry. This is representative of the large growth in the subject worldwide. The enormous progress in the Langlands programme in recent years, with its origins in the work of Taylor and Wiles on modularity, is at least partly responsible. The UK has a number of the world experts in various aspects of the Langlands programme. The increase of UK activity in representation theory (meaning here representations of p-adic groups and automorphic theory) is another consequence of this. Another area of arithmetic algebraic geometry in which the UK continues to be at the fore is the arithmetic of elliptic curves. One area in which expertise is currently in short supply is in the general theory of automorphic representations, and the same applies to the associated analytic theory of the Langlands programme. Since the departure of Dietmar and Hoffmann, the UK could benefit from more expertise in analytic aspects of automorphic forms. The UK has been traditionally particularly strong in analytic number theory. De- spite the departures abroad or retirements of leading figures, which has led to a gradual decline in numbers over the past 30 years, there has always been a small but constant core of experts, with Oxford being particularly strong. However there have been a number of recent appointments which have strengthened and enhanced the UK’s reputation in this area. Notable are those of Wooley in Bristol, and of Green in Cambridge; with the latter, the place of the UK in the internationally resurgent field of additive number theory is firmly established. Transcendental number theory, an area in which the UK used to be represented at the very highest level, has declined both internationally and locally, and with the retirement of Baker, there are no specialists in transcendental number theory remaining in the UK. The UK continues to have a small group of active researchers of Diophantine approximation, which has mainly been concentrated in York and East Anglia. The school of Margulis is under-represented, in comparison with the international picture. Computational number theory is an area in which the UK has always been at the forefront, and numbers working in this area have grown since the last international review. The emphasis is on the application of computation to arithmetic algebraic geometry, pushing the boundaries of what computations are feasible, and development of new algorithms, rather than on algorithmic complexity theory, as well as theoretical developments with a strong computational flavour. It may be

16 noted that leading computer algebra packages (MAGMA, SAGE) currently rely on algorithms for elliptic curves and L-functions developed by UK computational number theorists. Apart from the areas already mentioned there are others in the UK who work on the interface between number theory and other subjects. Three such areas in which there is world-leading expertise are logic, random matrix theory and cryptography. (Some of the last group work in Computer Science departments.) International collaboration is, as would be expected, of a high level. There have been formal international collaborations through at least two concurrent Marie Curie networks in recent years. Plans have been made for further network activities of this kind but the shift in emphasis in the Marie Curie programme has hampered their realisation. In 2005 the Heilbronn Institute for Mathematical Research in Bristol was opened, an initiative by GCHQ (Government Communications Headquarters) in partnership with the University of Bristol. The institute has created research opportunities for some 30 researchers with funding from GCHQ; a signiﬁcant proportion of these work in number theory or closely related areas. The age balance of the research community is good. Retirements have brought an inﬂux of young researchers. Across mathematics there has been a strong increase in the proportion of non-UK appointments, and in Number Theory the trend is particularly striking: 25 of the permanent researchers submitted to the 2008 RAE were appointed to their positions since 2001, and of these only 4 are UK nationals. However several recent appointments have been made to non-UK nationals who previously had completed UK PhDs. The return of Andrew Wiles to a position in Oxford in 2011 is of course a very welcome development. Reliable statistics for graduate student throughput in number theory are not available. In recent years at any one time there have been 65–75 postgraduate students spread across the country, with both UK and overseas students well represented. Demand for places is high. Many recent UK PhDs in number theory have gone on to positions in leading departments here and abroad.

17 International Review of Mathematics – Analysis Landscape

Author: John Toland (with contributions from David Preiss) Consulted: In writing this document the authors consulted numerous members of the research community.

1.Statistical Overview: In the 2008 RAE, Analysis comprised a quarter of all submissions but only 19% of early career submissions in the Pure Mathematics Unit. However, important activities in analysis, including some rigorous aspects of nonlinear PDE theory, were not returned as Pure Mathematics whereas some probability and not-so-pure dynamical systems were included. The tables below, which were derived from MathSciNet data, give rough estimates of the level of analysis activity in the UK relative to the rest of the world in 2009-10. Columns 5 and 9 in the first table indicate the UK output in MSC classifications as a percentage of world output for 2009-10 and 2000-10, respectively; the same data for the total UK analysis output are 2.6% and 2.5%, respectively. MSC Subject UK World UK /World % UK UK Insts RAE UK/World% Authors Nos. [ 2000-10 ] 26 Real Functions 16 825 1.9 16 12 3 [ 1.5 ] 28 Measure Theory 16 427 3.7 8 7 30 [ 4.3 ] 30 Complex Analysis 34 1553 2.2 21 14 11 [ 2.4 ] 31 Potential Theory 3 167 1.8 3 3 0 [ 2.4 ] 33 Special Functions 25 677 3.7 20 13 1 [ 3.7 ] 34 ODE 76 4141 1.8 77 36 17 [ 1.9 ] 25 PDE 250 7727 3.2 200 41 145 [ 2.6 ] 39 Functional Equations 6 847 0.7 7 7 1 [ 1.3 ] 40 Sequences 2 165 1.2 3 2 3 [ 1.4 ] 41 Approximation 5 674 0.7 5 4 2 [ 1.9 ] 42 Fourier Analysis 26 1278 2.0 26 13 22 [ 1.7 ] 43 Harm Analysis 8 214 3.7 8 5 1 [ 2.1 ] (abstract) 44 Integral Transforms 2 112 1.8 2 2 0 [ 1.4 ] 45 Integral Equations 7 302 2.3 15 8 1 [ 1.8 ] 46 Functional Analysis 83 1941 4.2 61 27 43 [ 3.7 ] 47 Operator Theory 51 2816 1.8 44 20 74 [ 2.4 ] 49 Calculus of Variations 36 1455 2.5 34 18 10 [ 1.8 ]

The next table is an amalgamation of data from various MSC classifications and sub- classifications related to (but not always a sub-classification of) the main one and the topic listed. (When estimating percentages of analysis outputs, numbers of papers etc., these additional data are included in the calculation.) MSC Subject UK World UK/World% UK UK Insts UK/World % 2009-10 authors [ 2000-10 ] 46 Operator and Banach 49 693 7.1 31 14 [ 5.6 ] Algebras 37 Ergodic Theory & 22 430 5.1 17 9 [ 5.9 ] Complex Dynamics 28 Fractals, GMT 17 300 5.7 13 11 [ 6.2 ] 35 Spectral Theory 14 377 3.7 20 11 [ 4.5 ] Data for the remaining part of functional analysis are

MSC Subject UK World UK/World% UK UK Insts UK/World 2009-10 authors % [ 2000-10 ] 46 Functional Analysis 34 1289 2.6 31 19 [ 2.7 ] (without algebras) The institutions producing the largest numbers of analysis papers in the last 10 years, ordered by numbers of papers, were: Oxford; Cambridge; ICL (all London is very much bigger); Leeds; Nottingham; Warwick; Edin burgh; Cardiff; Bath. In almost all areas of analysis th e number of published papers was roughly the same as in France, the main exception being PDE (UK about 1/4 of France in the last 10 years, about 1/3 in the last 2 years).

2. Subject breakdown: Since the probability landscape document will cover stochastic analysis and there is a separate numerical analysis landscape document, evidence from the RAE in 2008 and the tables above suggests t hat the analysis document should cove r the following topics (ordered roughly according to size):  Nonlinear PDEs (including calculus of variations, geometric analysis, nonlinear functional analysis etc) (MSC: 35, 45, 49, 58)  Operator algebras, Banach algebras and operator theory (MSC: 46, 47)  Linear PDE and spectral theory (MSC: 31, 35P)  Harmonic analysis (MSC: 42, 43, 44)  Real analysis (convexity, geometric measure theory (GMT), fractals, Banach spaces) (MSC:26,28. 40)  Ergodic theory (MSC: 28, 37)  Complex analysis (MSC: 30) Dynamical Systems has not been included here because it has components in various other landscape documents and the balance is elsewhere. Analysis work in ODEs includes a small number of good papers i n “pure ODEs” but is mostly subsumed in nonlinear functional analysis or spectral theory.

1.Discussion of Research areas1: By subject area, a frank assessment (both positive and negative) of past and current international standing of the field, supported by reference to highlights.

Nonlinear PDEs. Despite the investment of about £7M in the form of S&I awards in nonlinear PDE to Oxford an d Edinburgh, and a further Leverhulme grant to Warwick, this area remains weak relative to the high l evel of activity in t he rest of the world. However there is top qu ality activity in particular topics such as: mathematical aspects of material sciences (Oxford); calculus of variations (Oxf ord, Bath, Warwick, Sussex, Surrey, Swansea); nonlinear waves (Bath, Surrey ); homogenisation theory and applications (Bath, Oxford, Cardiff ); geometric PDEs (Warwick, S wansea, Cambridge, Imperial, Bath); dissipative PDE s (Oxford, Surrey, Warwick); stochastic nonlinear PDEs (Warwick, Edinburgh, York, Swansea); general relativity (Cambridge).

Kirchheim (Oxford), Topping (Warwick) and Dafermos (Cambridge) were awarded LMS Whitehead prizes in 2005 and 2009, for work in the calculus of variations, geometric PDEs and general relativity, Maz’ya (Live rpool, half-time) was awarded the 2009 Senior Whitehead prize for lifetime achievements in linear and nonlinear PDEs, and Dafermo s and Stuart (both Cambridge) shared the 2005 Adams Prize. Ball (Oxford) has received countless awards and honours for his contributions over many years.

Operator algebras, Banach algebras and operator theory. Internationally C*algebras is a huge field to which the UK made sig nificant early contributions. It con tinues to be involved, with high quality work in good journals (Winter’s 80 pages in Inventiones 2010 is an example). The community has link to other fields such as algebraic and non-commutative geometry, K-theory and conformal field theories, and has a significant influence worldwide in the training of postdocs. High q uality wo rk ha s been produced on non-self adjoi nt ope rator algebras. The study of Banach algebras is less popular but the UK has produced strong results on hypercyclic o perators and a solution of the long standing "identity plus compact" p roblem (Haydon (Oxford)). In operator theory, cutting edge work is bei ng done by individuals, for example on model theory, complex and harmonic analytic methods and operator theory methods for differential equations. Significant work is carried out in a number of (mathematically) very small in stitutions; a particularly interesting example is the London Metropolitan University. However, in spite of the high overall level of work being produced and the pre sence of individuals producing world-class rese arch, the UK is not heavily involved with the most strategically significant work being done internationally.

Linear PDE and spectral theory. Numerically, research in linear PDE is dominated by the spectral theory of linear elliptic operators including domain optimization problems. In addition there is high quality activity in function-space theory, pseudo-differential and Toeplitz operator

1 Only prizes awarded since the last IRM are mentioned.

19 theory, and the inte raction between integrable systems, Riemann-Hilbert theory and PDE boundary value problems. Loughborough, which will host Equadiff 2011 2, has a small group working on geometric and topological aspects of (global) analysis.

In spectral theory the UK is undoubtedly internationally leading and London, where there is a large concentration of both established and rising senior people, is probably the pre-eminent centre in the world. Moreover in the UK there are good links with the related areas of rigorous computational spectral analysis and mathematical physics. The editor-in-chief of the new Journal of Spectral Theory is based in London. Despite this concentration of established expertise, London spectral analysis is not having the impact that might be expected on graduate education in the UK and will struggle to renew itself from within.

Harmonic analysis. Harmonic analysis impinge s on re search not only in a large num ber of areas of a nalysis but also in applied mathematics (wavelets, si gnal & image processing etc) and structural algorithms for large data sets. In the UK its influence is not so pervasive and there are only two recognisable research groups, at Edinburgh and Birmingham, with distinguished individuals elsewhere, for exampl e in Cambridge. The UK g roups are ho wever strong, especially noteworthy is Edinburgh’s long-standing international collaboration with Tao (e.g. the mult ilinear Kakeya problem in Acta 2006) which continues to produce cutting edge results. The connections with PDEs have been strengthened by appointments to Edinburgh’s S&I Centre for Analysis and Nonlinear PDEs (CANPDE).

Some of the work of Green (Whitehead Prize 2005, EMS Prize 2008, FRS 2010) and Gowers in Cambridge is on the interface between discrete harmonic analysis and number theory.

Complex analysis. In the UK complex analysis is mainly focused on complex dynami cs of transcendental, rational and quasi-regular mappings, for which Rempe (Liverpool) was awarded an LMS Whitehead Pri ze in 2010. In 2004 Markovi c (Wa rwick) was awarded the same pri ze f or work at the interface of complex analysis and geometry. Th ere a re strong individuals working on classical complex analysis and a series of annual “function theory days” is well supported by existing and retired members of th e community. However overall, internationally, the UK’s presence and influence in complex analysis has noticeably narrowed. (There is no history of UK activity in the theory of several complex variables.)

Real analysis. The largest sub-area is fractal geometry, with its many connections, for example to dynamics, ergodic theory and applications. St Andrews is generally recognised as the main centre of the theory, but related work is done in about 12 universities, and the area is by now well established and healthy. There is a small, but world-leading, group working on dimensional aspects of Diophantine approximation theory in York.

There are 3-6 UK researchers whose work contributed to the study of Banach spaces. The most significant contributions are mentioned in the section on operator algebras above. Others study embeddings via infinite games, and differentiability problems.

The geometric measure theory population is even smaller (unless one counts related work, for example in geometric analysis) but highly significant internationally (Csörnyei ICM Invited Lecture 2010; Preiss, FRS 2004, Pólya Prize 2008).

Ergodic theory. In the UK, there is high quality activity in ergodic theory and in pure dynamical systems theory (see Complex analysis), areas which continue to have international prominence. Measured in terms of size and infl uence, this area necessarily follows the USA, France and Israel, where classical ergodic theory is tradition ally very strong. Ho wever, the vitality of the subject in the UK is evidenced by the strength of publications, the increase in the number of researchers working in th e area and t he n umber o f resea rch m eetings. Both managing editors of Ergodic Theory & Dynamical Systems (among whose editors is the 2010 Fields medalist Elon Lindenstrauss) are based in Warwick.

2 http://www.lboro.ac.uk/departments/ma/equadiff/

20 4. Discussion of research community . A brief description of demographic trends, e.g. composition, origin and volume of research student pipeline, age distribution of academic staff, etc.

Nonlinear PDEs. Since the late 1970s, John Ball (Oxford ) has had a strong influence internationally in the multi-dimensional calculus of variations, including related topics such as regularity and GMT. An S&I award to h is group OXPDE following the last IRM has expanded it with 6 a ppointments, including a p rofessorship in nonli near hyperbolic conservation la ws and a senior appointment (an ICM2010 invited speaker in Hyderabad) i n Navier Stokes theory. The chair appointment of Chen (with recent papers in the Annals) gives the UK a real presence in hyperbolic equations for the first time. OXPDE and CANPDE organise workshops and other activity throughout the UK. OXPDE, with 11 faculty, 7 postdocs and 12 research students is by far the UK’ s largest concentration of PDE activity, and with the backing of its department and university, and a good age balance, seems sustainable.

The study of PDE questions arising in geometry and topology has aroused huge international interest over the past 15 years, yet it was a grant from a private charity, the Leverhulme Trust, and the NSF (USA) that primarily supports the g eometric analysis activity at Warwick where there are 3 faculty, 5 postdocs an d 5 postgrads. T he setup looks fragile an d vulnerable to staff movements, with significant difficulties in attracting good UK graduate students.

The S&I award which created CANPDE in Edinburgh led to 3 lectureship appointments, 2 postdocs and a number of PhD students, but it suffered a serious loss of leadership when Kuksin moved to Paris. It has since secured three more substantial research grants and organised mini-symposia, crash courses and instructional workshops, and a large high profile international conference, to support PDEs across the UK.

Stochastic PDEs is a n important sub -area of nonlinear PDEs and the UK co mmunity, which produces world-class re search, may need ca reful m onitoring since som e of its groups are small and may disappear after the retirement/resignation of key people. Following recent retirements there is very little ODE theory or abstract nonlinear functional analysis in the UK.

Despite the large S&I investments, the area of nonlinear PDEs, particularly the pure mathematical side, is under-funded relative to the size of the community and by comparison with the international activity with which it competes.

Although OX PDE and CANPDE provid e some networking a ctivity and are now plan ning to hold an international conference in Oxford in July 2012, a regular national event (say, a biannual symposium) involving all researchers in the field of nonlinear PDEs scattered across the UK might be useful to knit the community.

Structural reasons for the continuing failure of the UK to build strength in this internationally essential area from within are discussed in Section 6.

Operator algebras, Banach algebras and operator theory. A ctivity remai ns hi gh a nd widely distributed with about 4 0 researchers at Aberyswyth, Aberdeen, Belfast, Cardiff, Edinburgh, Glasgow, Lancaster, Leeds, London Metropolitan, Nottingham and Oxford, with allied activity in Manchester, Swansea, Southampton and a strong framework of workshops. However, there is an overwhelming feeling in the community that the subject is in decline. A substantial number of the people who were hired in the 60s and 70s, when functional analysis flourished in the UK under the leadership of Bonsall, Johnson and Ringrose, have retired since the last IRM. In spite of several new (young) appointments to match some of the retirements there has been a steady, thoug h slo w (till n ow) decline in the number of permanent re searchers with interests in the area. Many more retirements are now imminent and the influence of this subject in the UK seems destined to decline further. This may be inevitable but the community feels it has been excluded from recent research council investment. Since the new S&I PDE centres and the recently-created d octoral training centre, Cambridge Centre for Analysis (CAA), seem not to cater for these interests, it is conscious that the universities that attract the best undergraduates are not teaching the subject at the right level.

21 Linear PDE and spectral theory. Research in linear PDE theory is based almost exclusively in the South, at Bristol, Cardiff, four London colleges and Reading. Overall numbers of researchers i n linear PDE s in the UK are largely u nchanged during the pa st 10 years an d although some senior figures have retired or left the field there remains a nucleus of strong individuals that will maintain UK strength in the area. However, existing personnel are somewhat less well distri buted geographically than before and the predominance of eastern- European trained researchers persists. (The organisers of the London Analysis Seminar are: 11 Eastern Europeans, 1 Dane and 1 from UK.)

Despite its notable strength, there is concern that the UK is not taking adequate measures to develop, or even to renew, its capacity in these important areas, through a lack of candidates for Ph Ds, b ecause what resources there are for research trai ning in analysis are diverting potential students to other areas. PhDs tend to leave the subject (often for financial mathematics) after graduation.

Harmonic analysis. In the UK harmonic analysis would seem to be roughly in steady state in terms of numbers, with slightly more younger researchers and slightly fewer senior researchers than a few years ago. The resignations of Cowling and Verbitsky from Birmingham and Volberg from Edinburgh in 2010 were serious losses. There are fewer Ph D students than before in Edinburgh, though there have been some in Birmingham; many of the better o nes come from o utside the UK. UK harmonic analysis activity needs to be a little larger and more diversified geographically and scientifically to be more viable and stable. The scarcity of UK trained postgraduate students is also a problem.

Complex analysis. Complex analysis in the UK has experienced a steady decline over several decades a nd the world-leading strength-in-depth in the theory of functions of one complex variable, once centred on Imperial College and Cambridge, has been depleted through retirement, non-replacements and lack of graduate students. Now the main strength is in dynamical systems at a few institutions such as the Open University, Liverpool and Warwick where there has been research council support, a few PhD studentships and research fellowships. Although there are distinguished individuals elsewhere, the subject overall is in decline, partly for structural reasons that affect all of analysis,

Real analysis. Many subareas are relatively small (even worldwide) and so heavily dependent on a few individuals and may significantly diminish when researchers change their area, leave or retire. This is what happened to set-theoretical analysis and convexity which seem to have almost disappeared from the UK. Although this was a natural development and not in itself a cause for concern, it would be a worry if steady erosion led to the complete disappearance of significant a reas of analysis f rom the UK. Recently su ccessful a reas, such as the study o f Banach spaces or GMT, could experience similar fates in the future.

Another matter fo r concern is the virtual non-existence of classical real a nalysis, a nd oth er areas that elsewhere serve as a first step in the careers of many analysts. Among various sub-areas, only the theory of fractals has an appropriately sound base with sufficiently well distributed activity to be considered safe. The development in the UK of new activity in the analysis of fractals with connections to harmonic analysis would improve the situatio n by strengthening both areas.

Ergodic Theory. Since the last IRM, there has been growth in the number of established and younger people through appointments and fellowships. The number of UK groups has also grown with Bristol and Surrey joining more established groups such as East Anglia, Liverpool, Manchester and Warwick. Interactions between the groups include an LMS funded series of One Day Ergodic Theory Meetings and larger conferences such as the current year-lon g EPSRC-funded Symposium on Ergodic Theory and Dynamical Systems at Warwick.

5. Cross-disciplinary/Outreach activities.

The Oxford PDE group is focused on problems such as phase transitions that that arise from material sciences and in mechanics. In Wa rwick work on geometric analysis is concerned with PDE problems from geometry and topology and there is PDE activity in Cambridge

22 related to general relativity. Obviously spectral theory addresses questions from quantum mechanics and in Oxford, Bath and elsewhere nonlinear PDE work focuses on problems with the equations of fluid mechanics. CANPDE in Ed inburgh has made it a priority to en courage transfer of knowledge f rom analysis to applications, for example in mathematical biology (recent workshop on cell migration and tissue mechanics) and risk analysis in oil recovery. There are strong relationships throughout t he UK between analysts workin g on theoretical PDEs, their computational counterparts and others working on discretized versions of continuous systems (lattice models). Analysis methods inform all of numerical analysis and significant developments in analysis lead to improved numerical algorithms. There are strong analysts in the UK working at these interfaces and in applications in biology, finance, surface sciences, optics, control theory etc.

6. Structural Issues.

In the responses to our extensive consultations one worry that was repeated over and over again concerned the infrastructure for analysis research in UK universities and the diminishing l evel of undergraduate exposure to analysis, much o f whi ch i s considered too difficult for mathematics undergraduates even at the very best universities. Undergraduates not exposed to the subj ect at the right le vel do not be come postgraduates. Consequently the subject is not producing strong graduate students and when positions become available they go to mathematicians fro m co ntinental Europ e (having a Habilitation or similar qu alification higher than a standard PhD) or the USA. No one, including senior people at the two S&I PDE centres who also report the effects of this t rend, have any idea how to a ddress this serious problem when universities, under pressure from funding agencies, insist on putting their priorities elsewhere.

Analysis underpins all of applied mathematics and mathematical physics, and lots of geometry, topology, combinatorics and number theory. However in the UK there is general concern that analysis is being phased out by universities in favour of so-called re al-world subjects with a higher capacity to win grants income. Further there is a widely held belief that the shift of funding away from re sponsive mode to priority research areas and from Doctoral Training Grants to Doctoral Training Centres, with funding for interdisciplinary research threatening t o supplant rather than reinforce funding for fundamental research, is already proving detrimental at the postdoctoral and postgraduate level.

Concentration of funding in the Research-Council-identified centres of excellence is proving neglectful of and damaging to the morale of other places where high quality research is being done, conspicuous examples being the linear PDE group in London and the wide functional analysis network with significant centres throughout the North. It is also felt tha t the docto ral training centre in Camb ridge (CCA) has the intention of diverting some of the best students away from fundamental aspects of analysis 3. Whether and what impact this CDT or the Warwick CDT (MAS DOC which is directed towards applications) will have on analysis in the UK remains to be seen. See http://www.epsrc.ac.uk/about/progs/maths/train/Pages/centres.aspx

Analysis is occupying an increasingly prominent place in international mathematical research. At ICM2010 in Hyderabad, the Chern and Gauss prizes and two of the Fields medals went to analysts yet only two analysts from the UK, Marianna Csörnyei and Gregory Seregin (both of whom were educated abroad) were Invited Speakers and there was not one UK Plenary Speaker in any discipline. The position of UK analysis internationally seems to be weakening as it strengthens in other countries.

3 At CCA a reader and a lecturer have b een appointed in kinetic theory and in comp utational harmonic analysis. T wo post-doc appointments sho uld be made this year. Eleve n students will start in October 2010 (5 UK, 5 EU, 1 R oW), of whom three show prior inclination towards pure/abstract analysis, three towards stochastics, five P DE and computational. The aim of CC A is to br oaden the knowledge of all eleven. Also, the pro gramme has been attractive to students with a mixture of interests whose ultimate direction is unclear.

23 INTERNATIONAL REVIEW OF MATHEMATICS LOGIC LANDSCAPE

Lead Author: Anand Pillay, with input from Alex Wilkie. Consulted: In writing this document the authors consulted numerous members of the research community.

1. Statistical Overview: Logic in the UK covers a wide area. There are more logicians in computer science departments than in mathematics departments. Computer science departments will be discussed briefly in section 6 below. But in so far as mathematics departments are concerned, the 2008 Research Assessment Exercise included around 35 “designated” logicians in 10 institutions, although there has been some movement and change since then (discussed below).The UK has internationally leading figures in all the “core” areas of mathematical logic (referred to in the table below) with the possible exception of set theory.

2. Subject breakdown: The table indicates the number of active researchers in designated sub-areas based on submissions to the 2008 RAE. But there have been many changes in the past couple of years.

Research Area Number of % of Total Researchers

Model theory 17 49% Proof theory, constructive 7 20% mathematics, and category theory Recursion theory (aka 3 9% Computability theory) Set theory 4 11%

Other 4 11%

3. Discussion of research areas: The traditional subareas of mathematical logic are model theory, proof theory, recursion theory and set theory. All these subareas are now firmly part of mathematics. Proof theory and set theory retain a close connection with the foundations of mathematics in the sense of Hilbert and Gödel. Recursion theory is the mathematical theory of “effectiveness” (or “computability in principle”). For many years the focus was on studying related structures (degree structures) and their “algebraic” properties. More recently the techniques and notions have also been applied to measure and randomness. Model theory is now somewhat removed from the “foundational” tradition. In addition to a rich “inner” theory (connected to what is called “stability theory”), model theory has many connections with and applications to other parts of mathematics, such as number theory, algebraic geometry, representation theory. There are also multiple connections of these subareas of logic to computer science. We restrict ourselves in this section to logicians in mathematics departments, and try to concentrate on highlights of current and recent activity.

(a). Model Theory. The past 10 years have seen the emergence of a major specifically “UK” school/trend in applied model theory. This is the attempt, initiated by Zilber (Oxford), to understand the complex exponential function from a model-theoretic point of view.

24 Schanuel’s conjecture on the transcendence degree of fields generated by a finite set of complex numbers and their exponentials plays an important role. Active participants in the project, in addition to Zilber himself, include Kirby (East Anglia), Macintyre (Queen Mary, University of London), and Wilkie (Manchester), and there are a few international participants. Zilber’s conjectures on the complex exponential function have also contributed to the formulation of what is currently called the Zilber- Pink conjecture in number theory, concerning intersections of algebraic varieties with sets of “special points”, and there are several international meetings on this topic planned throughout 2010-2011. A related striking development, with strong roots in the UK, is the application of ideas from o-minimality (in model theory) to diophantine-geometric problems, using fundamental work of Pila (Oxford) and Wilkie. It has led to the first unconditional proof, due to Pila, of the Andre-Oort conjecture. Another related development, but now emanating from the interface of “stability theory” and differential equations is a proof of function field analogues of Schanuel, more precisely Lindemann’s theorem, for “families”, possibly nonconstant, of semiabelian varieties by Pillay (Leeds) in collaboration with the number theorist Bertrand from Paris VI. The strength of the above (and related) work is reflected in the “elite” journals where the material is published: M. Bays, J. Kirby, A. Wilkie, A Schanuel property for exponentially transcendental powers, to appear in Bulletin of London Math. Soc. Luc Belair, A. Macintyre, T. Scanlon, Model theory of the Frobenius on the Witt vectors, Amer. J. Math. 129 (2007), 665-721. D. Bertrand and A. Pillay, A Lindemann-Weierstrass theorem for semiabelian varieties over function fields, Journal of American Math. Soc. 23 (2010), 491-533. J. Pila and A. Wilkie, The rational points of a definable set, Duke Math. Journal, 133 (2006), 591-616. J. Pila, O-minimality and the Andre-Oort conjecture for Cn, to appear in Annals of Mathematics.

A recent major international development in model theory has been the in-depth analysis of first order theories without the independence property, and applications to specific examples such as o-minimal structures and valued fields, one consequence being a recent radically new approach to rigid analytic geometry (Hrushovski- Loeser). Macpherson and Pillay from Leeds, have played leading roles at the “foundational level” in these developments, on both the pure and applied sides. D. Haskell, E. Hrushovski, D. Macpherson, Stable domination and independence in algebraically closed valued fields, Lecture Notes in Logic, CUP, 2008. E. Hrushovski, Y. Peterzil, A. Pillay, Groups, measures, and the NIP, Journal of American Math. Soc. 21 (2008), 563-596.

Pillay’s return from the US in 2005 has, among other things, strengthened the “pure” or “stability-theoretic” side of model theory in the UK, where traditionally the applied side of the subject has tended to dominate. Other major and continuing research themes, where the UK has important figures, are the model theory of modules (where Mike Prest, in Manchester, is the dominant figure and has established close links with both representation theory and category theory), simple groups of finite Morley rank (Borovik at Manchester being an international leader in this topic), homogeneous structures, their automorphism groups, and combinatorics (at the interface of permutation group theory and model theory this theme has strong roots in the UK and the leaders are Macpherson and Truss at Leeds and Evans at East Anglia). In addition to Wilkie, mentioned above, Tressl as well as postdoc Jones, make Manchester a major centre for o-minimality.

25 Koenigsmann in Oxford makes deep contributions at the interface of logic and Galois theory, and has recently obtained the strongest results around Hilbert’s 10th problem for the field of rational numbers. Recently published books representing the state of the art in current research are: T. Altinel, A. Borovik, G. Cherlin, Simple groups of finite Morley rank, AMS monographs, AMS, 2008. M. Prest, Purity, spectra, and localization, Encyclopedia of Mathematics and its applications, vol. 121, Cambridge University Press, 2009.

(b) Proof Theory. At Manchester, Jeff Paris directs a very original research group around “uncertain reasoning”, or “foundations of probability theory in the setting of mathematical logic”. A monograph summarizing the achievements of the past 10 years of work is in preparation, but among the best recent papers is: J. Landes, J. Paris, A. Venkovska, A characterization of the language invariant families satisfying spectrum exchangeability in polyadic inductive logic, Annals of Pure and Applied Logic, 161 (2010), 800-811.

Rathjen directs a group in “classical” proof theory at Leeds. Rathjen is a world leader in proof theory and constructivism. He has done ground-breaking work on the “ordinal analysis” of strong theories (see the reference below), and is currently also investigating themes with collaborators such as Harvey Friedman (Ohio) at the interface of proof theory and combinatorics. Rathjen is complemented at Leeds by Peter Schuster, an expert in constructive algebra, analysis, and set theory. Prominent papers include: T. Coquand, H. Lombardi, P. Schuster, The projective spectrum as a distributive lattice, Cah. Topol. Geom. Different. Categ. 48 (2007), 220-228. M. Rathjen, The art of ordinal analysis, Proceedings International Conference of Mathematicians, Vol II, EMS 2006, 45-69.

The themes of proof theory, combinatorics and incompleteness are also explored by Andrei Bovykin (Bristol), who together with his advisor Kaye (Birmingham) represents a continuation of the “model theory of arithmetic “ topic, once a central component of UK research in logic.

Cambridge remains a centre of proof theory, category theory (including categorical logic) and topos theory, led by Hyland, Johnstone (as well as Pitts in Computer Science). Within proof theory, work on Linear Logic continues at Cambridge, with strong links to work in Computer Science work around Game Semantics (especially at Bath and Manchester). A major development is a precise approach to the old notion of genus of a proof, based on Frobenius algebras. A major new direction within category theory is that of higher categories. Leinster (now in Glasgow) has made substantial contributions to the general theory. Cambridge has more recently focused on rich low-dimensional phenomena with work relating to operads and 2-dimensional representation theory (Lopez Franco). A striking achievement in category theory proper is Garner’s (Cambridge) refinement of Quillen’s small object argument in terms of natural weak factorization. Topos theory, led by Johnstone, continues to flourish in Cambridge and one looks forward to closer links developing with model theory. References: M. Fiore, N. Gambino, M. Hyland, G. Winskel, The Cartesian closed bicategory of generalised species of structures, Journal of the London Math. Soc. 77 (2008), 203 – 220.

26 R. Garner, Understanding the small object argument, Applied Categorical Structures 17 (2009), 247-285.

(c) Recursion theory (also known as Computability theory). In so far as the UK is concerned, the subject is based in Leeds, around the well- known and established Barry Cooper, and his former students Andy Lewis, and Charles Harris (postdoc). Although the centre of the subject is now situated in the US, the group in Leeds maintains a relatively strong reputation. There remain very basic, difficult and longstanding problems concerning structural properties of various “degrees of computability”: Turing degrees, recursively enumerable degrees etc. Lewis, a rising star in the field, is making fundamental contributions to these issues, by himself and with a wide range of top collaborators, as well as contributing and connecting these notions to Martin-Lof randomness. Cooper, in addition to working on hard technical problems, is devoting time to more speculative work situating the basic notions of recursion theory in broader scientific contexts. Outside Leeds, Philip Welch, a set theorist, in Bristol, has done influential work on infinite time Turing machines, a topic that Lewis helped introduce 10 years ago. There is also a close collaboration of Beggs (Swansea) with Tucker in CS at Swansea, on physical models of computation. References: S. B. Cooper, Extending and interpreting Post’s programme, Annals of Pure and Applied Logic, 161 (2010), 775-788. A. Lewis, A random degree with strong minimal cover, Bulletin London Math. Soc., 39 (2007), 848-856.

(d) Set theory. The set theory community in the UK is relatively small, consisting of Dzamonja and (recently appointed) Kolman at East Anglia, and Welch at Bristol. They are very active with a wide range of top international collaborators. At East Anglia research has focused on forcing, large cardinals and connections to other fields of mathematics (such as analysis and measure theory). Recent work by Dzamonja (referred to below) gives a counterexample to a long standing conjecture of Talagrand. Welch at Bristol focuses on Inner model theory, determinacy and reverse mathematics, but also works on models of computation (referred to in (c) above). In addition Welch has an ongoing and effective collaboration with philosophers of logic and mathematics (supported by the British Academy). References: M. Dzamonja and G. Plebanek, Strictly positive measures on Boolean Algebras, Journal of Symbolic Logic, 73 (2008), 1416-1432. S. Friedman, P. Welch, H. Woodin, On the consistency strength of the inner model hypothesis, Journal of Symbolic Logic 73(2008), 391-400.

4. Discussion of research community Oxford, Manchester, and Leeds remain world centres of mathematical logic. Cambridge retains its prominence, and East Anglia is rising fast. Queen Mary, University of London, has regained some of its early momentum. Activity at Bristol is increasing, with the recent appointment of Bovykin. There has been considerable movement and even expansion from 2003 up to 2010. Dominant figures such as Wilfrid Hodges (Queen Mary, London) and Stan Wainer (Leeds) retired. Macintyre moved from Edinburgh to Queen Mary, soon after which Ivan Tomasic was appointed there. Anand Pillay moved to Leeds from the University

27 of Illinois. Andy Lewis moved to Leeds on a Royal Society Fellowship and will continue there on a permanent position. Recently Peter Schuster was appointed in Leeds as a replacement for Wainer. Alex Wilkie moved from Oxford to Manchester in 2007, and Marcus Tressl was appointed in Manchester soon after. Jochen Koenigsman was appointed in Oxford in 2007, and in an interesting development reflecting the nature of contemporary model theory, Jonathan Pila, a number theorist, was recently appointed to the Oxford Readership in Mathematical Logic, vacated by Wilkie. Jonathan Kirby and Oren Kolman have recently been appointed to Lectureships in East Anglia. On the category theory side, Cheng and Gurski were appointed in Sheffield and Leinster in Glasgow (all of whom were doctoral students in Cambridge). There have been also a stream of postdoctoral appointees in logic, funded through the EPSRC (UK), European networks, and other sources. Currently there are around 11 such researchers. In addition there are around 50 research (Ph.D.) students in logic (24 of whom are at Leeds), and most of whom are from the UK/European Union. In spite of the comparatively low level of funding of core mathematics research in the UK, logic as a whole does very well: for example essentially all the logicians at East Anglia and Leeds are currently running EPSRC supported research projects. UK-based logicians are very visible at both the national and international level. For example Macintyre and Hyland are President and General Secretary, respectively, of the LMS (London Math. Soc.), Wilkie is President of the (international) Association of Symbolic Logic, and Rathjen is on the Scientific Committee of the Oberwolfach Research Institute.

5. Cross-disciplinary/Outreach activities. The discussions above already point to intrinsic connections between mathematical logic in the UK and other areas of mathematics and science (for example number theory and algebraic geometry in the case of model theory). We will discuss some additional interdisciplinary activity, sometimes at an early stage where success is not yet guaranteed.

Parallel to, and closely related to, his work on complex exponentiation, Zilber (Oxford) has been pursuing connections (sometimes rather speculative) between model theory, noncommutative geometry, and physics. A motivating insight is that many sophisticated and “exotic” constructions in model theory (e.g. Hrushovski constructions) produce mathematical objects that can and should be seen as belonging to noncommutative geometry. Among published papers on the topic are: B. Zilber, A class of quantum Zariski geometries, in Model Theory with Applications to Algebra and Analysis, (edited by Chatzidakis, Macpherson, Pillay, Wilkie), LMS Lecture Note Series 349, Cambridge Univ. Press, 2008. Zilber is collaborating with physicists in Oxford, and they share a joint student.

In Leeds a collaboration is beginning between Pillay and colleagues from integrable systems around aspects of differential equations. Key themes involve the use of model-theoretic tools in the Galois theory of (possibly nonlinear) algebraic differential equations, as well as in their classification. This work will be supported by the EPSRC and will involve shared students and postdocs.

Again in Leeds, an EPSRC supported project on homogeneous structures is underway, led by Truss and Macpherson. Part of the project involves applications to constraint satisfaction from complexity theory (in computer science). Considerable

28 interaction with computer scientists (such as Bodirsky from Paris) is expected, and has already begun.

Hyland, in Cambridge, collaborates with computer scientists (in fact a part of his work simply is computer science). Collaborative work with Plotkin and Power (referred to below) using category-theoretic approaches has transformed the understanding of computational effects in programming language theory. M. Hyland, G. Plotkin, J. Power, Combining effects: sum and tensor, Theoretical Computer Science 357 (2006), 70-99.

The British Logic Colloquium remains a coordinator of logic in Europe and helps preserves close links with computer science. UK-based logicians such as Cooper are prominent in networks such as “Computability in Europe” which organize regular international meetings where logicians of various persuasions and computer scientists can exchange ideas. Also UK logicians Evans and Macpherson have been coordinators of large-scale European research and training networks in model theory and other parts logic (MODNET, MATHLOGAPS, MALOA).

Logicians are also active in various consortia such as MAGIC which provide postgraduate mathematics courses through videoconferencing (Dzamonja and Kirby at E. Anglia, Pillay at Leeds). The need for such additional courses was highlighted in the 2003 International Review of UK Mathematics.

6. Logic in Computer Science Departments. There are Logic research groups in the following Computer Science Departments: St. Andrews, Bath, Birmingham, Cambridge, Durham, Edinburgh, Imperial, Kings College London, Leeds, Leicester, Liverpool, Manchester, Nottingham, Oxford, Queen Mary and Westfield, Swansea, and University College London. Traditional and ongoing areas at the interface of logic and computer science include automated reasoning, modal logics, semantics of programming languages, constructive logic, complexity theory, and program extraction from proofs.

Major new currents include applications of logical and structural methods to areas such as (i) Systems biology, (ii) quantum information and foundations of quantum mechanics, (iii) computational game theory and economics. Leading UK centres in (i) are Edinburgh (Hillston, Danos) and Microsoft Research Centre in Cambridge (Cardelli). In (ii) the quantum group at the Oxford University Computing Lab, led by Abramsky and Coecke, has some 30 members and has been instrumental in building a new international community using new kinds of logical and category-theoretic approaches. In (iii) Liverpool has a leading group Goldberg, Wooldridge). Other major developments include the introduction of separation logic, now influential in verification, with a leading group being Queen Mary (O’Hearn). UK strength in logical methods in databases and information systems has dramatically improved: leading groups are in Oxford (Gottlob, Horrocks) and Edinburgh (Buneman, Libkin).

29 INTERNATIONAL REVIEW OF MATHEMATICS COMBINATORICS LANDSCAPE

Lead Author: Peter Cameron, Peter Keevash Contributors/Consulted: In writing this document the authors consulted numerous members of the research community.

1. Statistical Overview:

Statistical information on combinatorics is difficult to come by. Many researchers in the field are not in mathematics departments, but may be labelled as computer scientists, statisticians, or even operations researchers in business schools. However, the British Combinatorial Bulletin provides an overview, and is fairly reliable.

Data compiled from the British Combinatorial Bulletin by James Hirschfeld shows that the number of staff in post working in Combinatorics in 2010 was double the figure in 1998 (up from 152 to 309), and the number of articles published or in press had increased by 50%.

A survey was undertaken for us by a statistics undergraduate threw more light on the demographics and will be referred to below.

2. Subject breakdown: The table indicates the number of active researchers in designated sub-areas (please state source, e.g. data extracted from submissions to RAE 2008) Research Area Number of % of Total in Researchers XXX

We have not attempted to answer this question. Combinatorics does not fall into neatly separated areas. However, the talks given at the biennial British Combinatorial Conference confirm the view that there has been growth in hypergraph theory and decline in design theory and finite geometry.

Combinatorics is a m ajor and important pa rt o f modern mathematics, w ith l inks to almost all parts of pure mathematics and a wide range of applications in statistics, computer science, operations research, and other areas. However, it fits somewhat uneasily into the usual assessment processes for mathematical sciences, for v arious reasons: many peopl e w ho do Combinatorics are not pa rt o f mathematics departments; much Combinatorics research is done by mathematicians who do not regard themselves as combinatorialists; and it is a subject where techniques have arguably more importance than big theorems. It is our hope that the document will be read with these comments in mind.

3. Discussion of research areas: By subject area, please provide a frank assessment (both positive and negative) of past and current international standing of the field, supported by reference to highlights as appropriate.

All areas of combinatorics are studied in the U.K. There are particularly strong traditions in graph theory, design theory, Ramsey and extremal combinatorics, and symmetric functions (established in part by people such as C. St.J. A. Nash-Williams, R. A. Fisher, R. Rado and I. D. Macdonald), but there is also considerable expertise in probabilistic combinatorics, combinatorial number theory, asymptotic enumeration (especially related to algebra), matroids, finite fields and finite geometries, group actions and representation theory, symmetric functions and orthogonal polynomials, coding theory, information security, algorithmic combinatorics, and computational complexity. Moreover, areas such as extremal hypergraph theory are developing rapidly.

A glance at recent publications in Combinatorics in Britain (a list of current papers is available at http://www.essex.ac.uk/maths/BCB/BCBFiles/2010/listc2010.pdf) shows papers on a wide range of topics in Combinatorics and links with other areas, including graph homomorphisms and constraint satisfaction, stability in extremal hypergraph theory, the Urysohn universal metric space, graph polynomials and statistical mechanics, primitive permutation groups, countable categoricity of first- order theories, sparse Ramsey theory, mathematical analysis of the AES and other ciphers, to mention just a few. The links to other parts of mathematics, and the wide range o f applications, ar e hinted at in this list. It is not clear to us that this highly interdisciplinary research is adequately judged by current research council protocols.

The first author would like to highlight his own recent work (with a widely dispersed international group) on synchronization, taking in automata theory, permutation groups, and graph homomorphisms.

There is less research now in combinatorial design theory and finite geometry than at the tiime of the last review; but a significant recent development has been the increased attention paid by groups such as Queen Mary to optimality criteria in design theory, linking this subject to a variety of topics in graph and network theory such as Laplacian eigenvalues, random walks, electrical networks, etc. A survey article in the last British Combinatorial Conference invited speakers' volume contained much new research and is already leading to new work in this area.

4. Discussion of research community Please give a brief description of demographic trends, e.g. composition, origin and volume of research student pipeline, age distribution of academic staff, etc.

The UK research community in Combinatorics has high international status at all levels. Strong groups at Cambridge, Oxford, London ( Queen Mary, LSE, Royal Holloway), together with newer groups established at Birmingham and Warwick contribute to this.

As older members of the community (Bollobas, Cameron, Welsh and others) approach or reach retirement age, there is a very strong group of active researchers

31 including Leader , Scott, Thomason, Kuhn O sthus, Riordan, Keevash, among m any others.

In co nnection w ith this exercise, we have conducted a comparison o f numbers of researchers over the last ten years and a survey of current researchers. All indicators show t hat t he su bject i s healthy; numbers are u p, and the current age distribution shows, after the expected peak for age 55+ and dip for 45-54, an even larger peak of young researchers. Many of these young researchers already have a high reputation, which bodes well for the future of the subject.

In the course of these changes, there has been some shift in subject-matter. There is now comparatively more work in extremal graph theory and hypergraph theory, at the expense of design theory and finite geometry (though the latter are still represented in our subject).

Areas of strength at the time of the previous review such as algebraic and probabilistic combinatorics continue to flourish. Indeed, many strong applied probabilists such as Grimmett, and algebraists such as Muller, produce work which can be regarded as combinatorics.

Currently there are many very talented foreign nationals working as combinatorialists in the UK. Our survey found people from Australia, Belgium, Bosnia, Brazil, Canada, Czech Republic, Denmark, France, Germany, Greece, Hungary, Ireland, Israel, Italy, Kenya, the Netherlands, New Zealand, Peru, Russia, South Africa, Sweden, Thailand and the USA.

This wealth of international researchers should be welcomed, both in the provision of talent not available among local researchers, and in raising the international profile of UK research when these researchers take back elements of their UK experience to their home countries.

We should not be complacent about the continuing ability of the UK to attract these talented r esearchers. While some are attracted by positive f eatures, such as the opportunity to work with an established UK mathematician who is a world leader in their field, others look to the UK because of difficulties in obtaining positions or funding in their own countries.

It is clear that the current regime of tightened visa regulations and a proposed cap on non-EU nationals employed in the UK threaten to have a seriously detrimental effect on the subject.

The previous International Review of Mathematics in 2003 recommended certain changes t o make UK P hDs more competitive internationally. T hese changes have been implemented successfully. We do not yet have evidence either that more foreign students are attracted to Britain, or that British PhDs are more competitive, as a result of the changes.

Our survey showed a small but increasing proportion of female combinatorialists, and a good ethnic mix (although a fairly large proportion of respondents preferred not to answer the question on ethnicity).

5. Cross-disciplinary/Outreach activities: e.g. describe connections with other areas of mathematics, science, engineering, etc.

32 Many combinatorialists work in computer science departments or business schools. Another important area is statistical design theory, where combinatorialists are involved with issues concerned with the efficiency of experiments, with spin-offs across science and engineering. Within mathematics, the interface between Combinatorics and Number Theory has led to the growth of the field o f Additive Combinatorics, which has made great strides on some old difficult problems. On the applied side, combinatorial methods have been crucial for algorithms in computational biology, a nd t he dr ive t o understand complex net works such as the web graph has brought combinatorial researchers in Graph Theory into contact with a wide range of other disciplines.

In most of these cases, mathematicians play a l eading, or at least independent, role in the research. Problems that arise in applications take on a life of their own in the hands of mathematicians and t he dev elopment of t hese ar eas feeds back into the applications.

The 2003 International Review mentioned that the number of mathematicians (especially combinatorialists) in Computer Science departments in the UK is low by international standards. This remains true. We make some further comments on this at the end of the document.

It is clear from the list of current publications that collaborations are not restricted to the UK or indeed t o any group of countries. T he st rategically important partners in Europe, USA, China, India and Japan are well represented, as are many other countries including Australia, New Zealand, Canada, Israel, Iran, the Philippines and South Africa.

One of the most important sources of collaboration is the biennial British Combinatorial Conference, an international conference that typically attracts 250-300 delegates, of whom more than half are from other countries. The Isaac Newton Institute in Cambridge provides a forum for international collaboration o n selected themes, including a program on Combinatorics and Statistical Mechanics in 2008.

The Combinatorics community in the UK is involved with many areas of new technology and societal challenges, including quantum information theory, scheduling t heory, and cryptography. In the last of these, the Information Security Group a t Royal H olloway ar e world l eaders, and have even f eatured i n t he recent bestseller “The Da Vinci Code”. A strong group in Bristol works on quantum computation.

Four of the fifty-odd institutions listed in the British Combinatorial Bulletin are industrial rather than educational. The researchers at these places consider themselves part of the Combinatorics community and attend conferences regularly. No doubt more could be done in this direction.

British combinatorialists have a strong tradition of work in areas such as radio frequency allocation, where groups at LSE, City and Oxford interact with the telecommunications industry. The Information Security group at Royal Holloway have close links with industry.

It has to be said that much of this research is not conducted under the auspices of mathematics departments. This, of course, is not really a handicap since the channels of communication between combinatorialists work well.

33 Another issue concerns outreach activities, which can make a major contribution to the health of the subject. Authors such as Robin Wilson are good at producing books that straddle the divide between the research community and the public. Also, Robin Whitty's “Theorem of the Day”, a t http://www.theoremoftheday.org/ , co ntains maybe more than its fair share of Combinatorics, which reflects the fact that the subject is relatively accessible t o non -specialists. T here se ems to be c onsiderable unfulfilled potential in using Combinatorics to further the public understanding of mathematics.

6. Further comments

The report of the 2003 International Review said,

“In view of the increasing demand for graduates in this area, especially from industrial, governmental, and academic employers, we are concerned that the supply of graduates in this area may not be adequate to fill the anticipated demand. In view of this demand, and the relative understaffing in this discipline in the UK, we feel a good argument can be made for building more capacity in Combinatorics here.

In addition, w e f eel that v aluable opportunities for t his discipline ar e bei ng missed because of the lack of strong ties to the theoretical computer science community, that are typically present i n universities outside of t he U K. Much of C ombinatorics now being developed world-wide has a strong algorithmic component, and as pointed out by the earlier International Review for Computer Science, this is a side of computer science which is not as strong in the U.K. as it should be.”

They also said,

“Most areas of Combinatorics are present in the UK, many of them at an excellent, sometimes outstanding level. Combinatorics is present or implicit, at a very high level, in many "other" subjects such as group or representation theories. Originality, openness and some spectacular achievements characterize UK Combinatorics. Demographic trends are nevertheless a cause for concern, since many of the current leaders are approaching retirement age.

In view of the increasing demand for graduates in this area, especially from industrial and governmental employers, it is worrisome that the supply of graduates in this area may not be adequate to f ill t he ant icipated demand. T his is also t rue for academic positions as well. In addition, it seems that valuable opportunities for this discipline are being missed because of the lack of strong ties to the theoretical computer science community (as compared with the situation in many other countries). A lot of Combinatorics now being developed has a strong algorithmic component and as pointed out by the earlier International Review Panel for Computer Science, this is a side of computer science which is not as strong in the UK as it should be.”

As explained ear lier, Combinatorics is a t hriving discipline in t he U K, but its health should be attributed to factors other than any specific initiatives taken after the last review: the points raised above remain valid today.

UK Landscape in Numerical Analysis

1 Statistical Overview cases there is related UK-based work being con- 1065 individuals working in 45 institutions were ducted by others and/or other institutions. submitted to the Applied Mathematics Panel of RAE2008. A rough estimate is that 100 of those 3.1 Differential Equations individuals from 20 institutions would regard themselves as working fully or largely in numerical anal- Nonlinear PDEs The UK has several world- ysis, of whom 15 were early career researchers. leading research groups working in various aspects About one quarter of the submissions contained of computational nonlinear PDEs. At Cambridge, more than four numerical analysis researchers. the group led by Markowich (hired from Vienna Other units of assessment also received submissions in 2006) is active in dispersive and highly oscil- that included numerical analysis activity. Overall, latory problems. Elliott (at Warwick since 2007) this represents a substantial increase in numerical leads a research group in surface PDEs and inter- analysis activity from RAE2001. faces that has made important contributions to the field of computational geometric evolution equa- 2 Subject Breakdown tions and PDEs on evolving surfaces. He also con- The greatest number of researchers are working tributed to the development and analysis of nu- in differential equations, followed by optimization merical algorithms in a range of problems involv- then numerical linear algebra then approximation ing nonlinear diffusion. Süli(Oxford) is active theory. in several areas of nonlinear PDEs including coupled Navier-Stokes/Fokker-Planck systems arising 3 Discussion of Research Areas in non-Newtonian fluid mechanics (with Barrett, The UK has been internationally leading in Numer- Imperial and Schwab, ETH) and in nonlinear solid ical Analysis research since the early days of dig- mechanics and materials science where (with Ort- ital computers. Linear algebra, optimization and ner) he is engaged in OxMOS “New Frontiers in approximation theory are long-standing areas of the Mathematics of Solids” (led by Ball, Chap- strength. Research in (partial) differential equa- man and Süli),and OxPDE: the “Oxford Centre for tions has grown in volume and strength consider- Nonlinear Partial Differential Equations”, directed ably over the last two decades and is the largest by Ball. Others active in the broad area of com- subarea. The UK is internationally competitive on putational nonlinear PDEs include Barrenechea all the major fronts within numerical analysis. (Strathclyde), Budd (Bath), Burman (Sussex), Kay Three worldwide trends in numerical analysis (Oxford), Jimack (Leeds), Lin (Dundee), Mac- and computational mathematics over the last 10– Donald (Oxford), Nürnberg (Imperial), Ramage 15 years have also been seen in the UK. First, (Strathclyde), Stinner (Warwick), Styles (Sussex) the area has become more integrated with core ap- and Wendland (Oxford). The field as a whole is a plied mathematics, especially through the organi- growing one within the UK, with several key senior zational structures of some of the larger groups. appointments having helped maintain competitive- Second, research straddling the different subareas ness with other leading countries. has grown significantly, for example at the linear al- Adaptivity The construction and mathemat- gebra/optimization, linear algebra/PDE and (most ical analysis of adaptive algorithms for nonlinear recently) PDE/optimization interfaces. Third, nu- PDEs continues to be an area of strength in the merical analysts are increasingly engaging in inter- UK. Ainsworth leads research in computational disciplinary research, for example in biology, bioin- PDEs at Strathclyde, much of it focussed on adap- formatics, data assimilation, engineering, materi- tivity and a posteriori methods. Houston (Notting- als science and networks. The hiring in the UK ham) has made important contributions to the de- of researchers trained abroad has had a noticeable velopment and analysis of adaptive finite element influence on some of these trends. Nevertheless, methods, particularly the hp-version discontinuous the UK numerical analysis community remains less Galerkin finite element approximations of nonlinear integrated with the broader mathematical and sci- PDEs that arise in compressible and incompress- entific research landscape than in other research- ible fluid flow problems. Other important contribu- leading countries. tors to this area include Budd (Bath, in connection The following overview of the current research with the numerical approximation of singularity landscape, and opportunities for the future, cannot formation), Dedner (Warwick, hyperbolic PDEs), cover all areas of activity, nor name all the active Georgoulis (Leicester, convection-reaction-diffusion researchers involved. We mention selected key re- problems), Jimack (Leeds, space-time adaptivity searchers, and institutions, but note that in most for solidification), MacKenzie (Strathclyde, Stefan

35 problems) and Lakkis (Sussex, norm-based a pos- the UK, with leading research groups at several teriori methods). This field is thus well-represented institutions. Iserles (Cambridge) continues to be within the UK. active in this area and his research agenda con- Solution of Linear Systems Arising in cerning Lie group methods, Magnus series and re- PDEs The UK is strong in this important field, lated topics has spawned a number of significant re- with research leadership provided by Wathen (Ox- search problems, together with research groups in ford) and Silvester (Manchester), with many con- the UK and worldwide. Leimkuhler (Edinburgh) tributions aimed at saddle-point problems and re- leads research at the interface of numerical inte- lated systems rising in fluid mechanics and opti- gration, computational mechanics and statistical mization, and the group led by Graham and Spence mechanics and has developed world-leading exper- (Bath) studying discretization of boundary value tise in the construction of thermostats. In com- problems. Other active researchers in this and re- putational ODEs Cash (Imperial) is active in the lated areas include Kay (Oxford), Loghin (Birm- development of code for singularly perturbed two- ingham), Marletta (Cardiff), Powell (Manchester), point boundary-value problems and differential al- Scheichl (Bath) and Duff (RAL, and partly through gebraic equations, emphasizing the importance of his team at CERFACS, Toulouse). An EPSRC conditioning. Many of his codes are widely used, network in the area would give the community a e.g., in Maple 14. Hill (Bath) has made significant greater UK cohesiveness; an INI programme would contributions to the analysis of nonlinear stability enhance its international identity. of discretized ODEs, using control-theoretical tech- Stochastic Problems This is an area of com- niques and within the framework of general linear putational mathematics that has grown in impor- methods. Iserles (Cambridge) has played a major tance significantly over the last decade, primarily role in establishing a theory for the integration of because of the increasing use of stochastic mod- highly oscillatory ODEs and this too has led to new elling in many realms of science and technology. research questions and emerging junior researchers. The UK is well-represented, with several inter- This field has, to a large extent, been invigorated nationally leading research teams. The group of by links to broader trends in multiscale numerical Stuart (Warwick) has studied design and analy- analysis (see the next item), which provides a nat- sis of numerical methods for SDEs and SPDEs, ural path for the future development of the area. focussing particularly on long-term behaviour, er- Multiscale Problems There is a growing godicity and links with Markov chain-Monte Carlo. trend worldwide to study a variety of problems More recently this group has been active in the which possess two or more different scales under Bayesian approach to inverse problems. The multi- a single broad umbrella, and to look for common level Monte Carlo (MLMC) method of Giles (Ox- themes across different application areas, as evi- ford) has provided a step-change in the computa- denced, for example, by the successful SIAM jour- tional efficiency of certain calculations required in nal Multiscale Modeling and Simulation introduced financial mathematics, and the basic idea appears in 2003. This has led to an interest in develop- to have far wider applicability: it is being extended ing and implementing numerical methods which by researchers in Germany, the UK and the USA account for, or exploit, these scales. The UK to other stochastic settings. At Strathclyde the re- has developed a range of expertise in this broad search groups of D. J. Higham and Mao are very ac- area, and was recently host to an LMS-EPSRC tive in the numerical approximation of SDEs, jump Durham Symposium in the field (led by Graham diffusions and the chemical master equation. There and Scheichl from Bath, together with Hou). Some is also significant effort in the numerical solution of of the research in this area has appeared under (primarily elliptic) PDEs with random coefficients, previous items; in particular some of the work of where the key issues are often related to linear alge- Markowich, Süli,Barrett, Graham, Stuart, Iserles bra and preconditioning (see the references to Gra- and Leimkuhler has ramifications in the multiscale ham, Powell, Scheichl and Silvester in the previ- context. Additional work not yet mentioned in- ous item). Tretyakov (Leicester) is also very ac- cludes the group of Chandler-Wilde in Reading, tive in the numerical solution of SDEs and (with working on high-frequency wave-scattering (and Milstein) has written an important research mono- including T. Betcke and Langdon) and that of graph in this area. Others active in this broad area Ainsworth (Strathclyde) who has worked on dis- include Lakkis (Sussex), Lord (Heriot-Watt), Lythe persion relations in (high-frequency) wave propaga- (Leeds), Shardlow (Manchester) and Voss (Leeds). tion, and on the adaptive multi-scale modelling of The field is thus healthy and growing in tandem masonry structures. There is also research activity with broader trends in modelling. in the use of lattice Boltzmann models of fluids at Geometric Integration and ODEs Geomet- Oxford (Dellar) and Leicester (Gorban, Levesley). ric integration remains an area of strength within The community of researchers engaged in multi-

36 scale computation has thus emerged naturally from (with Brunner, Newfoundland). Iserles (with Brun- within existing research groups present in the UK. ner and Nørsett, Trondheim) has developed meth- In particular the EPSRC funded network “Compu- ods for the computation of spectra of non-normal tation and Numerical Analysis for Multiscale and highly oscillatory integral operators. Multiphysics Modelling” played an important role 3.2 Approximation Theory in establishing this community. Inverse Problems On the applications side, For the past 20 years there has been a significant ac- interest in inverse problems is a significant growth- tivity in UK approximation theory related to kernel area, worldwide. This is driven by many factors based approximation methods in Euclidean space including medical and military applications, design and on compact manifolds. Starting with Pow- and optimization, and the integration of large data ell (Cambridge) and Light (Leicester) in the early sets with mathematical models. Computational 1990s, this has been continued by Baxter and Hub- challenges within the field are significant. The bert (Birkbeck), Levesley (Leicester), Wendland British Inverse Problems Society provides a coher- (Oxford) and Hesse (Sussex). These researchers ence to the community as a whole and has played are well-integrated with the international approxi- a role in a number of international meetings held mation theory community. More recently this has within the UK over the last decade. The inverse become more application based, with developments problems group of Lionheart and Dorn at Manch- in practical algorithms for solving partial differ- ester plays a significant role within the UK and the ential equations, and approximating difficult data innovative work of Dorn, using level set methods sets. Davydov (Strathclyde) has worked in such for inverse imaging problems, has strong links to kernel methods, but his work is more directed to- research in computational nonlinear PDEs. Lesnic wards polynomial approximations such as are useful (Leeds) applies various computational techniques, for the solution of PDEs. Goodman (Dundee) has including meshless and boundary element methods. also worked in multivariate approximation theory, The data assimilation group at Reading, led by in box splines, multiresolution and wavelet theory. Nichols, is involved in a range of geophysical inverse A notable UK development on the compu- problems (mainly arising from weather forecasting) tational side of approximation theory has been and drawing on a range of techniques from numer- the growth of the Chebfun software system by ical linear algebra and control theory. The work Trefethen (Oxford) and collaborators since 2002. of Stuart (see item on Stochastic Computation) is Chebfun exploits a mix of old and new algorithms also directed at computational aspects of inverse of polynomial and rational interpolation and ap- problems, and in particular at using the Bayesian proximation and quadrature to produce a system approach to quantify uncertainty. Others working that feels like MATLAB but computes with func- in inverse problems include Arridge (UCL), Chen tions instead of numbers or vectors, including the (Liverpool), Freitag (Bath), Johansson (Birming- solution of differential equations. ham) and Potthast (Reading). The community en- Recently, the UK approximation theory and gaged in the numerical analysis of inverse problems mathematical signal processing community has in- is thus a rather small one when compared to the vested in “sparsity based” research, as part of the fairly broad research goals represented and com- attempt to tackle the “data-deluge” of informa- pared with international competitors. tion. The EPSRC Network on multiScale Infor- Integral Equations Research in this area is mation RePresenation and Estimation (INSPIRE) well-represented in the UK, especially in relation has brought together members from the mathemat- to previously mentioned work on multiscale and ics, computer science, statistics, and electrical en- inverse problems, much of this activity developing gineering communities. Active in this area from from a 5-month programme at the Newton Institute the numerical analysis community within the UK in 2007 on Highly Oscillatory Problems (organised are Cartis and Tanner (Edinburgh), who (along by Engquist, Fokas, Hairer and Iserles). In addi- with M. Davies) have founded the Edinburgh Com- tion to the aforementioned work of Betcke, Chen, pressed Sensing Group http://ecos.maths.ed.ac.uk. Chandler-Wilde, Langdon, Lesnic, Johansson and This group is composed of eleven active researchers, Potthast, there is significant research in high fre- the largest in the UK in this area and one of the quency wave propagation via the Bath group of largest in the EU. It is part of an EU FP7 FET- Graham and Smyshlyaev (now at UCL) and on Open program “Sparse Models, Algorithms and integral equation methods for the wave equation Learning for Large-Scale Data” which has partners and time-dependent Maxwell equations by Davies at Queen Mary London, INRIA, EPFL, and Tech- (Strathclyde) and Duncan (Heriot-Watt). This lat- nion. The research in this area is inherently inter work has motivated other work on the numeri- terdisciplinary and partially addresses the concern cal analysis of first kind Volterra integral equations stated in IRM2004 about insufficient connection be-

37 tween the numerical analysis and computer science mathematics and Network Science: for example, by “algorithms” communities. Estrada and D. J. Higham (Strathclyde), Spence, There is also excellent activity in a number of and Grindrod (Reading). There is an excellent op- other areas, for instance Sabin (Cambridge) and portunity for the UK to build on existing compu- Goodman (Dundee) in computer aided geometric tational strengths and compete with a highly ac- design, Iserles (Cambridge) in approximation on tive and data-driven international movement that Lie groups, and Shadrin (Cambridge), famous for is currently dealing with issues of high-dimensional, the solution of the de Boor Conjecture, in univari- heterogeneous and time-dependent network data. ate spline theory. An EPSRC Matrix and Operator Pencil Net- 3.3 Numerical Linear Algebra work (2009–2011) led by Marletta (Cardiff) and Levitin (Reading) is enhancing links between lin- The UK continues to have a small but world-leading ear algebra, operator theory and engineering, and effort in this area. The Manchester group of N. J. includes among its members a number of strong re- Higham and Tisseur has made significant advances cent UK appointments in spectral theory. in nonlinear eigenvalue problems, especially in an- Although relatively small, the numerical linear alyzing conditioning and stability of linearizations algebra community provides international leader- of polynomial eigenvalue problems. Tisseur has de- ship across the spectrum of theory, algorithms and rived tools and theory for eigenvalue and mapping software. problems with Lie and Jordan algebra structure. 3.4 Optimization Higham’s work on theory and algorithms for matrix functions has had strong influence on software Since IRM2004, UK research in optimization has (e.g., in MATLAB) and has led to him writing the continued to flourish. Three leading universi- first research monograph on this topic. Dongarra ties (Birmingham, Edinburgh and Southampton) (Manchester, part-time since 2007) has made ma- now have sizable optimization groups, while in- jor contributions to parallel numerical algorithms teractions between individuals scattered elsewhere in linear algebra and to open source software pack- have grown. Most significantly, cross-area collab- ages and systems. oration between optimization, linear algebra and At the Rutherford Appleton Laboratory, Ari- differential equations, particularly in the area of oli, Duff, Hogg, Reid, Scott and Thorne work on simulation-based (or PDE-constrained) optimiza- both direct and iterative methods (and are now tion, is commonplace—three recent international looking at hybrid schemes) for large, sparse linear workshops (Durham, Oxford and Sussex) attest to systems, developing theory as well as production- this. There is also now an established biennial quality software made available through the HSL series of Birmingham conferences on optimization Library. Because this group is outside the Univer- and linear algebra organised by the IMA. sity sector there is a consequent lack of graduate Current research in the UK has focused on the- throughput. Research on preconditioning for iter- oretical and practical investigations into interior- ative methods is represented at RAL and in sev- point methods for linear, semi-definite, conic, con- eral universities (see “Solution of Linear Systems vex and nonconvex problems (Birmingham, Cam- Arising in PDEs” above). Spence (Bath) has ob- bridge, Edinburgh, Oxford, Southampton, RAL), tained results on iterative methods for large-scale the solution of (large) problems for which some or eigenvalue problems arising from discretizations of all unknowns are required to take integer values PDEs, notably having introduced (with Freitag) (Birmingham, Dundee, Edinburgh, RAL), meth- “tuned preconditioners”. ods for stochastic optimization (Edinburgh, Ox- Since completion of his monograph on pseu- ford, Southampton) problems involving “comple- dospectra (2005), Trefethen (Oxford) has devel- mentarity” constraints (Birmingham, Cambridge, oped methods for approximating eA and f(A)b Dundee), structural optimization (Birmingham, by contour integration, exploiting connections with Southampton), second-derivative SQP methods approximation theory. (Dundee, RAL), the efficient approximate solution The emerging theme of Network Science offers of (NP) hard combinatorial problems (Edinburgh, a common framework in which to model, analyse Imperial), multiobjective optimization (Birming- and summarize data sets from a diverse range of ar- ham, Southampton), support vector machines, eas in science, technology and even crime. Graph (Dundee, Edinburgh), the complexity of noncon- theory and linear algebra come together to offer vex optimization (Edinburgh, RAL), derivative-free tools for analyzing the type of sparse, unstructured optimization (Cambridge, Southampton), steepest but non-random connectivity patterns that arise. descent methods (Dundee), financial optimization Currently there is a limited amount of UK activity (Imperial, Oxford), and algorithms for infinite- at the intersection between applied/computational dimensional problems for which (some of) the con-

38 straints are differential equations or variational in- Other key meetings include: LMS Durham sym- equalities (Birmingham, Cambridge, Dundee, Im- posia on Numerical Analysis of Multiscale Prob- perial, Oxford, RAL, Sussex, Warwick). Upcoming lems (2010) and Computational Linear Algebra areas such as optimization over semi-algebraic sets for Partial Differential Equations (2008); EPSRC- and robust optimization are less well represented, Warwick Symposium Challenges in Scientific Com- but work is underway here too. puting in 2008–2009 (seven workshops and one con- A continuing trend in the UK is to match ference); and IMA Conference on Numerical Lin- theoretical developments in optimization with the ear Algebra and Optimization (2007 and 2010). design and implementation of software. Gener- The Newton Institute has also supported a num- ically this is available in the HSL and NAG ber of numerical-analysis related themes, including libraries, and in general-purpose packages such a 5 month program on highly oscillatory problems as FilterSQP, HOPDM, LANCELOT/GALAHAD in 2007. The EPSRC funded network on Com- and PENOPT. Many of the key developments– putation and Numerical Analysis for Multiscale the DFP and BFGS secant updates, augmented and Multiphysics Modelling (see above) ran sev- Lagrangian and SQP algorithms, and the non- eral workshops and seeded collaborations. There linear conjugate-gradient method—occurred either have also been several relevant LMS-EPSRC in- entirely or partly in the UK and as a direct structional courses aimed at graduate students, in- consequence of the software needs of both com- cluding a long running series in Numerical Anal- mercial and academic sectors. More recent key ysis (now Computational Mathematics and Scien- research—the trust-region and cubic regularisation tific Computing, most recently in Durham, 2010) paradigms and their extensions, methods for large- and, in 2007, a stand-alone short course on Mul- scale linear, quadratic and conic programming, tiscale Methods. Two recently established EP- large-scale interior-point and SQP methods, and SRC Centres for Doctoral Training include signifi- the filter and funnel approaches—has continued cant components in computational mathematics (at this trend. Current computing developments are Cambridge and Warwick) and the EPSRC-funded having profound implications for optimization, and Taught Course Centres have also involved many UK researchers are evaluating and exploiting both courses in numerical analysis. massively-parallel HPC and more modest multicore Individual Items of Esteem Higham and Tre- and GPU systems. fethen were elected FRS in 2007 and 2005, respectively, and Trefethen was elected to the US 4 Strengths National Academy of Engineering in 2007. Don- Editorial and professional service UK numerical garra, Duff, Hammarling, Higham and Trefethen analysis punches well above its weight in this re- are ISI Highly Cited Researchers. In international gard. 11 UK numerical analysts number among prizes the 2005 SIAM Dahlquist Prize was awarded the boards of the SIAM journals SIMAX, SINUM, to D. J. Higham, the 2007 SIAM J. D. Crawford SIOPT, SIREV and SISC. The UK has major in- Prize to Stuart and the 2010 SIAM Richard C. volvement in SIAM through (in 2010) Duff (SIAM DiPrima Prize to MacDonald (Oxford), and Elliott Chairman of Board), Higham (SIAM Vice Pres- and Markowich were awarded a Humboldt Prize in ident at Large) and Trefethen (SIAM President- 2009–2010. Duff, Gould, Fletcher, D. J. Higham, Elect), and new SIAM Student Chapters in Edin- N. J. Higham, Stuart and Trefethen were elected burgh, Manchester and Oxford, while the SIAM SIAM Fellows. Recent UK awards are the LMS De UKIE Section remains very active. Editor-in- Morgan Medal to Morton (2010), LMS Whitehead Chief positions are held by Iserles (Found. Comput. Prizes to Ainsworth (2004) and Tisseur (2010), the Math., Acta Numerica), Gould (SIAM J. Opt.), LMS Fröhlich Prize to N. J. Higham (2008), and Ralph (Math. Prog. Series B), and N. J. Higham a Philip Leverhulme Prize to Tanner (2008). N. (SIAM Fundamentals of Algorithms book series). J. Higham held a Royal Society Wolfson Research The UK-based IMA Journal of Numerical Analysis Merit Award, 2003–2008. Süliwas elected Foreign (est. 1981, with current EiCs Iserles and Süli)con- Member of the Serbian Academy of Sciences and tinues to thrive, with a 2009 Impact Factor placing Arts (2009) and to a Fellowship of the European it in the top 10% of Applied Mathematics journals. Academy of Sciences (2010). Stuart and Süliare co-editors of the Oxford Univer- Research Funding EPSRC’s Grants on the Web sity Press Monograph Series in Numerical Mathe- lists 96 grants totalling £54 million as of 31-8- matics and Scientific Computation. 10 with “Numerical Analysis” as “Research Topic” Conferences, Networks and Summer Schools (though, like the totals for most subareas of math- The highly successful Biennial Dundee meetings in ematics, this includes approximately £17 million of Numerical Analysis have been taken over by Strath- generic postgraduate training grants and the block clyde and the Leslie Fox Prize continues biennially. grants for the Isaac Newton Institute and the Inter-

39 national Centre for Mathematical Sciences). This interfaces between different parts of numerical anal- total exceeds those for most other areas of the ysis. Research in PDE analysis has grown af- mathematical sciences, but it does include a num- ter the last IRM through a number of EPSRC- ber of grants held by researchers outside the nu- funded initiatives and developing this further to merical analysis community. integrate with, and build upon, the potential in Notable recent major individual grants include the Numerical Analysis PDE community is a key an EPSRC Career Acceleration Fellowship for T. to competing internationally. The interface be- Betcke (2009) and an EPSRC Leadership Acceler- tween computational PDEs and optimization is a ation Fellowship for Tisseur (2010), both of 5 years rich research area, worldwide; both communities duration. Stuart was the first UK Mathematician are strong within the UK but links between them to be awarded a European Research Council Ad- are at an early stage. vanced Investigator Award (2008–2013). Research at the interface of statistics and nu- IRM2004 noted the need for increased support merical analysis, including uncertainty quantifica- for numerical analysis (“to remain at the level tion, is an area of growing importance worldwide presently achieved in Numerical Analysis much where the UK has world-leading strengths which more will be required”—page 27). The EPSRC could be further built upon, both on the theoreti- initiatives that have impacted on the area are all cal side (for example the work of Tanner at the indirectly related to high performance computing. terface of approximation theory and statistics, and The 2008 Science and Innovation call had as one the integrated research groups of Roberts and Stu- of the 5 solicited areas “Numerical Analysis and art at Warwick) and in applications such as data High Performance Computing: Software Develop- assimilation in the geophysical sciences, where the ment”; of two proposals in this area that reached recently (NERC-funded) National Centre for Earth the full proposal stage, one was funded: “Numerical Observation provides a significant driver, as do the Algorithms and Intelligent Software for the Evolv- interests of oil companies. ing HPC Platform” involving Edinburgh, Heriot- The UK has been very successful in building Watt and Strathclyde (£4.5M). An EPSRC sandpit library software on top of research in numerical lin- “Extreme Computing” in January 2010 led to two ear algebra and optimization over several decades NA/HPC projects being funded at around £1.5M (NAG, LAPACK, HSL, LANCELOT/GALAHAD in total. etc.). There is also new UK expertise in finite el- 5 Weaknesses, Threats and Oppor- ement software (DUNE, via Dedner at Warwick) and further opportunities exists to build further on tunities the substantial research output in differential equa- Whilst UK research in numerical analysis is strong, tions. it does not play the central role in applied mathe- The numerical analysis group at RAL is funded matics that it does in major research competitors. directly by EPSRC (from the Mathematical Sci- A strategy aimed at ensuring that the best numeri- ences and other programmes) and has recently re- cal analysis informs large-scale scientific computing ceived a further 4 years funding from September and mathematical modelling will have long-term 2011. The long-term future of the group, and of benefits to the science and engineering communi- development of the HSL Library is, however, un- ties as a whole. certain. To realize these benefits a computational in- Support for MSc courses is no longer available frastructure is needed that allows for the use of from EPSRC. Yet MSc courses in Numerical Anal- state-of-the-art computer systems, including visu- ysis (and more generally Applied Mathematics) alization, and training of researchers at this inter- have been of significant benefit to the health of the face. While many universities do attempt to pro- subject, PhD recruitment, industry and commerce. vide computational resources and training there is MSc funding needs to be considered within the no national strategy or funding mechanism in this contexts of the Bologna framework for European area. Individuals have to fight within their institu- higher education and EPSRC support for Doctoral tions to maintain minimal support. Funding to sup- Training Centres. port software development has always been prob- Finally, the age distribution of the community lematic. The 2006, 2008 and 2010 EPSRC HPC is healthy, thanks to a steady stream of lecture- Software Development calls, are welcome excep- ship appointments over the last decade. However, tions, but these are open to applications from all with future new academic appointments likely to areas and the majority of proposals do not involve be increasingly strategic it will be important for a significant numerical analysis component. numerical analysts to exploit fully links with other A significant opportunity exists to grow inter- disciplines and with other areas of applied mathe- faces with other areas of mathematics, and to grow matics.

40 INTERNATIONAL REVIEW OF MATHEMATICS STATISTICS LANDSCAPE

Lead Author: Mike Titterington Contributors: David Firth, Peter Green, Byron Morgan, Chris Skinner, Andy Wood

1. Statistical Overview:

The strengths of UK statistical research are in the development of methodology and computational tools for handling topical applied-field problems. The resulting methods have often turned out to be world-leading and generic even if the original applied problem had appeared specialised. Some of the applied areas, such as genomics and bioinformatics, climate research, ecology and environmental science, are of very high public interest and the UK is prominent in all of these. Many of the recent methodological and computational advances follow the Bayesian approach and again the UK is at the forefront. More traditional areas are still active, such as aspects of medical statistics, design of experiments, survey sampling and non-Bayesian methodology. The level of interest in the UK in the most theoretical parts of mathematical statistics is comparatively low, again a traditional state of affairs. The RAE 2008 returns for the most relevant Unit of Assessment (UoA22), which also includes operational research and much of probability, show that about 265 FTE statisticians were submitted but it must be borne in mind that a significant number of statistical researchers were submitted under other UoAs. The RAE 2008 UoA22 Subject Overview Report noted that about 25% of those submitted were 'Early Career Researchers', having been in-post for at most 4 years. Information gathered by the Committee of Professors of Statistics (COPS) suggests that the number of academic staff (including those not submitted to the RAE and likely to include probabilists) is currently about 500. COPS 2010 percentages for the age ranges ( <30, 30-39, 40-49, 50-59, 60+) are (3.4, 31.1, 27.4, 24.4, 13.7), corresponding to about 60 individuals aged 60+. The RAE 2008 returns for UoA22 reported about 450 research student FTEs in 2006-7, representing a student/staff ratio greater than 1 for UoA22. Further details and trends are discussed in Section 4.

2. Subject breakdown: The table indicates the number of active researchers in designated sub-areas Research Area Number of % of Total in Researchers Statistics

Bayesian statistics 45 16 Further generic 90 32 methodology Social Statistics 30+ 12 Design/Clinical trials 30+ 12 Biomedicine, etc. 80 28 Source of data : RAE 2008 data for UoA22 supplemented by rough lower bounds for some research areas gathered from university webpages and data for other UoAs.

3. Discussion of research areas:

BAYESIAN STATISTICS

Generic Bayesian methodology : The UK is strong in research in Bayesian methodology, particularly for generic models not tied to specific areas of application.

41 The specific topics we choose to highlight here are inference for stochastic processes and the analysis of mixture-like models.

For stochastic processes Roberts (Warwick) leads an important group, especially on inference for partially-observed diffusions, time series, filtering and control; Blackwell's work in Sheffield is motivated by problems in ecology and environmental science. There is wide interest in mixture modelling, latent variables, hidden Markov models and related models, including the work of Holmes (Oxford), Hurn (Bath), Wilkinson (Newcastle), Godsill (Cambridge), Green (Bristol), Nobile (Glasgow) and Richardson (Imperial). Titterington (Glasgow) has studied variational Bayes methods, particularly for mixture and related models. This work intersects with much recent Bayesian activity, mainly in computer-science departments, under the heading of 'machine learning'. Another leading figure is Brown (Kent), on multivariate analysis.

Bayesian theory : As for some other areas of statistics, underpinning rigorous theory and foundational questions for Bayesian statistics are less intensively pursued in the UK than in some other countries, but there are some notable exceptions. Currently important topics include Bayesian nonparametrics and the theory of prior distributions.

Cutting-edge theory for Bayesian nonparametric inference, including time series and regression models, is developed by Walker and Griffin (Kent) and Steel (Warwick). Sweeting's (UCL) forte is the relationship between Bayesian and frequentist inference and the effects of alternative prior specifications, and O'Hagan (Sheffield) and Garthwaite (Open) do leading work on the elicitation of expert prior knowledge.

Selected Bayesian applications : Bayesian methods are becoming ubiquitous in applied statistical methods in many domains; this work is omitted here where the main purpose and benefit of the research is to the substantive application domain. Here we highlight computer models and biostatistics; other important applications of the Bayesian approach, pursued in the UK, are mentioned elsewhere in the document.

Uncertainty in computer models, especially when used in geophysical systems, climate modelling, etc., are a major interest in the UK, involving O'Hagan and Oakley (Sheffield), M. Goldstein, Craig and Seheult (Durham), Rougier (Bristol) and Sahu (Southampton) as part of the activities of the EPSRC-supported Managing Uncertainty in Complex Models project. Bayesian methodological advances in medical statistics and biostatistics more generally include high-profile research by Spiegelhalter and Thompson (Cambridge), Matthews (Newcastle), Abrams (Leicester), Welton (Bristol), Grieve (KCL) and Best and Ashby (Imperial).

Bayesian computation : In the UK there are particular strengths in Monte Carlo methods, including MCMC and Sequential MC (particle filtering), with some interest in perfect simulation. Whether or not computational inference is the goal, this work is inherently statistical because of the stochastic nature of the algorithms.

The major research-leaders are Roberts (Warwick), Fearnhead (Lancaster) and Andrieu (Bristol): Roberts' group has pioneered exact simulation for diffusions, in particular exploiting retrospective sampling; Fearnhead has made major contributions to sequential Monte Carlo methods and other filtering tasks, and to statistical genetics; Andrieu works on adaptive Monte Carlo methods, on methodology combining MCMC and particle methods, and on advanced auxiliary variables methods. Other prominent figures include Green (Bristol), Nicholls (Oxford) and Kendall and Thonnes (Warwick), as well as Beaumont (Bristol) and Stumpf (Imperial) on Approximate Bayesian Computation, especially in systems biology.

42 FURTHER GENERIC METHODOLOGY

Parametric statistical theory : 'Classical' theoretical research on general methods of inference remains important and active worldwide, but the UK does not have large concentrations of work in this area. A few notable contributions are mentioned here.

Copas (Warwick), with Eguchi, has done important work on inference robust to model specification and sampling assumptions. Liu (Southampton) is an international authority on simultaneous inference, and Mardia and others at Leeds have contributed strongly to the emerging theory of inference based on 'composite' likelihood. Kosmidis (UCL) and Firth (Warwick) lead the way on inference via penalised likelihoods, notably for generalised nonlinear models. A. Wood (Nottingham) and Young (Imperial) do cutting-edge work on bootstrap methods and asymptotic approximations.

Time series, chaos and dynamical systems, wavelets, longitudinal data : These topics are crucial in signal processing and other time-related applications, and the following contributions are of a high international standard.

Yao (LSE), an ongoing collaborator of Jianqing Fan, is prominent in volatility function modelling and nonparametric function estimation. Other time-series research includes Lawrance's (Warwick) work on statistical aspects of chaos, Bhansali's (Liverpool) work on intermittency and long-range dependence and Cai's (Plymouth) work on forecasting. A notable example of time-series research in Econometrics is Shephard's (Oxford) collaboration with Ole Barndorff-Nielsen. Walden (Imperial), Olhede (UCL) and Fryzlewicz (LSE) work at the intersection of time series analysis, wavelets and signal processing. Nason (Bristol) and co-workers have led the way on the estimation of evolutionary wavelet spectra and on multiscale methods for graphs and irregular data structures in several dimensions. Recent and current research activity in longitudinal data analysis and related areas such as event-history analysis includes work of Diggle (Lancaster) and Pan (Manchester). Henderson (Newcastle) and colleagues recently published a well- received RSS discussion paper on drop-out in longitudinal data.

Functional data analysis, nonparametric function estimation, smoothing and resampling methods : These topics cover a range of activities of a nonparametric rather than parametric nature.

Innovative applications of functional data analysis include the work of Faraway (Bath) on human movement, Dryden (Nottingham) and Browne (Bristol) on modelling proteomic mass-spectometry data and Aston (Warwick) on the analysis of linguistic pitch. Brown (Kent) and colleagues have developed wavelet-based methods for various medical applications in which functional data arise. Samworth (Cambridge) co- authored a successful RSS read paper on shape-constrained density estimation. Zhang (Bath) works in a variety of topics in nonparametric function estimation including local polynomial modelling and varying coefficient models, S. Wood (Bath) is well known for computational algorithms for generalised additive models and penalised regression splines and Park (Lancaster) works on local additive models, nonparametric function estimation and nonparametric Bayesian inference. Computational aspects of smoothing are treated by Currie (Heriot-Watt) and Taylor (Leeds).

Multivariate analysis including graphical models, causality and classification : This is a sub-area in which internationally excellent contributions are being made both by very distinguished researchers and by outstanding younger people.

43 Lauritzen (Oxford) has recently studied graphical models with edge and vertex symmetries; Dawid (Cambridge) has worked on causality and influence diagrams; Cox (Oxford) contributes in a variety of areas including multivariate dependence structures and causality. Graphical models and causality also underlie the Bayesian research of Smith (Warwick) and Didelez (Bristol), the latter with medical applications in mind. A major recent focus, in the UK and elsewhere, and motivated by application areas such as bioinformatics, is on high-dimensional problems; there was a 2008 Newton Institute programme on this topic. Pleasingly, some of the most important contributions are being made by younger people: Tanner (Edinburgh) has collaborated with David Donoho on aspects of high-dimensional geometry, with applications to high- dimensional data analysis and signal processing; Meinshausen (Oxford) has worked on the lasso and false discovery rates in high-dimensional settings; Samworth (Cambridge) has collaborated with Peter Hall on high-dimensional classification and with Jianqing Fan on high-dimension feature selection. Further strong contributions are those of Steel (Warwick) and Jones (Open) on multivariate distribution theory, and of Krzanowski and Hand (Imperial) on clustering.

Spatial statistics, image analysis and shape analysis : In all these topics the UK makes world-leading contributions.

Diggle (Lancaster) and co-workers have made important advances in the analysis of spatial and spatial-temporal modelling, including the modelling of point processes, focusing mainly on medical applications. In image analysis, Aston (Warwick) works on MR and PET imaging, and Kent and Mardia (Leeds) have developed intrinsic random processes to model deformations of images. The Leeds group are also prominent in the statistical analysis of shape, another major centre being at Nottingham (Dryden, Le, Preston, A. Wood); recent activity includes new work on projective shape analysis and modelling shape change by fitting smoothing splines in shape space. Other major contributors include Bowman (Glasgow), Kume (Kent) and Jupp (St. Andrews).

SOCIAL STATISTICS

Activity in this field has been stimulated since 2005 by ESRC funding of ‘nodes’ (at Bristol, Imperial, Lancaster, Warwick and the Institute of Education, London) of its National Centre for Research Methods. The particular work highlighted below does not include a large body of research in Econometrics. However, there has been increasing confluence between social statistics and microeconometric methodology, in which the UK has leading expertise, notably the Centre for Microdata Methods and Practice, led by Andrew Chesher at the Institute for Fiscal Studies/UCL.

The established focus on survey sampling at Southampton has been advanced by Berger, Pfeffermann and Skinner, stimulated by official statistics applications. Durrant, Shlomo and Brown (Southampton), Lynn (Essex) and Plewis (Manchester) have concentrated on problems with non-response, and statistical disclosure limitation work has been a focus at Southampton (Skinner, Shlomo, Forster) and Manchester (Elliot). Small-area methods are developed at Southampton (Pfeffermann, Tzavidis) and, with a Bayesian slant, at Imperial (Best, Richardson). The leading research of Deardon, Micklewright and McDonald (Institute of Education, London) combines administrative and survey sources, and Firth (Warwick) has developed a remarkably accurate method for predicting election outcomes using exit polls. Leading contributions have been made on multilevel modelling by Steele, Browne and H. Goldstein (Bristol), on missing data issues by Carpenter and Kenward of the London School of Hygiene (LSHTM), on social network analysis by Snijders (Oxford),

44 on categorical-data modelling by Bergsma and Kuha (LSE), Forster and Smith (Southampton) and Francis (Lancaster) and on latent variable modelling by Moustaki (LSE), Rabe-Hesketh (Institute of Education) and Pickles (Institute of Psychiatry).

DESIGN OF EXPERIMENTS AND CLINICAL TRIALS

These topics have a long tradition in the UK and continue to generate internationally recognised research.

For Design of Experiments there are concentrations of expertise at Queen Mary University of London (Bailey, Bogacka, Gilmour and others) and Southampton (Lewis, Woods and others) together with Wynn (LSE), Torsney (Glasgow) and links to industrial organisations such as GlaxoSmithKline. The 2011 Newton Institute research programme should provide a substantial boost to UK work in the topic. Clinical trials continues to be a fertile research area, both for basic methodology and for direct practical involvement. There are major centres at LSHTM (Pocock), Oxford (Altman, Darby, Peto), Imperial (Ashby) and Queen Mary (Cuzick). Important work is done by Jennison (Bath) on sequential trials, by C. Donnelly (Imperial) on controlled field trials, by Bolland and Todd (Reading), by Chetwynd and Whitehead (Lancaster) and on crossover trials by Senn (Glasgow). Large-scale practical involvement occurs at the Robertson Centre (Glasgow) and the MRC Clinical Trials Unit (London).

STATISTICS IN BIOMEDICINE, ECOLOGY AND THE ENVIRONMENT

Epidemic Modelling, Statistical Epidemiology and Survival Analysis : Interest in these topics is not widespread but there are pockets of undoubted excellence.

Epidemic modelling is prominent at Nottingham (Ball and O'Neill), with further contributions from C. Donnelly (Imperial), Cox (Oxford), Browne (Bristol) and Bailey (Exeter) on foot-and-mouth disease, from Lyne (Kent), from Farrington (Open) and from Clancy (Liverpool). Additional key contributors in statistical epidemiology include Diggle (Lancaster), Greenhalgh and Robertson (Strathclyde) and Best and Richardson (Imperial). Survival data analysis is less actively pursued than in the USA but important work is done by Kimber (Southampton), on additive hazards models by Gandy (Imperial) and on leukaemia survival by Henderson (Newcastle).

Statistical Genetics and Bioinformatics : Statistical genetics and bioinformatics are highly topical, internationally. They are also prominent in the UK, where world-leading contributions are being made.

The Wellcome Trust Centre for Human Genetics in Oxford is a major research base for statistical genetics, with leadership provided by P. Donnelly. Topics investigated there include genome-wide association, haplotype mapping and many others. Other leading figures include Balding (UCL), Tavare (Cambridge), Griffiths (Oxford), Macaulay (Glasgow) and Fearnhead (Lancaster). Statistical research in bioinformatics is widespread, two representative topics being alignment (Green (Bristol), Mardia (Leeds) and Hein (Oxford)) and microarray problems (Richardson (Imperial) and Gilks (Leeds)). Much of the underlying methodology is Bayesian. Systems biology is prominent in, for example, Newcastle (Wilkinson), Kent (Morgan and Ridout, on prion modelling in yeast cells), Nottingham (Ball on ion-channel modelling), UCL (Girolami), Bristol (Beaumont) and Imperial (Stumpf).

45 Climate research, extremes, ecology and environmental statistics : These topics are growing in importance in the world at large and commensurately in statistical research, including in the UK.

The UK's involvement in statistical aspects of climate research has grown out of past-and-current leading work on extremes, by Tawn (Lancaster), Anderson (Sheffield) and Walshaw (Newcastle), but new directions have been established by Rougier (Bristol), in developing efficient probabilistic predictions of regional climate. Ferro and Jolliffe (Exeter) are also making important contributions. Related to this is research on environmental statistics, particularly on rainfall and hydrology by Isham, Chandler and Northrop (UCL) and on pollution by Shaddick (Bath) and Scott and others (Glasgow). A current programme at the Newton Institute should lead to substantial advances. With its 10-year EPSRC/NERC support, the National Centre for Statistical Ecology has made a major impact. The Centre encompasses research from several universities and research institutions, including Kent (Morgan), St. Andrews (Buckland), Bath (S. Wood), Sheffield (Blackwell), Bristol (Browne) and Glasgow (Bowman). The many topics covered include estimation of biodiversity, effects of climate change, marine ecology, process-based ecological dynamic models and models for multi-species population dynamics. The Centre is co-directed by Buckland and Morgan. There is further important work, on radiocarbon dating by Scott (Glasgow) and Buck and O'Hagan (Sheffield), on the estimation of endangered species and cetacean density at the St. Andrews Centre for Environmental and Ecological Modelling (Buckland, Thomas, Borchers, King), and on the management of fish populations by Gettinby and Gurney (Strathclyde).

TOPICS REPRESENTING OPPORTUNITIES FOR MORE INTENSIVE UK INVOLVEMENT : UK statisticians enjoy high international profiles in all the above areas, but they do not cover the full gamut of statistical research of high topicality.

A bare list of some areas of strong current interest that represent opportunities for increased UK involvement includes machine learning, agent-based modelling in social science, citation analysis, theory for analysing high-dimensional data, astrostatistics and topological statistics.

4. Discussion of research community

Trends in staff numbers suggest a concentration of expertise, which has its advantages and disadvantages for the UK sector as a whole. In RAE 2008, UoA22 (Statistics and Operational Research) included 388.8 FTEs of Category A staff from 30 submissions as compared to 386.6 FTEs from 46 submissions in RAE 2001 and 417.7 FTEs from 55 submissions in RAE 1996. Most of the submissions omitted in 2008 were those in the lower 'tail' of the 2001 submissions and, in compensation, some of the retained submissions involved significantly more people than in 2001. The subject area of 'Statistics' (as opposed to Probability and Operational Research) contributed about two-thirds of the volume of material submitted in 2008 and a rough count of statisticians submitted to UoA22 came to 265 individuals, about 68% of the total. One factor behind the concentration of expertise, and a direct response to the findings of the previous Review, was the award of three Science and Innovation awards to Statistics, totalling over £10M. The awards enabled the creation of CRiSM (the Centre for Research in Statistical Methodology) at Warwick, of SuSTaIn (Statistics underpinning Science, Technology and Industry) at Bristol and of the Cambridge Statistical Initiative. All three centres of excellence have the aim of protecting and revitalising the UK's strength in research into statistical methodology motivated by currently challenging applications, and a number of capacity-building, permanent academic positions have been filled along with the provision of postdoctoral and PhD

46 opportunities. The Warwick and Bristol centres especially have organised many research workshops. However, in spite of these valuable initiatives, it is of concern that it is still often difficult to fill vacant posts, even chairs in strong departments, bearing in mind the substantial number of existing staff aged over 60; see Section 1. COPS data, likely to include probabilists, suggest the following trends, or lack of, over the last few years: the percentages of lecturers, senior lecturers/readers and professors have been fairly consistent at about 42%, 27% and 31%, respectively; about 22% of academic staff have been female; and the age distribution of academics has been fairly steady; see Section 1 for the current distribution. The RAE 2008 UoA22 Subject Overview Report notes that, between 2001 and 2007, research funding, especially from the various research councils, rose steeply, with annual research income in UoA22 increasing from about £3M to about £10M. In addition, the numbers of doctorates awarded and research students registered increased substantially, the latter by over 70%. The percentages of students funded by OST/OSI Research Councils, Other UK sources, Institutional/self-funded, Overseas sources and Other sources were, respectively, about 37%, 11%, 22%, 14% and 16%, a distribution that was fairly stable throughout the period. However, there is serious concern about trends since 2008, within the research councils, to reduce the level of commitment to the funding of postgraduate students, and to reduce the funds getting through to departments for responsive-mode projects. A feature of recent years has been the development of several courses for research students introduced to encourage development of the next generation. The Academy for PhD Training in Statistics (APTS) is a collaboration of nine UK research groups that organises four residential two-course weeks each year for first-year students. There are also statistics components in the material of the Scottish Mathematical Sciences Training Centre (a 'virtual' centre) and the London Taught Course Centre. A further substantial resource is the EPSRC-supported Statistics and OR Doctoral Training Centre at Lancaster. Statistics research has been very strongly represented in recent years at the Isaac Newton Institute in Cambridge. The Institute's schedule for 2006-2011 includes 7 research programmes in Statistics or with substantial statistical content out of a total of 31, whereas there were barely 2 out of 32 in the preceding 6-year period. Topics included Design of Experiments, Bayesian Nonparametrics, High-dimensional Data, Climate Modelling, Genome Resequencing and Biological Sciences more generally.

5. Cross-disciplinary/Outreach activities:

The pervasive influence of UK Statistics research in biological science, physical science, social science, medicine, engineering and many areas of everyday life is spectacularly illustrated in Section 3. Areas not mentioned there include finance and actuarial research (see another Landscape but including Hand (Imperial) and Rogers (Cambridge)) and forensics, spearheaded by Aitken (Edinburgh) and Lucy (Lancaster). Links with applied mathematics are increasing, MASDOC at Warwick and MMBNOTT at Nottingham being just two examples. Facts in the RAE 2008 UoA22 Subject Overview Report reinforce this outstanding picture: the journal articles submitted for assessment were published in 324 journals, of which nearly 200 came from beyond the mathematical sciences; advice about one or more outputs was sought from 13 other RAE sub-panels; research funding came from a wide range of research councils and charities, along with UK industry and commerce. In addition, as mentioned in Section 1, a substantial number of statisticians were included in the RAE submissions for other UoAs such as Pure and Applied Mathematics, Economics & Econometrics and Epidemiology & Public Health. Internationally excellent statistics research is also carried out at non-university, applications-oriented institutes such as the MRC Biostatistics Unit in Cambridge and Biomathematics and Statistics Scotland (BioSS).

47 INTERNATIONAL REVIEW OF MATHEMATICS 2010: PROBABILITY LANDSCAPE

1. Statistical Overview: The determination of the boundary between probabilist and non-probabilist is necessarily arbitrary, especially since sophisticated probabilistic methods are now pervasive and influential through much of science and engineering as well as substantial parts of pure mathematics. Probability is an intrinsically diverse subject with permeable borders. With this caveat, a count derived from institutional web pages (mathematics, statistics, and some CS departments) estimates size of the UK academic community of probabilists at 139 full-time tenured researchers. This represents a substantial increase on the numbers reported in the previous international review (approximately 100), and reflects strong numerical growth since then. In the last three years this community has started a new series of national meetings focussing on research topics of intense international interest; Warwick Spring 2008 (69 participants, international speakers on SLE, coagulation and fragmentation, random matrices, SPDE), Bath Autumn 2009 (100 participants, international speakers on random networks, random planar maps, random polymers, sharp mixing thresholds); further meetings are planned for Oxford (2011), Warwick (2012) and Cambridge (201x). The attendance figures correlate loosely with our estimate of community size.

2. Subject breakdown: The table indicates the number of active researchers in broad sub-areas; data extracted from departmental webpages as above; sub-areas based loosely on the Probability Landscape document for the previous IRM (note that actuarial mathematics and stochastic finance in general are re-located in the Financial Mathematics landscape document). Classification is highly subjective, and takes no account of researchers working in more than one sub-area. However it does reflect the strong emphasis placed on stochastic analysis by UK probabilists. Research Sub-area Number of Researchers % of Total in Probability Stochastic analysis 78 56% Spatial probability 12 9% Discrete probability 17 12% Applied probability 32 23%

3. Discussion of research areas: We now survey UK research highlights in the period since the last International Review. The account is necessarily compressed, as the subject of probability is broad, fertile and very varied. A count for 3 top international probability journals (Annals of Probability, Annals of Applied Probability, Probability Theory and Related Fields) shows UK involvement in published papers at a consistent 10% with considerable annual fluctuation, frequently in collaboration with world experts from outside of the UK. UK population is about 5% of total for the comparable group of US, Europe, Japan; so this is broadly satisfactory, and the following summaries confirm a good spread of strong work with substantial international cooperation at high levels of excellence. However there are significant areas of concern, mentioned in passing below. Stochastic analysis: This traditionally means mathematical investigations of continuous-time stochastic systems that are primarily concerned with developing and refining analytical tools and insights, though it is now key to almost all probability, both applied and pure. It is a historical UK strength, and now thrives in a context of intense international activity; new tools are maturing and finding economically important applications, while interactions with numerical analysis are of growing significance. Much of its vitality derives from a characteristic combination of mathematical rigour and specific concrete examples. At the analytical end of stochastic analysis, Wang and co-workers have made strong contributions to functional inequalities closely linked to properties of random processes [1]. Path-space analysis is of ongoing international interest and [2] develops differential form theory for path space in case of curvature of the underlying manifold.. Developments in the now-classical topic of Lévy and Lévy-type processes have focussed on theory required by new applications, either directly in applied probability, or arising from more complex stochastic models within which Lévy processes are naturally embedded. Active UK research groups centre around Bath, Manchester and

48 Swansea; highlights include introduction of the so-called scale function as an analytical tool [3], the introduction of the quintuple law [4], and an authoritative study of small-time properties [5]. [6] introduces a topological group context. Lévy process theory is now a major computational tool in applied probability and mathematical physics [7], and the UK contribution has been central to this. Coagulation and Fragmentation lies at the intersection of stochastic analysis, discrete probability, and applications such as genetics, being the study of structures exhibiting random division and merging. [8] is a significant review of the state of the art of this rapidly developing subject. [9] presents a striking extension of Aldous' scaling limit for the Erdös-Rényi random graph. [10] is a tour-de-force resolving a natural technical problem which had previously appeared out of reach. [11] discusses fragmenting Brownian flows. Rough Path theory originated in the UK with work of Lyons [12]; this provides a powerful tool for extending stochastic analysis methods to non-Markovian and even deterministic contexts [13,14], as well as generating significant computational and numerical methods in a range of applications. High-level contributions while working in the UK have been made by: Cass, Caruana, Friz, Gyurko, Hambly, Litterer, Lyons, Victoir, Qian; there is also an outstanding recent contribution from Hairer concerning deterministic nonlinear vector PDE. Contributions to Stochastic partial differential equations (SPDE) include rates of convergence for discrete approximations [15], ergodicity properties for 2-D stochastic Navier-Stokes equations [16], development of stable manifold theory for SPDE [17], and foundations of stochastic hyperbolic PDE [18]. A recent programme of the Isaac Newton institute (www.newton.ac.uk/programmes/SPD/) was devoted to this fundamental topic, ranging from challenging issues of well-posedness through to many applications. The Filtering problem (estimation of the unknown present state of a dynamic system based on noisy observations) concerns a specific semi-linear SPDE of great importance, and UK workers have contributed both to theoretical and numerical analysis: examples include robust representation of the fundamental filtering solution [19], a new particle method [20], and exact sampling methods [21]. [22] addresses issues of incorrect initial data for a discrete-time filtering problem. While Mathematical Finance forms a landscape document of its own, there is a substantial overlap with probability and stochastic analysis in particular, in questions of stochastic control and optimal stopping, and in careful applications of stochastic calculus. As instances of this work we cite the discovery of a remarkable stopping region for a Brownian optimal stopping problem [23], a surprising use of strict local martingales to model bubbles in finance [24], and a penetrating discussion of the use of convex cone theory in dealing with transaction costs [25]. There is a compelling need to increase the supply of highly trained probabilists so as to raise the level of academic insight and analysis in this area, which is of great economic importance to the UK. The UK has historical strengths in applications of probability to Mathematical Physics, which in return have frequently led to significant developments in stochastic analysis itself. Recent highlights from the UK include applications of Nelson's stochastic mechanics to derive Kepler's laws [26], work on Anderson models [27,28], contributions to Glauber dynamics for particle systems [29], and the first mathematically rigorous result relating Yang-Mills measures to the Yang-Mills energy [30]. In summary, stochastic analysis is an area of strenuous international activity. The UK is well-connected to many of its recent developments with a leading position in some; however there is little or no UK presence in the topics of SLE, random planar maps, and Gaussian free fields. These latter topics are of intense international interest (witness recent Fields medals) and the lack of UK involvement is a matter of concern. Spatial probability: The treatment of probabilistic structure extended over space brings challenges related to limitations of the spatial Markov property and resulting phase transition phenomena; thus there are strong interactions with mathematical physics in the form of statistical mechanics. Percolation has been a UK strength over the medium term; recent successes include proofs of exact values for critical points for 2-D percolation on Voronoi and other tessellations (eg [31]). Grimmett produced a magisterial exposition of the important related topic of random cluster models [32], while [33,34,35] analyze diffusion on random structures. Stochastic Geometry considers the theory of random patterns, and is a topic which partly originated in the UK. Recent UK activity centres around geometric networks: [36] uses Poisson line processes to derive surprising and significant results concerning statistical measures of efficiency for planar networks, while [37] demonstrates unexpected distributional limits for the length of random on-line and directed neighbour graphs. A central limit theorem for Gaussian polytopes [38] is a key contribution to a line of investigation following the classic Rényi- Sulanke work of the 1960s, which is now developing rapidly using new analytical tools.

49 The theory of random matrices, and its links to interacting particle systems and growth models, has grown rapidly in recent years. The most fundamental progress has been made using techniques of asymptotic analysis allied to powerful combinatorial tools, rather than via probability. However the associated topics of random matrix processes, non-colliding processes, and intertwining have been the subject of sustained work by UK probabilists in collaboration with others, for example [39], [40]. Other important probabilistic work in this topic concerns relations with growth models [41] and combinatorial probability related to random Young diagrams [42]; here there is a link to the next area. Discrete probability: Interest continues in the conundrum of how to control rate of convergence to equilibrium for the kind of finite (but large) Markov chain Monte Carlo problems met in practice: general situations are known to lead to issues in algorithmic complexity, so the puzzle is to provide useful characterizations of amenable situations. [43] produces a treatment of graph colouring which covers updates controlled by systematic rather than random scan. [44] weakens the notion of geometric ergodicity to that of “variance bounding”, with useful links to ordering properties. Older work on optimal scaling is now being extended to infinite-dimensional situations with an eye to a wide range of applications [45]. Spin glasses and combinatorial optimization is a topic of intense recent activity, spanning probability, statistical physics and theoretical computer science. UK presence is limited here, but we mention work on spectral heuristics for random graphs [46]. A new approach to threshold behaviour for the classic Random Graph is provided by [47]. Normal and other probability approximations using Stein's method have been developed significantly in recent years, with applications, for example, to the Mann-Wilcoxon-Whitney statistic [48]. Applied probability: Increasingly all parts of probability are being driven by real and challenging applications. However in this section we survey UK activity in some strongly established applications. Genetics requires sophisticated probability to meet the demands of previously unimaginable amounts of data derived from advances in DNA sequencing, often involving high-dimensional spaces of trees or coalescent phenomena. Probability theory feeds this not only directly through mathematical developments but also by providing highly trained researchers who work directly with data. [49], motivated by modelling of evolution in a spatial continuum and resolving “the pain in the torus” (Felsenstein's famous phrase), introduced a new class of measure-valued processes whose rich mathematical structure is now being uncovered. In contrast, following studies of genome rearrangement, [50] discovered surprising results about random transposition random walk. The topic of Stochastic networks and queues can be characterized as the stochastic modelling of resource- constrained systems. In the current period UK research has been strongly driven by applications to varieties of communications networks and this has formed a recent semester at the Isaac Newton Institute (www.newton.ac.uk/programmes/SCS/). Examples of UK work include: analytical simplifications arising under “fair” resource-sharing policy for pricing and congestion control [51,52], large-deviations and the “single big jump principle” [53], and use of random graphs to model topological structure of networks [54]. [55] makes an interesting connection to random matrix theory. To set against this, it is a concern that UK research lags significantly behind that of the US in the strategic area of modelling of energy networks. UK contributions to probabilistic aspects of Epidemics range from control strategies for vaccination [56], approximations related to the requirements of statistical likelihood theory [57], consideration of susceptibility processes for model networks with both local and global connections [58] [59], right through to specific applications to head-lice [60].

4. Discussion of research community We have already mentioned the new series of national meetings, building on a earlier tradition of national meetings and vital for building a coherent community of probabilists which aims to perform at the highest international standards. The strategic importance of this kind of meeting includes: strong opportunities to bring in international experts to promote international engagement of UK community, promotion collaboration and coordination of research activity within UK, enabling research strengths to be disseminated across UK rather than confined to a few centres. About 40% of the community of UK probabilists originate from the UK; age distribution is roughly 22% young, 27% 10-20 years in post, 52% senior (not including about 10 retired but very influential and active researchers). These informal estimates conceal a deficit of working UK probabilists in the last 10 years of career. The community has seen success in attracting support at a more junior level (5-year EPSRC research fellows);

50 however this deficit suggests the community should seek to attract some world-class research leaders to the UK, particularly as many of the very strongest UK probabilists work abroad (eg: Aldous, Barlow). We estimate research students in probability to number around 36 UK and 100 non-UK (crude and approximate figures derived from inspection of department websites). The low number of UK PhD students is a matter of concern, bearing in mind the strong demand for all kinds of probability PhDs (for example from finance); in particular this is a limiting factor in capitalizing on the many opportunities for applications which require real depth of probabilistic training. For this reason it is vital that probability is strongly represented at centres producing substantial numbers of graduates in the mathematical sciences. Influenced by recommendations of the previous International Review of Mathematics concerning the need for greater breadth amongst UK Mathematics PhDs, the community has put substantial effort into developing PhD training weeks, often based around probability research centres (P@W at Warwick, Prob-L@B at Bath, the Swansea node of WIMCS) and complementary to the series of national meetings. These training weeks have made it possible for UK Probability PhD students (and post-docs) to attend courses given by major figures on the world probability scene (Aldous, Kurtz, Stroock, Fukushima, Bramson, …). Two new Centres for Doctoral Training commencing operation this Autumn have a strong probability component (MASDOC at Warwick, CCA at Cambridge); the Oxford Man Institute has also funded a number of outstanding international graduate students in stochastic analysis.

5. Cross-disciplinary/Outreach activities: The new Centres for Doctoral Training, CCA and MASDOC mentioned above should lead to intensified contacts between probabilists on the one hand, and on the other respectively mathematical analysts and applied mathematicians. Additionally HEFCW have funded WIMCS for Welsh mathematics, and the Swansea node in particular represents significant interaction between probability and the rest of mathematics. A major example of interdisciplinary activity is to be found in genetics. For instance, Peter Donnelly (Oxford), now director of the Wellcome Trust Centre for Human Genetics, started as a probabilist and has written seminal papers on measure-valued diffusions. Crucially, the community has been highly successful in maintaining close links with different parts of the biology community, being an integral part of the International HAPMAP consortium (a successor to the human genome project). Not only does the desire to understand biology lead to rich classes of biological models, but mathematics directly influences our understanging of biology. For example [61] identified a motif in DNA sequences associated with recombination hotspots, leading to the discovery of a gene responsible for recombination and ensuing experimental research. Although UK research in this topic is at the leading edge, it depends on a few individuals, largely based in Oxford. Continuing advances in genome sequencing will further intensify the need for probabilistic input; there is a pressing need for highly trained probabilists, resourced so as to be able to spend substantial time interacting with theoretical biologists. There are many more potential opportunities for highly productive cross-disciplinary activity, reflecting both the spread of probabilistic techniques across science and the increasing sophistication of the mathematical tools being deployed in probability. To take advantage of these it would be necessary to create time and opportunity for highly trained probabilists to work in conjunction with first-class scientists and engineers in the corresponding areas. The demands on time and energy of the present UK academic culture of continuous and short-term research assessment do not improve the prospects for such developments.

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53 INTERNATIONAL REVIEW OF MATHEMATICS MATHEMATICAL ASPECTS OF OPERATIONAL RESEARCH LANDSCAPE

Lead Author: Professor Richard Eglese, Lancaster University Contributors/Consulted: Professor Jeff Griffiths, Cardiff University

1. Statistical Overview: This Landscape Document has been informed by returns of a questionnaire on behalf of research groups in the Mathematical Aspects of Operational Research in the following universities: Brunel, Cardiff, Edinburgh, Kent, Lancaster, Nottingham, Salford, Southampton, Strathclyde, Warwick.

These returns will underestimate the total research activity in this area as there are many researchers whose work could be classified under Mathematical Aspects of OR who are working in smaller groups or in a department whose main focus is in a different area (e.g. researchers at Cambridge and Oxford).

The last International Review of Operational Research took place in 2004. The questionnaire returns suggest that since the beginning of 2004 there has been significant research activity in Mathematical Aspects of Operational Research with a total of over £32 million in research funding including £27.2 million from Research Councils. In terms of Research Council funding, Nottingham has a prominent position with £11.7 million, Lancaster received £5.0 million and other groups with over £1 million over this time period are Cardiff, Salford, Southampton, Strathclyde and Warwick. Other sources of research income since 2004 are made up of £1.4 million from Industry, £2.3 million from the Public Sector and £2.0 million from other sources.

2. Subject breakdown: An initial estimate of the number of active researchers was made by extracting data from RAE 2008 submissions to the Statistics & Operational Research Sub-panel of the main Mathematics Panel.

Research Area Number of % of Total Researchers Mathematical aspects of Operational research Stochastic Modelling 65 16 Deterministic Modelling 21 5

These data were extracted from RAE 2008 submissions to the sub-panel for the Unit of Assessment (UoA) ‘Statistics & Operational Research’. The percentages quoted in the table relate to the number of active researchers in the designated research areas of OR compared to the total number of researchers submitted to the sub-panel. The criterion for inclusion of an active researcher was that at least one paper was submitted in an area deemed to be OR. In the report of the sub-panel, it was stated that ‘the volume of submissions to the UoA under-estimates the size of the active research community’. A major reason concerned the fact that the sub-panel received a large number of incoming cross-referrals from other sub-panels, particularly from the sub-panel ‘Business and Management’, which were mainly of a mathematical nature. Data is not available relating to these submissions.

Another guide to the size of the research community in the mathematical aspects of OR comes from the returns to the questionnaire outlined in Section 1. These are

54 summarised in Section 4 and cover 171 researchers, though this figure will also underestimate the total.

It is also likely that the relative proportion of researchers in stochastic and deterministic modelling would be radically different if data on submissions to all sub- panels were available.

3. Discussion of research areas:

The UK boasts a critical mass of internationally competitive research in (deterministic) optimisation, some of which is internationally leading. Optimisation is a broad field with sub-fields in continuous optimisation, discrete (combinatorial) optimisation and in heuristic methods which are needed when exact algorithms fail to provide an optimum within the required computing time.

Specialists in optimisation can be found in most of the research groups listed in Section 1. Internationally leading research at Lancaster applies polyhedral theory and convex analysis to NP-hard problems. Optimisation is one of the research themes of the LANCS Initiative (described in Section 4), particularly involving researchers from Lancaster and Southampton who are starting to explore lessons that can be learned from both continuous and discrete methods for the benefit of both. This has led to a proposal for a four-week programme at the Isaac Newton Institute on Polynomial Optimisation.

At Warwick, important advances have been made in optimisation methods linked with problems that have been studied by Operational Researchers for many years, such as the Capacitated Facility Location Problem. Theoretical advances in the study of efficient frontiers in Data Envelopment Analysis have been produced.

The Optimisation group at Strathclyde has made significant contributions in a variety of theoretical areas, ranging from LP (linear programming) and LCP (linear complementarity problems) algorithms to IP (integer programming) methodologies and practical applications of mathematical programming in various settings.

Researchers need to develop effective heuristics for optimisation problems where exact methods fail. The LANCS initiative has a strong focus on heuristic approaches which is led from the ASAP group in Computer Science at Nottingham. The group has established itself over the last 12 years as a leading pioneer in the innovative exploration of computational search methodologies at the interface of Computer Science and OR. More recently, the group has focussed on the investigation of hyper-heuristics, which can be defined as heuristics to choose or generate heuristics. There is clear evidence that this work has been influential. Three recent papers have, according to ISI’s essential science indicators, been in the top 1% of cited papers in both Mathematics and Computer Science. Two significant application areas for this work have been airport operations and nanobiotechnology. The ASAP group can reasonably claim to be pursuing a world leading research agenda in these areas. Additionally, Southampton is also designing algorithms for airport operations with support from EUROCONTROL, while many groups house research activity aimed at the development of heuristic algorithms for application in many areas including transport & logistics, cutting & packing, scheduling and timetabling.

As Whittle’s paper ‘Applied Probability in Great Britain’ in the 50th anniversary (2002) edition of Operations Research testifies, the UK has historic strength in stochastic OR. It remains the case that there is internationally competitive research in many sub-fields, and internationally leading research in some. A concern is the lack of

55 critical mass in some sub-fields with progress heavily dependent on the work of a relatively small number of outstanding individuals, several of whom are close to retirement.

Internationally leading research at Lancaster has deployed Lagrangian relaxation and mathematical programming to develop novel approaches to the design and evaluation of heuristic policies for complex stochastic dynamic programmes (DPs). Several major developments of Gittins index theory have been achieved. Lancaster also has a developing capability in approximate DP. The area of stochastic optimisation in the UK has been strengthened by the recruitment under the LANCS Initiative of an internationally leading expert in stochastic programming, with a particular focus on modelling and applications. LANCS now has a major research focus on decision-making under uncertainty and has begun to expand capability in the area. Beyond LANCS, there is important research activity in stochastic optimisation at Brunel and Edinburgh.

Researchers at Cardiff and Lancaster have secured major advances in the development of analytical and numerical approaches (both exact and approximate) to the performance analysis (both transient and dynamic steady state) of time dependent queueing networks. This work is closely linked to applications, most especially in healthcare. Healthcare modelling is a strong research area.

A recent retirement has considerably weakened the UK’s contribution to the mathematical underpinnings of simulation modelling, though it remains very strong in its practical application. At Warwick there has been progress in estimating the warm- up time needed for simulation runs which has been implemented in commercial software.

The Risk and Reliability group at Strathclyde has made internationally leading contributions in dependability modelling research, through developing modelling frameworks that integrate expert judgement with operational data, deeper understanding of the relationship between expert judgment and model parameters through probabilistic inversion theorems, characterisation of the distinct role expert judgment has within reliability modelling as opposed to risk assessment, alternative metrics to subjective probabilities for capturing expert judgment and construction theorems for graphical partial dependency models. The primary area of application for this material has been in availability of systems modelling which has resulted in award winning papers.

Forecasting research is an area which has seen a strong UK research contribution. The Lancaster Centre for Forecasting has enhanced its international profile in both forecasting and marketing analytics (including data mining). Since 2004, the Centre has worked mainly on the use of statistical time series analysis to understand and improve forecasts made in the context of supply chain planning. Further contributions to this area come from researchers in other groups, including Salford. This is an area where an upcoming retirement could significantly impede future progress.

Although Financial Mathematics is another area where a Landscape Document is being prepared, there are members of the OR research community whose main application area relates to financial applications. The group at Edinburgh hosts one of the largest clusters of academic researchers in consumer risk assessment in the world and has acquired an international reputation in the area. Research by this cluster was the first to introduce macroeconomic variables into survival scoring models so enabling them to be stressed as required by the Basel II Accord and the first to produce international comparisons of consumer credit constraints using

56 microdata. The biennial credit scoring and credit control conference organised by the cluster is established as one of the leading conferences in the area and involves 260 practitioners and academics from 15 countries. Brunel also has significant research in the area of Finance, particularly involving stochastic optimisation.

There is growing interest in the development of new areas such as Mathematics in Sport, where there are more researchers, notably at Salford.

The 2004 International Review of OR noted that there were a number of research areas in which OR in the UK was not strong. Those which relate to the current review include supply chains, the integration of economic agency and game theory into OR research on auctions, pricing and regulation, revenue yield management, service management and data mining. These areas remain relative weaknesses, though there has been considerable progress in revenue yield management where there is now a significant UK contribution from research at Lancaster and Southampton.

4. Discussion of research community

There have been three particularly significant developments in the research community involving EPSRC since the last International Review of OR in 2004.

First, there was the introduction in 2006 of NATCOR, a national taught course centre in OR (http://www.natcor.ac.uk/ ). NATCOR is a collaboration between six universities (soon to grow to nine) to develop and deliver taught courses in OR to PhD students. The initiative to develop such taught course provision came from EPSRC and is part of a drive to deepen and broaden PhD studies in the mathematical sciences in the UK. The centre is funded by EPSRC until September 2011 and is supported by the Operational Research Society which is a collaborating partner. The six collaborating universities to date have been Brunel, Cardiff, Lancaster, Nottingham, Southampton and Warwick. These are soon to be joined by Edinburgh, Kent and Salford. NATCOR courses have been taken by students from nearly 30 universities, with an average course attendance of more than 80 students.

NATCOR’s programme includes five week-long advanced courses covering material in the following areas: Combinatorial Optimization, Convex Optimization, Heuristics & Approximation Algorithms, Simulation and Stochastic Modelling. The provision is delivered over a two-year cycle so that it is possible for PhD students to attend each of the courses over the first two years of their doctoral studies. NATCOR also facilitates a research student conference where PhD students can present and discuss their research. EPSRC has recently commissioned an independent review of doctoral training courses and NATCOR received a very positive assessment (details are not yet public at the time of writing). The courses have been very well received by both students and supervisors.

Second, there have been major EPSRC research grants relating to Mathematical Aspects of Operational Research which have been made as part of the Council’s Science and Innovation programme. In particular, the LANCS Initiative (http://www.lancs-initiative.uk.ac ) has been funded to build and maintain national research capacity in foundational OR. LANCS is built on a collaboration between four universities: Lancaster, Nottingham, Cardiff and Southampton. EPSRC granted the award of £5.4 million to support the development of research capacity at these universities. Together with support from the universities themselves, the development means that during the period 2008-2013 a £13 million investment will have been made. During the initial period of the award, additional research appointments have been made at junior and senior levels and this is not yet complete. The publicity for

57 the Initiative has helped to attract highly qualified applicants and some excellent appointments have been made from the UK and overseas. It is perhaps still too early to tell what results the Initiative will have in terms of new research outputs and research impact in applications.

Further, at Warwick, the Centre for Discrete Maths and Its Applications (DIMAP) (http://www2.warwick.ac.uk/fac/cross_fac/dimap/ ) was established in March 2007 and funded, like LANCS, as part of EPSRC’s Science and Innovation scheme. The Centre builds on collaboration between the Department of Computer Science, the Warwick Mathematics Institute and the Operational Research & Management Sciences Group in the Warwick Business School.

Both of these Science and Innovation awards have provided resources to sponsor additional doctoral students in Mathematical Aspects of OR. However, most PhD students sponsored so far will not yet have completed their theses.

Third, the EPSRC recently awarded a Centre for Doctoral Training (CDT) in Statistics and OR to Lancaster University (http://www.stor-i.lancs.ac.uk/ ). The Centre, called STOR-i, has received a £4.3 million grant from the EPSRC and is also supported by a number of industrial collaborators. STOR-i aims to attract high quality doctoral students through enhanced stipends, a high level of industrial engagement and a training programme which incorporates a dedicated MRes programme in the first year of study. Students will thereafter work on research topics at the interfaces between statistics, OR and industry. An outstanding first cohort of ten UK students has been recruited to start in October 2010.

In addition to EPSRC support, HEFCW awarded a grant of £5 million in 2006 to establish the Wales Institute of Mathematical and Computational Sciences (WIMCS). One of the five research clusters is in OR and is led by Cardiff.

These changes to the research scene since the last International Review demonstrate that there has been significant new investment which is designed to improve and support the quantity and quality of research in the Mathematical Aspects of Operational Research.

The questionnaire returns showed that at the end of the academic year (June 2010) there were in total (at least) 240 students registered for PhDs in the Mathematical Aspects of OR. It remains a challenge to recruit UK PhD students and the large majority of current PhDs come from outside the UK. The evidence to date from initiatives like the STOR-i CDT is that highly qualified PhD students can be recruited to strong programmes which are industrially engaged when stipends are sufficient.

The age profile and seniority of research staff is based on a total of 171 staff categorised in the questionnaire returns and is shown in the table below:

Age Range (years) ≤ 30 31-40 41-50 51-60 ≥61 Total Professors 0 2 10 15 11 46 SL & Readers 0 8 13 7 2 37 Lecturers 5 19 3 1 0 45 Research Staff 13 6 1 0 0 43 Total 18 35 27 23 13 171

The profile shows that the demographic trends that were a cause of concern in the 2004 review have changed and the profile is currently closer to what might be regarded as standard for a discipline.

5. Cross-disciplinary/Outreach activities:

Operational Research as a discipline has its origins in cross-disciplinary research into military operations by teams of scientists, so it is not surprising that much of the research takes place in a multi-disciplinary environment. The main research groups are located in different departments and faculties of their universities and include Mathematics, Business & Management Schools and Computing Schools. Examples of cross-disciplinary research collaboration are shown for example in the DIMAP Centre at Warwick (with Computing) and the STOR-i Doctoral Training Centre at Lancaster (with Statistics) that were described in the previous section.

The emphasis within Operational Research of analytical methods to support better management decision-making means that much of the research has been motivated by practical problems in industry and other organisations. Thus there are many examples of outreach activities in the form of research collaborations with industry and public sector organisations. For example, Strathclyde has a well-established Risk Consortium that involves ten industrial partners. Strathclyde was also involved in the creation of Simul8, a company producing leading OR software and collaborating with academic groups. Cardiff, Lancaster, Nottingham and Southampton have close contacts with the Health Service and have held research workshops at which Health Service Professionals have participated. Cardiff is involved in the Health Modelling Centre Cymru (HMC2) where there is substantial input from OR, but also areas such as image processing and fluid dynamics.

Nottingham’s research involves different disciplinary areas including bioinformatics and healthcare scheduling. Moreover, they have been able to translate their cutting- edge techniques into hree viable spin out companies: Eventmap Ltd (automated timetabling), Aptia Solutions Ltd (cutting and packing) and Staff Rostering Solutions Ltd (healthcare duty rostering). In addition, their runway scheduling software will be implemented at Heathrow Airport in late 2010 and other implementations are planned. The Nottingham group also lead an institute called the Inter-disciplinary Optimisation Laboratory for i nter-disciplinary appl ications of OR including research with physicists and chemists working on nanoscience.

As well as national contacts, many of the contacts are local; for example, Warwick has applied research projects with Coventry City Council. The Smith Institute for Industrial Mathematics and Systems Engineering is working with the LANCS Initiative to help find new industrial partners and to identify promising topics for research collaborations. Most research groups also have collaborations with overseas researchers in all parts of the world.

While this document has focussed on Mathematical Aspects of OR, developments in areas such as decision support and problem structuring are also helping to make OR relevant to decision makers and problem owners. Examples given in this section show that activities associated with the Mathematical Aspects of Operational Research have a high research impact.

59 INTERNATIONAL REVIEW OF MATHEMATICS MATHEMATICAL PHYSICS LANDSCAPE

Mathematical Physics in the UK includes a number of domains, listed I-VII below, and is strongly interdisciplinary, especially between mathematics and physics – one of its strengths. I Particle Physics Phenomenology: concerns making detailed predictions of the Standard Model of Particle Physics and constructing models for physics at higher energies, including implications of particle physics for cosmology and astrophysics. II Lattice Field Theory: concerns both analytic work and numerical simulations, and links to statistical mechanics. The aim is to calculate accurately the properties of hadrons within the QCD framework. III Formal High Energy Theory: includes efforts to construct a unified theory of all interactions including gravity, addresses major problems within the Standard Model, such as quark confinement, and develops new techniques in quantum field theory, including conformal field theory, integrable field theory, and string theory. There are links with I and II, and areas of pure and applied mathematics. IV General Relativity: concerns the solution of Einstein’s equations and the mathematical and physical properties of solutions and their singularities, and matter fields (classical and quantum on curved space-time). There are strong links with III via string theory. V Cosmology: concerns the detailed modeling of cosmological solutions with strong links to IV. VI Quantum Chaos and Random Matrix Theory: concerns the classical and quantum spectra of complex dynamical systems linking with several areas of mathematics, especially number theory. VII Quantum Information and Quantum Computing: concerns the foundation of quantum mechanics and applications of quantum mechanics in computation, with links to physics and computer science. Specific subject areas with strong mathematical content will be described below. These lie mainly in areas III-VII, while I and II, though linking strongly with mathematical work, is mostly carried out within physics departments. A few illustrative highlights are included but space does not permit more.

1) String Theory, Supergravity and Geometry: String theory, with its non-perturbative completion M-theory, is the leading candidate for a unified theory of all interactions including gravity. Its ideas and techniques have become theoretical tools for understanding strongly coupled gauge theories and condensed matter systems. These applications are a growing field in their own right and are included in (4) below. Fundamental aspects of string theory and space-time geometry are areas funded primarily by EPSRC. There are strong links with mathematics especially algebraic geometry and topology: many of the key contributions to the historical development of string theory have been made by UK scientists working in mathematics departments and UK mathematicians such as Atiyah, Donaldson, Hitchin, Segal and Joyce have had important impact on string theory. Recent highlights include the work of Bagger and Lambert (King’s) on world-volume theories for membranes in M-theory (over 400 citations since 2008). There are strong mathematics groups in this area in Cambridge, Durham, Edinburgh, King’s College London. Over the last ten years, string theory has benefited from over 25 new hires at faculty level ensuring the UK is well placed to sustain its strong position. 2) String Phenomenology and String Cosmology: This concerns the analysis of the observable consequences of string theory for particle physics and cosmology. It is primarily an STFC funded area, although there are also strong connections to the geometrical aspects of string theory contained in (1). The UK has significant strengths in both areas (1) and (2) and the combination of EPSRC and STFC funding for these areas is critical to maintain this.

60 3) Perturbative Gauge Theory and Scattering Amplitudes: Despite the established status of non- abelian gauge theory as the theoretical framework of the Standard Model, the means to calculate within it is severely limited. At high energies, strong interaction processes can be analysed in perturbation theory, and such calculations are vital for the elimination of background effects at the Large Hadron Collider (LHC), but explicit computation in terms of Feynman diagrams is cumbersome. Recent years have seen the development of powerful new techniques that bypass the need for calculating individual Feynman diagrams. Methods used include string theory techniques and twistor theory. UK scientists have played an important role in these developments with highlights being the work of QMUL-Physics on one-loop (maximally helicity-violating) amplitudes and the continuing contributions of Oxford-Mathematics in the development of twistor methods. 4) Strongly-Coupled Gauge Theory and the AdS/CFT Correspondence: Present understanding of the strong interactions at low energies is more limited. The description in terms of the non-abelian gauge theory QCD is well established, the basic low energy properties can, so far, only be investigated in numerical simulations. The question of confinement and mass gap was marked in 2000 when it was announced as one of the Millenium Prize problems of the Clay Foundation. There has been significant recent progress with the advent of the AdS/CFT correspondence, a far reaching but so-far unproven conjecture relating strongly-coupled gauge theory to a weakly-coupled theory of gravity; previously intractable problems in one theory can be mapped into accessible problems in the other. This has led to an unprecedented convergence of ideas and interests within lattice gauge theory, string theory and general relativity. UK scientists have played key roles in these developments. For example, the proposal of a world-sheet action for the superstring in Anti- de-Sitter space by Tseytlin (Imperial-Physics), with Metsaev, was a vital step leading to an exact solution for the string spectrum in this background. Starinets (Oxford) is one of the pioneers in applying the AdS/CFT correspondence to the quark-gluon plasma as probed in experiments at the Relativistic Heavy Ion Collider (RHIC). UK scientists have also contributed to gravitational aspects of the AdS/CFT correspondence, in (9) below. 5) Conformal Field Theory and Integrable Quantum Field Theory in (1+1)-Dimensions: Conformal field theory has received critical input from a number of UK theoretical physicists and mathematicians during its lengthy history, and work continues in many directions. One is the classification, calculation of correlation functions and partition functions, exploring conformal field theories defined on restricted domains, analysing models containing defects and discrete dynamical systems. The relationships between conformal field theory and SLE are ongoing, as are the applications of conformal field theory to two-dimensional physics, mostly motivated by physical phenomena in condensed matter and statistical mechanics, the latter represented strongly in Oxford-Physics and elsewhere. Integrable quantum field theory models in two dimensions are widely studied and there are outstanding issues associated with models containing one or two spatial boundaries, or defects; groups at Brunel, Cambridge, City, Durham, King’s and York have contributed significantly. Techniques developed within integrable quantum field theory (for example, the Thermodynamic Bethe Ansatz, exact scattering matrices and bootstrap techniques), have become tools to study the AdS/CFT correspondence. The discovery in Durham-Kent-Torino that techniques developed within integrable models could be used to study the spectra of certain Schrodinger operators via the ‘ODE/IM correspondence’ was surprising, illuminated an established area in another field, and led to the settling of outstanding conjectures. 6) Classical integrable systems and their applications: Since the 1960s the UK has contributed extensively and influentially in the area of integrable systems, and there are groups distributed across the UK working within this broad topic (Brunel, Cambridge, Cardiff, Durham, Edinburgh/Heriot-Watt, Glasgow, Imperial, Kent, Leeds, Loughborough, Queen Mary, Swansea and York). There are links to many other branches of mathematics, especially algebra (including ‘quantum groups’), differential geometry, algebraic geometry and special function theory, and to physics, especially statistical mechanics and condensed matter. Connections between these groups are excellent and there are several mini-conference series. The UK input from physicists and mathematicians in discovering the properties of integrable and other closely related systems has been highly significant and includes, besides numerous applications, the celebrated ADHM construction of instantons, generalized separation of variables, and a variety of approaches to constructing multi-solitons and multi-monopoles. The study of skyrmions, both classical and

61 quantum, with applications in nuclear physics and knot theory, continues unabated, as does the study of multi-monopoles and vortices. Many ideas developed in these areas have stimulated new mathematics and some have been adopted by string theorists. 7) Quantum Chaos and Random Matrix Theory: These are both exceptionally strong in the UK. The subject of Quantum Chaos was essentially founded by Berry (Bristol-Physics). The UK remains the world leader, both at the pure mathematical as well as at the theoretical physics ends of the subject. Bristol is the main centre, but there are strong groups in Brunel, Cardiff, Lancaster, Loughborough, Nottingham, Royal Holloway and UCL. Many of these groups have grown significantly in the past five years. Recent research highlights include understanding how periodic orbit correlations govern spectral statistics (Sieber-Richter pairs), and the semi-classical theory of resonance states in open quantum systems (relating to the Fractal Weyl Law - one of the great challenges of modern spectral asymptotics). The UK is also one of the international leaders in Random Matrix Theory. Here the main groups are in Bristol, Brunel, Imperial College, Lancaster, Nottingham, Queen Mary and York. Again, there has been very considerable growth over the past five years. The main research highlights include significant contributions to the development of the Riemann-Hilbert method and applications to the theory of the Riemann zeta-function and other L- functions. 8) Mathematical Relativity and exact solutions: A significant feature of recent developments in mathematical relativity is the move towards looking at solutions of Einstein's equations from the point of view of PDEs rather than the geometry of space-time. A key result was proving the nonlinear global stability of Minkowski space. The analogous question for black hole space-times has stimulated recent work including significant results by Dafermos (Cambridge DPMMS). Closely related is the analysis of initial data sets, for example the introduction of a geometrical invariant characterising Kerr initial data. The gluing result of Corvino and the extensions of this by Crusciel (Oxford – now Vienna) have also had significant impact. Twistor theory originated with Penrose as an alternative description of GR but in recent years has had a significant impact in the area of integrable systems, perturbative quantum field theory, and Twistor-String theory. There is work on the nature of gravitational singularities in both Oxford and Southampton. There is also important activity in the area of QFT on curved spacetimes with a strong group in York using C* algebras and microlocal analysis. Finally, there is also work on alternative approaches to quantum gravity such as spin foam models. MacCallum is a co-author of the definitive book on four-dimensional solutions and with others in the UK has developed algorithms to classify solutions. Solution generating techniques have developed in Loughborough. 9) String inspired gravity theory: The AdS/CFT correspondence has resulted in new interest in areas of classical GR, for example, the study of Penrose limits in the context of plane-wave backgrounds in string theory, initiated in 2002 by Figueroa-O'Farrill (Edinburgh), Hull (Imperial- Physics), and Papadopulos (King’s), together with Blau (Trieste). This paper and subsequent work lead to a period of rapid progress in the computation of both the string and dual gauge theory spectra. Starting from the AdS/CFT, Rangamani and Hubeny (Durham) with Minwalla (Tata) and others have developed a precise map between general relativity and fluid dynamics that promises important insights for both fields. Inspired by string theory, higher dimensional black holes, possessing a rich structure have been explored at Cambridge as well as elsewhere. Black string solutions were shown to be unstable and it is conjectured that this instability leads to a number of exotic solutions such as the rotating black ring and the recently discovered ‘black saturn’ solution. 10) Gravitational Waves and Numerical Relativity: There is considerable international activity in constructing gravitational wave detectors but accurate models of the wave sources are needed to extract signals from the noise. There is collaboration with experimenters and data analysis in Physics departments at Cardiff, Glasgow and Birmingham. Mathematically, the interesting work involves developing theories of general relativistic (multi-fluid) hydrodynamics and general relativistic elasticity as well as studying the dynamical stability of the evolution equations. There is a large group at Southampton but otherwise work in this area is rather patchy in the UK. Numerical relativity has made significant progress in the last decade again motivated by the need to produce accurate gravitational wave templates. The work on orbits of black holes and black hole mergers is of note. An important ingredient was an analysis of mathematical structure of Einstein's equations and their reformulation as a symmetric hyperbolic system. The UK NR effort involves only a small

62 number of people but has made a significant contribution to mathematical formulations, relativistic hydrodynamics and critical phenomena. 11) Mathematical Cosmology: One of the major areas of activity in cosmology involves the study of non-linear cosmological perturbations. Significant effort has gone into the development of gauge invariant quantities to describe these as well as the use of renormalised perturbation theory to understand large scale structure. There is related research into structure formation involving multi- field inflation as well as ekpyrotic alternatives. There is significant activity in superstring cosmology involving brane inflation as well as cosmic superstrings. There are strong links with high-energy physics in connection with dark matter and dark energy, both in the early universe and at later times, as providing an explanation of apparent late-time acceleration. There is also related work on modified gravity. The main groups working on mathematical cosmology are in Cambridge and Portsmouth but there is also significant work at Oxford, Sussex, Sheffield, Nottingham and QMU. 12) Quantum Information and Quantum Computing: The research area of theoretical quantum information science, comprising quantum computing and information, quantum cryptography and communication, and related foundational areas is currently one of the most vibrant and high profile of all areas of scientific research internationally. It is the theoretical basis of cutting edge 21st century technology, in which it is becoming possible to design quantum states and manipulate them in a controlled fashion for specific purposes, with applications on all scales from nano-circuit design to global secure quantum communication networks linked by satellite. Since the creation of this subject, the UK has been, and continues to be, acknowledged as a principal world leader, with some outstandingly strong research groups, for example at Bristol, Cambridge, Edinburgh/H-W, Oxford, and several pioneers have been based in UK universities throughout their careers; some of the most important algorithms demonstrating that quantum computation is exponentially more powerful than classical computation for significant tasks were developed by UK researchers, for example quantum algorithms for solving sets of linear equations by Harrow, Hassim and Lloyd; similarly several of the most important quantum cryptographic protocols were devised by UK mathematicians and physicists, for example a provably secure quantum key distribution devised by Barrett, Hardy and Kent. The field brings together theoretical and applied physicists, mathematicians, computer scientists, engineers, materials scientists and computer scientists. Research groups are found in all the relevant departments, and several UK universities have created cross-disciplinary quantum information research centres with members in several departments. Quantum information science has a major impact on fundamental theoretical physics, giving profound new insights into our most fundamental physical theory, new ways of processing experimental data, and a new theoretical language in which to frame and answer questions. There has been a recent influx of important new techniques from mathematical physics and UK workers pioneered some of the most significant connections. Similarly, in issues of theoretical computer science, there has been much development of the understanding of the relationship between quantum and classical computational complexity, including world-leading UK contributions. Quantum cryptography is now a well-developed technology where some of the most significant early and recent theoretical and technological developments were made by UK researchers. Discussion of Research Community: theme III has expanded rapidly in the UK over the last ten years, with about fifty new posts at junior faculty level in the subject areas identified above. UK universities have attracted some of the very best people to junior faculty positions in mathematical physics, with the majority coming from abroad; there have been some retirements but mathematics departments have made a large investment. Commensurate levels of funding for postdocs and graduate students are needed to maximize the benefit of this investment and consolidate the UK's strong international position in formal theory, now second only to that of the US. There is a very strong tradition of high quality mathematical research in GR/Cosmology in the UK going back to Bondi, Hoyle, Sciama, Penrose and Hawking but the last twenty years has seen a substantial growth worldwide in gravitational physics not reflected in permanent appointments in the UK. The field is more diverse now yet some areas rely on a few key researchers, some of whom are approaching retirement. Thus, although the UK remains strong in this area and continues to produce world-class research, new investment is needed to maintain this position. Mathematical relativity in the area of PDEs will grow worldwide but much of this will take place in mathematics departments. It is not so clear that this will take place in the UK without a strategic initiative.

63 Numerical relativity and source modeling are also areas where the UK makes significant contributions with limited resources. Cross-Disciplinary/Outreach Activities: New connections between different areas of high-energy theory as well as new applications of formal theory to concrete physical problems have developed. Ideas from string theory have been applied in diverse areas: condensed matter theory, quantum information and fluid dynamics. The AdS/CFT correspondence has led to fruitful exchanges between scientists working on classical gravity quantum field theory, with the UK taking the lead in some of these. String theory has significant interactions with mathematics, particularly algebraic geometry: the development of topological string theory has lead to new methods of calculating topological invariants of Calabi-Yau manifolds. Donaldson and Thomas (Imperial) have been especially influential. There is interaction between GR and mathematics in nonlinear PDEs and geometrical functional analysis. EPSRC grants (in mathematics and physics departments): current, identified by theme

Theme 1 2 3 4 5 6 7 8 9 10 11 12 Total

People (∗∗∗) 21 27 18 56 50 55 27 14 14 7 16 41* 346

EPSRC (#) 13 1 3 5 6 3 10 3 3 1 0 3 51

Value (£k) 2,890 194 548 1,691 1,564 722 2,217 737 992 222 0 1,010 12,787

∗∗∗ The approximate numbers of people/theme are a guide; estimated numbers per area are given below. • Support for graduate students is via Doctoral Training Packages and Postgraduate Teaching Centres. Relevant meetings have been held regularly at INIMS (Cambridge), and ICMS (Edinburgh). • There are currently 12 Postdoctoral Research Fellows (2 in physics, 10 in mathematics); 9 Advanced Fellows (3 in physics, 6 in mathematics) and 2 Senior Fellows (1 each in physics and mathematics). STFC grants (in mathematics departments): Awarded for the period 2008-2013 (the numbers of graduate students currently supported are given in the final column);

Cambridge-DAMTP 4,693,249 7 Durham-Mathematical Sciences 2,273,989 4 Heriot-Watt + Edinburgh-Mathematics (33,915 + 179,688) 213,603 King's College London-Mathematics 1,454,207 1 Liverpool-DAMTP 1,794,525 2 Sheffield (Mathematics and Statistics) 1 Sussex-Physical Sciences and Mathematics 267,046 1 Portsmouth (ICG - Mathematics) 1,431,613 2 Queen Mary-Mathematics (Astronomy) 98,534 Southampton-Mathematics 424,637 1 ------Totals £12,651,403 19

• STFC supports IPPP (Durham), BUSSTEPP (variable location), and several Fellows. Also, the themes receive support from the Royal Society (including RSURFs), the Leverhulme Trust and the European Commission but current data on these is not readily available.

64 Appendix - Statistical Overview: The approximate numbers of permanent faculty working in these broad areas across physics and mathematics departments in the UK are recorded below (based on submissions for the 2008 RAE and checked with current web-sites).

Mathematics Physics University I II III IV V VI VII I II III IV V VI VII Bristol 1 9 3 1 2* Brunel 1 4 5 Cambridge 7 2 11 11 3 2 2* Cardiff 3 1 1 1 City 4 Coventry 1 Durham 13 5 14 1 Edinburgh+HW 8 3 7 3 13* Glasgow 6 4 3* Imperial 7 2 10 1 4 2 Kent 3 King’s 8 3 2 Lancaster 3 1 Leeds 5 1 3 Liverpool 5 3 3 Loughborough 5 1 2 Manchester 2 4 5 1 Newcastle 2 2 Nottingham 3 5 1 4 1 2 Oxford 3 3 1 4 2 3 3 8* Plymouth 4 Portsmouth 4 4 Queen Mary 4 1 3 7 4 1 1 Reading 1 Royal Holloway 2 2 1 Sheffield 1 1 2 Southampton 6 5 2 3 St Andrews 2* Strathclyde 2* Sussex 1 1 3 Swansea 4 4 9 UCL 1 York 5 3 1 1 1*

Totals 14 7 106 44 4 30 11 37 15 42 6 25 4 41*

Note: The (*) entries in column VII indicate that the number of people is spread beyond physics (for example, chemistry, computer science and materials science); space does not permit more precision.

Note: The numbers cannot be regarded as entirely accurate since they have been taken from several sources that might not be regularly updated. Double-counting has been avoided as far as possible.

65 INTERNATIONAL REVIEW OF MATHEMATICS NONLINEAR DYNAMICAL SYSTEMS AND COMPLEXITY LANDSCAPE

Lead Author: R.S.MacKay Contributors/Consulted: D.S.Broomhead, C.J.Budd, A.R.Champneys

1. Statistical Overview: The scope of this overview is restricted mainly to those aspects of Nonlinear dynamical systems and Complexity which were presented to the Applied Maths subpanel of RAE2008; aspects of Dynamical Systems theory presented to the Pure Maths subpanel can be expected to appear under Analysis. Furthermore, applications of dynamical systems theory to areas that have their own landscape documents have not been covered (except where the emphasis is on the dynamical systems techniques); these areas include Mathematical Biology and Medicine, much of integrable systems (under Mathematical Physics), quantum dynamics (under Mathematical Physics), Mathematical Finance, fluid dynamics (under Theoretical Mechanics and Materials), and stochastic processes (under Probability). On the other hand, Control Theory is included here, because it is not clear it would be covered elsewhere.

Nevertheless, some highlights in nonlinear dynamical systems and complexity outside the Applied Maths RAE2008 submissions or in overlap with the areas of other landscapes will be mentioned.

The period covered is Jan 2004 to Sep 2010.

Bifurcations & Pattern formation 10 7 Piecewise smooth dynamics 10 7 Hamiltonian dynamics 10 7 “Pure” dynamical systems theory 10 7 Integrable systems 10 7 Control theory 10 7 Complexity Science 10 7 Applied dynamical systems 30 22 Computational dynamics 10 7 Other 30 22

This breakdown should be taken with a pinch of salt because the areas are neither disjoint nor exhaustive and some researchers are active in more than one area or mainly active in some other landscape area (as mathematicians we should be able to devise a more appropriate breakdown method, perhaps analogous to a partition of unity for a manifold).

The figures are based on a subjective classification of the researchers submitted to subpanel 21 of RAE2008, leaving out those who were considered to be more

66 appropriate for one of the other landscape documents, resulting in a total of 140 researchers. Thus they do not count relevant researchers who were submitted to other subpanels of RAE2008, nor subsequent newcomers.

Bifurcations & Pattern formation: This topic continues to be a major strength, represented principally at Bath, Bristol, Cambridge, Imperial, Leeds, Manchester, Nottingham, Oxford, Southampton, Surrey, Warwick, and Heriot-Watt. Problems addressed have included understanding of the bifurcations of travelling waves in pipe flow illuminating the transition to turbulence, and snaking behaviour in phase separation problems modelled by high-order PDEs such as the Swift-Hohenberg equation. There is strong overlap between this activity and work in Analysis.

Piecewise smooth dynamics: There has been a resurgence of activity in this area, led mainly by Bath, Bristol, Exeter, Manchester and Southampton. It has led to extensive classification of the bifurcations associated with flows when their limit sets interact with surfaces across which the system has a jump. There are close links between this activity and control theory in the study of hybrid and asynchronous systems, as well as many application areas in engineering (such as robotics), biology (neuroscience) and physics (impacting systems). The UK is highly productive in this area.

Hamiltonian dynamics: Represented mainly at Bristol, Brunel, Imperial, Leeds, Loughborough, Portsmouth, QMUL, Surrey, Warwick, the UK continues to be productive in this area. Particularly interesting is an Imperial-Warwick collaboration on a mechanism for large excursions in action variables for Hamiltonian systems of more than two degrees of freedom. There is also much work on mechanics with symmetry, led by Surrey, with applications for example to satellites and underwater vehicles.

“Pure” dynamical systems theory (including Ergodic theory): Although the main activity here was submitted to Pure Maths in RAE2008, there was also significant research submitted to Applied Maths, from Exeter, Imperial, QMUL and Surrey. Particularly significant among these are results on rates of mixing for flows and on statistical limit laws for chaotic systems, developed principally in Surrey.

Integrable systems: Most of the activity in this area is best classed under landscape 10: Mathematical Physics, so comments here are on only those aspects for which dynamics is at the centre. It is represented broadly, e.g. Bath, Cambridge, Imperial, Kent, Leeds, Loughborough, Reading, Heriot-Watt. Although very active, one has the impression that this community is becoming more isolated and it would benefit from reconnection to the mainstream of dynamical systems.

Control theory: Although active in Bath and Exeter and many engineering departments, traditional control theory is disappearing from most mathematics departments (some remains in Birmingham and Sheffield). Within Mathematics there appears to be scope to further develop the interface between traditional control theory and dynamical systems on the one hand and uncertainty modelling, optimisation and stochastic analysis on the other. In particular, the University of Manchester has recently created a Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) working at the boundaries between mathematics,

67 computer science and control engineering, which has led to an increase in research on adaptive control, control of stochastic and hybrid systems, and control seen as an application of purer work in iterated function systems.

Complexity Science: Complexity science aims to understand the behaviour of large numbers of interacting units and to exploit this understanding in applications. The main sites in the mathematics of complexity science are Bath, Bristol, Cranfield, Imperial, KCL, Manchester, Oxford, QMUL, Reading and Warwick. Complexity Science is a trendy subject that offers both opportunities and hazards. The main opportunities are for mathematical development of ideas of potentially far-reaching relevance. Particular areas of complexity science which show great promise for fundamental scientific advances are the study of multi-scale systems, statistical physics, the analysis of complex networks, and the analysis, classification and identification of emergent behaviours. Important applications of these studies can be found in the physical sciences (e.g. meteorology, photonics, crystals), chemistry (polymer behaviour), engineering (complex materials, dynamic information networks, large infrastructure systems, unmanned aerial vehicles), biology (synthetic biology, flocking patterns, cell dynamics, genetics, disease modelling, neuroscience), computer science (human computer interaction, cloud computing), and the social sciences (demographics, crowd dynamics). This is also a field in which there is a degree of general public interest and it is an area of international growth with many large international meetings. The main hazards are that the subject as a whole is dominated by a lot of well meaning but often loose thinking (e.g. transfer of concepts from dynamical systems to social science as metaphors), with perhaps rather heightened expectations of what might be achieved in the social sciences and management. One does not want to inhibit creative thought and dialogue, yet the subject would benefit from more self-criticism. Although much of the work has little novel mathematical content, engaging with it provides a tremendous opportunity for mathematicians to contribute useful results. Mathematicians who get involved in the looser ends of complexity science would be well advised to enunciate clear principles to avoid the risk of being sucked down if there is a general intellectual backlash against it (cf. artificial intelligence, catastrophe theory). These comments should not, however, detract from the very serious science that is involved in much of the harder end of research in complexity science, and the opportunities that the subject as a whole presents to mathematics.

Applied Dynamical Systems. There is a wide range of research in applications of dynamical systems in the UK, much of which is left to other landscape documents (notably mathematical biology and medicine). So under this topic are covered just a selection of areas or approaches that might not feature elsewhere. Problems in dynamical systems arising in applications (for example in photonics or mechanics) often include noise, delay, hysteresis, and involve a mixture of both discrete and continuous dynamics. Various universities such as Bath, Bristol, Exeter, Manchester, Southampton and Strathclyde are active in pursuing research into such applications, leading to industrial collaborations in for example the electricity supply industry (Bath), the electronics sector (Exeter, Southampton), aerospace and automotive sectors (Bristol), and robotics (Warwick).

Computational dynamics. There is significant overlap between dynamical systems and computation/simulation. On the one hand, numerical methods are now designed to capture the long-term ergodic behaviour of systems, and ideas from dynamical systems have found a powerful application in this design; the resulting methods are of extreme importance in molecular dynamics, meteorology, computational chemistry and celestial mechanics. On the other hand, techniques from dynamical systems theory are also used heavily in the error analysis of such methods and form a central

68 plank in the body of methods called geometric integration. The UK is a world leader in this field, notably Bath, Cambridge, Edinburgh, Warwick and Dundee. Large-scale simulations of the dynamics of complex systems are often being made without sufficient theoretical understanding, and it is important that more mathematicians work in these areas. In addition, there is a significant expertise in numerical continuation and bifurcation analysis of attractors and manifolds, led by groups at Bristol, Surrey and Nottingham.

Others. There is a huge amount of activity in applications of dynamical systems and complexity science to biology and medicine, e.g. systems biology, neuroscience, epidemiology, ecology, evolution, but these are expected to be covered in landscape 13. There is significant work in quantum dynamics, but that is left to landscape 10: mathematical physics.

4. Discussion of the research community Please give a brief description of demographic trends, e.g. composition, origin and volume of research student pipeline, age distribution of academic staff, etc.

Overall the research community is healthy and vibrant, with a good number of centres of excellence. It is especially encouraging to see a large number of young and talented mathematicians working in the areas of dynamical systems and complexity. There have been a few retirements and departures, but many young people are coming onto the University mathematics staff in this area, notably at Bristol, Manchester and Warwick, and the usual fluxes at other career stages.

A number of large, well funded, research and training institutes have been set up, which have stimulated mathematical research into dynamical systems and complexity in the UK through research programmes, workshop/visitor programmes and the training of significant numbers of research students and PDRAs. Notable amongst these are: (1) the creation of EPSRC-funded doctoral training centres in Complexity Science at Bristol and Warwick, with an annual intake of about 10 students each (also at Southampton but with an emphasis on simulation rather than on mathematics), and (2) the EPSRC-funded research centres such as the Bath Institute for Complex Systems BICS, the Manchester CICADA (Centre for Interdisciplinary Computational and Dynamical Analysis), and the Bristol Centre for Applied Nonlinear Mathematics (BCANM).

5. Cross-disciplinary/Outreach activities: e.g. describe connections with other areas of mathematics, science, engineering, etc.

The UK produces several of the leading journals in dynamical systems, e.g. Ergodic theory and dynamical systems, Dynamical systems, Nonlinearity. The UK has hosted some of the major international events in the area, e.g. Dynamics Days Europe: Loughborough and Bristol, and European conference on complex systems: Oxford and Warwick. Dynamical systems feature on the main visitor programmes in the UK, e.g. Newton Institute programmes on Discrete integrable systems, Highly oscillatory problems, Painlevé equations and monodromy, Dynamics of non-equilibrium systems, Pattern formation; Durham symposia: Dynamical systems and statistical mechanics; ICMS: Probabilistic limit laws for dynamical systems, and Warwick symposium years: Mathematics of complexity science and systems biology. In addition, centres like BCANM, BICS and CICADA have run a large number of specialist workshops and conferences on interfacial aspects of dynamical systems and complexity science. The IMA ran a conference on Mathematics in the Science of Complex Systems in 2007.

69 The London Mathematical Society supports several relevant UK networks, e.g. PANDA (Patterns and nonlinear dynamics), London Dynamical Systems group, CoSyDy (Complex System Dynamics), Classical and Quantum Integrability.

UK academics are prominent in a number of international networks, bodies, journals and conferences involved with dynamical systems and complexity. Notable amongst these are the SIAM activity group in Dynamical Systems and the 2011 SIAM Conference in Dynamical Systems, both of which have chairs from the UK, and the Complex Systems Society. The SIAM Journal of Dynamical Systems (SIADS) has two associate editors from the UK.

Mathematicians in nonlinear dynamical systems and complexity science interact strongly with researchers from a wide range of other areas, notably biology and medicine, engineering, physics, chemistry, in which the applications of dynamical systems are natural and immediate. Many of these have already been listed. There are now also links with the social sciences (in economics, sociology and psychology), although these are largely in an early stage and long-term mathematical benefit remains to be realised.

Dynamical systems and complexity science also have many industrial applications, with Bath, Bristol, Exeter, Reading, Strathclyde and Southampton particularly active in this area. These include meteorology, climate change, traffic dynamics, aerodynamics and handling, electronics, photonics and electricity supply. Companies such as Airbus are very interested in using complexity theory in describing the systems of systems encountered in modern aircraft. There are also emerging applications in retail and health.

Dynamical systems and complexity science have played an important part in promoting mathematics to the general public. For example, the BBC Horizon programme had a complete feature this year on the subject of Chaos, and Chaos Theory also appeared in the BBC Programme ‘It’s only a theory’ and in the Royal Institution Christmas Lectures. Dynamical systems have played a role in public Mathematics and Art events, notably a crochet version of the Lorenz manifold produced by Bristol. In 2010 the Royal Society Summer Exhibition included an exhibit on ‘Living in a complex world’ which described many aspects of complexity and attracted 40 000 visitors. The European Conference on Complex Systems in Warwick last year included a public session on the challenges of global change.

70 INTERNATIONAL REVIEW OF MATHEMATICS THEORETCAL MECHANICS & MATERIALS SCIENCE LANDSCAPE

Lead Authors: Alexander Movchan, University of Liverpool and Nigel Peake, University of Cambridge.

Consulted: In writing this document the authors consulted numerous members of the research community.

1. Statistical Overview: In RAE 2008 there were 850.5 full-time equivalent staff submitted to the Applied Mathematics Unit of Assessment. Of these staff, we have identified 167 (i.e. 19.6% of the total) who work either exclusively or predominantly in fluid mechanics (including astrophysical fluids) and 66 (8% of the total) who work in mathematical models of solid mechanics. The RAE identified strong fluids groups in 21 institutions and solid groups in 18 institutions.

2. Subject breakdown: The table indicates the number of active researchers in designated sub-areas. Data extracted from submissions to RAE 2008. The classification by subject area has been completed by us based on declared research interests (on personal web pages etc), with fractions (no smaller than 0.5) where individuals covers more than one area.

Research Area Number of % of Total i n Researchers Fluids Fluids (note rounding) Astrophysical fluids & MHD 32 19 Geophysics & Environment flows 25 15 Stability & large Re flows 18.5 11 Free boundary problems 19 11 Low Re flows, microfluidics 9 5 Multiphase & reacting flow 8 5 Waves 28 17 Biological fluid flows 15.5 9 Vortex dynamics + turbulence 12 7 Number of % of Total i n Solids Researchers Solids

Waves in solids, mathematical 22 33 models of fracture, asymptotic analysis of multi-scale systems Nonlinear solids, phase 24 36 transition, crystals with defects, homogenisation Plasticity, fatigue, dislocations, 9 14 numerical modelling, atomistic simulations Liquid crystals, bi-stable 11 17 systems, fluid-solid interaction

71 3. Discussion of research areas: By subject area, please provide a frank assessment (both positive and negative) of past and current international standing of the field, supported by reference to highlights as appropriate

Applied Mathematics research in the UK is extremely strong, as confirmed by the recent results of RAE2008. In particular, theoretical mechanics is well represented in UK universities as a highly developed research area of Applied Mathematics. Historically there has been greater activity in fluid mechanics than solid mechanics, but both disciplines maintain a very strong and vigorous international profile.

FLUIDS

One (no doubt imperfect) way of measuring the volume and quality of research output from the UK in fluid mechanics is by analysing the country of origin of papers published in the Journal of Fluid Mechanics (JFM), the leading international journal in the field. The UK is consistently the second most prolific country after USA, and ahead of France in third place. In 2009, 83 papers with a UK corresponding author were published in JFM, compared to 167 from USA and 46 from France. The UK figure has increased steadily in recent years (for instance, in 2006 the corresponding numbers were: 73 UK, 145 USA, 49 France). Also of interest is the JFM acceptance rate for the UK; the JFM average is that about 49% of all submitted papers are accepted, while for the UK only this figure was 66.94% in 2009, compared to 63.98% for USA and 63.89% for France. These figures underline both the size and the quality of the UK’s fluids research output.

Astrophysical fluids + MHD This is a major activity in the UK, with several very strong groups including those in St Andrews, Leeds, Newcastle, Cambridge, Sheffield & Exeter. According to RAE2008 the subject has grown significantly in recent years, and is clearly a hot topic, stimulated by new observational and computational experiments. It appears that much of the funding in this area comes from STFC, but connections with other areas of mathematics (e.g. topology, dynamical systems) are strong. Particular research highlights include: a detailed study of the nature of the solar tachocline [1]; the nature of magnetic reconnection in three dimensions [2]; disc and planet formation [3]; pioneering work on numerics for relativistic MHD, which has triggered rapid recent developments [4]; and a model of sunspots [5]. Going forward, continued vitality will depend on young researchers trained in interdisciplinary fields such as nonlinear MHD, plasma theory and planetary dynamics (cutting across several Research Councils). Training in numerical methods, and continued access to internationally leading high performance computing is absolutely key in this area.

Geophysics & Environment flows This is another area in which UK research remains exceedingly active, with strong groups including those in Cambridge, UCL, St Andrews & Exeter. The topics which drive much of this research are climate change and the prediction of natural disasters. Activities range from the theoretical and modelling activities for simplified systems, to large-scale climate and weather prediction. Mathematicians working in this area are particularly well-connected with scientists in other disciplines and with practitioners. Several departments have close links with the UK Meteorological Office and European Centre for Medium-Range Weather Forecasting; this includes Exeter (Thurbon and others); Reading, (Belcher); Surrey (Roulstone) and UCL (Johnson). With regard to earth sciences, the Institute of Theoretical Geophysics in Cambridge (a joint venture between mathematics and earth sciences departments) specialises in a range of phenomena from polar studies (including close links with British Antarctic Survey) to vulcanology. Particular recent research highlights include: the work of Vanneste in Edinburgh on modelling large-scale motion in the atmosphere/ocean [6] (in 2009 Vanneste won the prestigious Adams Prize for his work in this area, see http://www.maths.cam.ac.uk/news/3.html ); the collaboration between Exeter and the Met Office on a new `dynamical core’ for their atmospheric prediction model, to become operational in 2011/12 [7]; and work in Cambridge on the fluid mechanics of underground carbon sequestration. There is strong interaction of the community with the various NERC Institutions; for instance, Andrew Wilmott, the Director of the Proudman Oceanographic Institute in Liverpool, collaborates with mathematicians from Keele, Reading and Manchester.

Stability & large Reynolds number flows The UK has traditionally been the world-leader in this area, through the work of such distinguished individuals as Stewartson, Stuart, Smith and others. Current activity is found in many departments, with particular highlights in Bristol, Imperial, UCL and Manchester. One area of international leadership is the study of pipe-flow transition, especially the theoretical work of Kerswell in Bristol [8] on identifying finite-amplitude transitional structures and parallel experimental work by Mullin in Manchester [9]. The work of L.N. Trefethen in Oxford [10] on pseudospectra and transient growth has also proved very influential, and a recent application has been found in turbomachinery flow [11]. Although activity in Stewartson’s triple deck theory has been much reduced in recent years, there are still many examples of outstanding UK research in high-Reynolds number asymptotics. For instance, Wu at Imperial has developed highly detailed theories of

72 nonlinear critical-layer instability [12]; while Hall & Sherwin [13] have developed an asymptotic description of the role of streamwise vortices in shear-flow transition. A striking result concerning the confinement of a rotating- disk boundary layer has been found by Healey (Keele) [32].

Waves Traditionally this has also been a strong area in the UK, and there is still considerable activity in both nonlinear and linear regimes, taking in both highly applicable aspects (such as the interaction of water waves with coastal defences in collaboration with several government agencies, aircraft noise with Rolls-Royce and underwater vehicle noise with Thales, DSTL and QinetiQ) and more theoretical considerations (such as the existence of trapped modes and the solution of scattering problems via advanced analysis). The activity is found in many departments, with centres of excellence including UEA and Surrey (nonlinear waves), Manchester and Brunel (linear waves, scattering ) and Loughborough (water waves, trapped modes). Clearly much of the effort described under Geophysics & Environment flows above is also concerned with waves. Turning first to highlights of nonlinear wave theory: Vanden-Broeck (UCL) is a major figure in water wave research[14]; Bridges at Surrey works on many aspects of nonlinear wave theory, using an especially wide range of mathematical and numerical techniques [15]; Korobkin at UEA works on violent flow and high-impact waves [16]. In linear wave theory, developments in diffraction theory have been made by Smyshlyaev (now UCL), Craster (Imperial) [17] and Biggs (Reading) on the development of so-called `embedding formulae’ to calculate diffraction coefficients for complex shapes. There are also close links also with numerical techniques for high- frequency scattering developed by the group of Chandler-Wilde in Reading. The group in Loughborough has made important advances in the detection of trapped modes, for instance around floating structures [18].

Biological fluid mechanics In the UK the level of activity in mathematical biology has increased very significantly in recent years, supported by a number of initiatives from EPSRC and the biological and medical research councils. Biological fluid mechanics plays a major role, with significant centres of activity including Nottingham, Oxford, Manchester and Glasgow, but with high quality activity found in several other institutions. By its nature this work is highly interdisciplinary, and mathematicians work closely with medical scientists and biologists. Recent large-scale initiatives spanning several academic departments include the Centre for Mathematical Medicine and Biology at Nottingham and the Centre for Mathematical Biology in Oxford. There is very high-quality research on the fluid dynamics of blood vessels and airways, which encompasses both modelling and large-scale computation [19]. In Cambridge, the appointment of Goldstein has led to the establishment of a very active interdisciplinary group working on biofluids at the cellular level [20].

Vortex dynamics + turbulence This area is taken to include turbulence modelling, CFD and fundamental analysis of the Euler & Navier-Stokes equations. There are certainly outstanding individuals working in the UK in these areas, including JCR Hunt (UCL), Gilbert (Exeter) on vortex dynamics [21], Dritschel (St Andrews) whose work includes many computational aspects [22], Gibbons & Holm (Imperial) and Sergein, Niethammer & Vasseur (Oxford) working on analysis of the Navier-Stokes equations [23], and the promising new appointment Okitani (Sheffield). However, when compared to the USA, France and Germany the UK mathematics community seems underrepresented in all these areas. It is certainly the case that much CFD work is conducted in UK engineering departments (especially related to aeronautics), and there is significant strength within UK applied mathematics in large-scale computation (e.g. in astrophysics & meteorology, described above), but more needs to be done, especially with regard to simulating turbulence. There also needs to be more activity at the interface between fluid mechanics and mathematical analysis. The 2003 International Review of Mathematics identified UK analysis as an area which needed to expand, and since then significant EPSRC resources have been made available (including approximately £8M funding for Doctoral Training Centres in analysis at Warwick and Cambridge). Care will be needed to ensure that this renewal of UK analysis continues to translate into additional activity studying Navier- Stokes.

Free boundary problems, thin films This remains a very active area in the UK, with especially strong activity in Oxford, Nottingham, Birmingham & Strathclyde. Much of this work is driven by application areas in industry and biology. In Oxford there is an exceptionally broad strand of work in this area, involving several senior figures including J. & H. Ockendon, Chapman and Howell. Applications here include paint coating, tear-film dynamics, thin-film rupture to describe the dynamics of foam [24], and the role of thin water layers in violent impact problems such as ship-slamming[25]. In Strathclyde, the group led by Wilson is very active on a number of aspects, including rivulet flows (with application to aircraft icing) and Marangoni (surface tension gradient) phenomena. Another highlight is a project modelling oil film cooling in a turbine engine by Riley & Cliffe (Nottingham) sponsored by Rolls-Royce [26]. In Birmingham there is also considerable activity, especially with regard to very challenging issues of moving contact lines [33], including work with Kodak.

73 Low Reynolds number flow, microfluidics. The UK has a strong community working on highly viscous and complex flows, with larger centres including Manchester, Imperial and Cambridge. Much of the work in this area is motivated by industrial application, but there are increasingly many applications now arising in biology, including low-Re swimming, airway closure and cell motility. In Manchester, Juel, as part of Manchester Centre for Nonlinear Dynamics (a joint venture between mathematics and physics), works on both experiments and theory, including viscous interfacial instabilities [27]. In Leeds, Kelmanson applies computational, asymptotic and symbolic algebra techniques in coating flows and rhelogy [28]. Viscous flow work in Cambridge has included micro-hydrodynamics, polymer rheology and non-Newtonian fluid dynamics (Hinch & Rallison, collaborating with a number of companies, including Shell, Pilkingtons and Domino ink jet printing), liquid-jet instability (Lister) [30] and low-Reynolds number swimming (Goldstein, see above). Crowdy’s work at Imperial on the use of complex-variable techniques in fluid mechanics has yielded significant results in several areas, including recently in low Reynolds number swimming close to a wall [31] (his work on the multiply connected Schwarz-Christoffel formula was a subject of an article in the Times newspaper 4/03/08). In Sheffield, Rees is involved in a number of inter-departmental collaborations, including with chemical and biological engineers.

Multiphase and reacting flows. This category seems relatively small at present in the UK, but is in fact an area of rapid growth, including many exciting cutting-edge applications. There is an exceedingly broad range of high- quality interdisciplinary activity in Birmingham, much of it inspired by links with industry, including work on bubble dynamics [34] (with QinetiQ & Schlumberger) and on the fluid dynamics of fuel cells [35] (with EoN) - here mathematicians take leading roles in the Doctoral Training Centre in Hydrogen Energy and the Centre for Mathematical Modelling and Chemical Engineering. There is increased UK activity in granular flows, with strong groups including Gray and Harris at Manchester and McElwaine (in close collaboration with the Swiss Federal Institute for Snow and Avalanche Research in Davos) in Cambridge. Others have also worked in this area, including Hinch [29], Huppert and Kerswell on granular collapse. Another exciting area attracting increased UK activity is tissue growth and regenerative medicine (e.g. Waters in Oxford), in which issues of fluid flow through porous media are crucial. Combustion remains a smaller subject within mathematics, but with excellent individuals who collaborate closely with engineering colleagues. Dold (Manchester) works on modelling fire as part of the multidisciplinary Manchester Fire Research Centre.

SOLIDS

The bulk of solid mechanics research produced in Mathematics departments of UK universities is relatively small compared to the research in fluids. As indicated in the Table of Section 2, 58 individuals across 18 institutions work predominantly in the area of theoretical solid mechanics. The quality of research is high. Among very successful research directions in Solid Mechanics are the formation of microstructure in solids and homogenization (Oxford, Bath, Warwick), fracture mechanics, strain gradient plasticity and homogenization theory (Cambridge, Liverpool, Manchester), waves in solids (Imperial College), phononic band gap structures in structured media and models of solids with defects (Liverpool), asymptotic models of plates and edge waves (Manchester, Brunel, Glasgow, Keele), non-linear elasticity theory and applications in bio-mechanics (Oxford, Glasgow, Keele), theoretical mechanics industrial research (Nottingham), and continuum models of phase transition in alloys and liquid crystals (Strathclyde University, Oxford).

The spread of Solid Mechanics researchers across UK institutions can be summarised as follows:

Aberystwyth: Mishuris (Singular integral equations, mathematical models of fracture, models of viscoelastic media) Bath: Zimmer, Sivaloganathan (Cavitation, fracture in nonlinear solids. Phase transition in crystals. Modelling of martensitic transformations) Brunel: Kaplunov, Lawrie, Mikhailov, Nolde, Pichigin, Warby (Waves in solids. Asymptotic approximations in elastic plates. Prestressed solids. Fatigue. Computational modelling for viscoleastic materials) Bristol: Porter, Chenchiah, Robbins, Slastikov (Fluid-solid interaction. Waves in plates. Shape memory effects in polycrystals. Liquid crystals) East Anglia: Scott (Thermoelastic waves in thermomechanically constrained solids) Glasgow University: Ogden, Coman, Hanghton, Lindsay, Lou, Roper, Watson (Finite elasticity. Biomechanics. Shell Theory. Fluid-solid interaction, fluid-driven fracture, phase transition in materials science) Imperial College, London: Atkinson, Craster (Localization in elastic waveguides. Wave propagation and non- destructive testing. Fracture) Keele: Fu, Rogerson, Heckl, Blyuss, Chapman (Surface and edge waves. Long-wave asymptotics for thin structures. Waves in structures. Localized vibrations in plates and rods. Phase transition. Fluid-solid interaction.) 74 Liverpool: Guenneau, Mazya, Movchan A., Movchan N., Selsil (Waves in structured elastic media. Modelling of photonic and phononic band gap structures. Mathematical models of fracture. Singular perturbation analysis for solids with irregular boundaries.) Loughborough: Khusnutdinova, Archer, Kenny, Smith (Nonlinear models of dislocations. Layered elastic waveguides. Materials modelling. Plastic deformation. Atomistic simulations.) Manchester: Abrahams, Gray, Harris, Parnell, (Dynamic elasticity. Wiener-Hopf analysis. Homogenization. Waves in prestressed solids. Fundamental modelling of granular materials and plasticity. Nottingham: Parry, Soldatos (Finite elasticity, anisotropic media, models of crystals with defects.) Oxford: Ball, Capdeboscq, Chapman, Howell, Niethammer, Ockendon, Ortner, Suli (Nonlinear analysis. Multi-scale asymptotics. Inverse problems. Dislocation theory. Coagulation-fragmentation, coarsening. Plasticity. Computational modelling of fracture and quasi-continuum methods.) Strathclyde University: Osipov, Grinfeld, Lamb, MacKay, Mottram, Murdoch, Osipov, Sonnet, Stewart (Phase transition, liquid crystals and systems switching between bistable states, models of systems with chiral structure.) Surrey: Bevan (Nonlinear elasticity, calculus of variation) Sussex: Taheri (Nonlinear elasticity, calculus of variation) University College London : Smyshlyaev (Homogenization, waves) Warwick: Friesecke, MacKay, Theil (Nonlinear elasticity. Gamma convergence in multi-scale systems. Frenkel- Kontorova models for lattice systems. Elastoplastic microstructures)

The UK publications in theoretical solid mechanics are at the forefront of world class research. This research area has exciting prospects for the future, which include modelling of multi-scale smart materials characterized by unusual static and dynamic response, challenging problems in non-linear analysis and a wide range of interdisciplinary research involving UK industry. An outline of the main four research sub-areas, shown in Table of Section 2, is as follows:

Waves in solids, mathematical models of fracture, asymptotic analysis of multi-scale systems. This is a very active area of mathematical modelling, with strong activities in Aberystwyth, Bristol, Brunel, Imperial College London, Keele, Liverpool and Manchester. Highlights include development of novel methods of solution for commutative and non-commutative (n x n) matrix Wiener-Hopf systems and their applications (Abrahams [36]), wave propagation in pre-stressed composites (Parnell [37]), novel solutions for 3D edge plate waves problems (Zernov and Kaplunov [38], Fu, Brookes [39]), a new class of crack front waves in an anisotropic medium (Willis, Movchan [40]), analysis of a new class of matrix weight functions for interfacial cracks (Piccolroaz, Mishuris, Movchan [41,42]) , new models of Bloch waves and dynamic localisation in structured plates and periodic waveguides (Evans, Porter [43], Movchan, Movchan, McPhedran [44], Adams, Craster, Guenneau [45]), a new class of uniform meso-scale asymptotic approximations for solids with defects (Mazya, Movchan [46]). This area of research generates quality interdisciplinary projects involving mathematicians, physicists and engineers from UK, Europe and overseas. Most recent events include a research international programme of French Groupe de Recherche (GDR) on Etude de la propagation ultrasonore en milieux non homogenes en vue du controle non destructif, with the UK section coordinated by I.D. Abrahams (Manchester) and M.J. Lowe (Imperial College London); this successful programme has led to a series of workshops and large symposia between mathematicians, engineers and industrialists in the area of ultrasonic waves in non-destructive evaluation of materials. A major Marie Curie European four year programme on the modelling of adhesive joints in heterogeneous elastic solids was held in Liverpool, and stimulated efficient research interaction between mathematicians and engineers in UK, Poland, Italy, Israel; in particular this has led to a substantial advance in analysis of several classes of functional equations of the Wiener-Hopf type used in models of interfacial fracture and dynamic fracture in heterogeneous discrete systems (see, for example, [41,42,47,48]).

Nonlinear solids, phase transition, crystals with defects, homogenisation. This area of research is highly advanced in the UK, and the excellent Oxford group led by J.M. Ball is very influential. In addition to the interdisciplinary links with the physics and materials science this area includes a substantial amount of rigorous non-linear analysis, which makes this direction attractive to pure mathematicians. Some of the research highlights in this area include new variational models for solid-solid phase transitions with surface energy allowing for both smooth and sharp interfaces (Ball, Mora-Corral [49]), analysis of local minimizers and planar interfaces in a phase-transition model with interfacial energy (Ball, Crooks [50]), a hierarchy of plate models derived from nonlinear elasticity by Gamma- convergence (Friesecke, James, Muller [51]), novel solutions in cavitation problems in non-linear elasticity (Sivaloganathan [52]), analysis of singular minimizers of strictly polyconvex functionals (Bevan [53]), novel mathematical models for crystal structures with defects (Parry [54]), hyperelastic modeling of arterial layers with distributed collagen fibre orientations (Ogden [55]), post-bifurcation analysis of thin-walled hyperelastic tubes (Fu, Pearce, Liu [56]). Research in non-linear elasticity is supported by good research infrastructure, and one important highlight is the EPSRC-funded critical mass project OxMOS: New Frontier in the Mathematics of Solids, which strengthens the

75 mathematical underpinning of the modern materials science and supports research and educational links between mathematicians, materials scientists, engineers and biologists. A special mention should also be given to the research group led by R Ogden at Glasgow University, where Ogden and his colleagues have developed novel approaches to modelling of biological tissues based on modern methods of non-linear elasticity (from 2010 Ogden is also affiliated with the School of Engineering at the University of Aberdeen).

Plasticity, fatigue, dislocations, numerical modelling, atomistic simulations. Traditionally the bulk of research in this area is channelled through the engineering and materials science departments in UK universities. However, the interdisciplinary links have led to a range of challenging mathematical formulations, which have been successfully developed by applied mathematicians. Some of advances in this research area include continuum and discrete models of dislocation pile-ups (Ockendon [57]), analysis of particle-size segregation in dense granular avalanches (Shearer, Gray, Thornton [68]), a generalisation of the plastic potential model in the mechanics of granular materials (Harris [69]), localized direct boundary-domain integro-differential formulations for incremental elasto-plasticity of inhomogeneous body (Mikhailov [58]), computational modelling of thermoforming processes (Warby [59]), multiscale modelling of nano-indentation (Kenny [60]). The Oxford group led by J.R. Ockendon at Oxford Centre for Industrial and Applied Mathematics (OCIAM) and Oxford Centre for Collaborative Applied Mathematics (OCCAM) has developed a series of interdisciplinary research projects on mathematical models of dislocations which provide a strong link between applied mathematics and materials science.

Liquid crystals, bi-stable systems, fluid-solid interaction. This is a very successful research area which is rapidly expanding and attracting expertise of pure and applied mathematicians, physicists and engineers. The largest UK group developing the theory of liquid crystals is at Strathclyde University. Recent significant results on modelling of liquid crystals were also obtained in Oxford. Some of research highlights in this area include orientable and non-orientable line field models for uniaxial nematic liquid crystals (Ball, Zarnescu [61]), a substantial research monograph on the static and dynamic continuum theory of liquid crystals (Stewart [62]), modelling of reflection of light at structured chiral interfaces (Osipov, Bedeaux, Vlieger [63]), the study of a flexoelectric switching in a bistable system (Mottram, Davidson [64]). For studies of fluid-solid interaction it is important to mention the work of C.J. Chapman, the author of “High Speed Flow” [65], and collaborators at Keele; some recent examples of Chapman’s work include analysis of the forced vibration of elastic plates under significant fluid loading [66], and the study of energy paths in edge waves [67].

FLUID-SOLID INTERACTION

There is considerable UK activity in fluid-solid interaction, and much of it has been described in the separate Fluids and Solids sections above. However, it is worth reemphasising the very strong interplay between the two fields. Areas of activity include: flow in blood vessels and airways, in which the full coupling between the fluid and the collapsible elastic tube is crucial [19]; cell motility [20]; vibration of marine structures underwater [66]; remote sensing and ultrasonics, involving wave propagation in both the solid and the adjacent fluid media; properties of granular media; wave diffraction in both solids and fluids; and dynamics of liquid crystals.

4. Discussion of research community Please give a brief description of demographic trends, e.g. composition, origin and volume of research student pipeline, age distribution of academic staff, etc.

Inward recruitment of senior researchers. In recent years the UK theoretical mechanics community has been successful in attracting some outstanding overseas researchers into permanent jobs. From the US recent appointments include Ray Goldstein & Paul Linden to Cambridge, Alain Goriely to Oxford and Demetrius Papageorgiou to Imperial. Given the current funding outlook in this country, and adverse publicity over visas and pensions, there must be concern that any such inward flow is under threat. PhD recruitment. RAE2008 reported that the number of PhD studentships in applied mathematics rose gradually over the period 2001-2006. The information about how many of these studentships were held in theoretical mechanics was not available, although it is reasonable to assume that the proportion of studentships is at least in line with the proportion of staff entered in RAE2008, and may well be higher as a result of industrially-linked funding (such as EPSRC CASE). In seeking information about this document, however, a number of colleagues complained to us that current funding mechanisms do not allow the UK to attract the very best research students from around the world (for instance, due to EPSRC limits on full funding for only home students). There are at present very few ways of funding the best overseas students to come, and as a consequence the UK is at a significant disadvantage to our overseas competitors. We feel that this issue requires urgent attention. Age profile. RAE2008 identified 183.5 early career researchers in the applied mathematics unit as a whole, of whom about 40 work in fluid mechanics & astrophysics (i.e. 22% of the total, a proportion which is roughly in

76 line with the proportion of fluids/astro researchers in the whole submission) and about 9 in solid mechanics. This is a good position. By the same token, however, we can expect a significant number of retirements in the near future and, in common with other subject areas, there is considerable uncertainty about the level to which these individuals can be replaced in the current economic climate. PhD education. The UK community has done a great deal to improve the training of its PhD students since the last International Review, with the taught-course centre initiative proving very successful. There is also increased provision for training in scientific computation, but from the point of view of theoretical mechanics we feel that even more is required in this direction, especially given the centrality of computation in so many of the areas described above. The recent internship scheme, in which PhD students are offered three-month placements in industry as part of their training, is a very positive development, but we wonder if more can be done, including placements in appropriate laboratories overseas to enhance knowledge transfer into the UK (including in scientific computation).

5. Cross-disciplinary/Outreach activities: e.g. describe connections with other areas of mathematics, science, engineering, etc.

New technological challenges and recent funding initiatives from EPSRC and other research councils and funding agencies have stimulated interdisciplinary research in a number of areas, including: modelling of nano-scale composite materials and complex media such as liquid crystals; smart mechanical structures which possess specially designed properties for static as well as dynamic responses; dynamic meteorology; astrophysics; and fluid mechanics in biology and medicine. Applied mathematicians working in continuum mechanics contribute significantly to these exciting areas of research. A major £3M EPSRC research award was allocated to support a research centre in nonlinear partial differential equations at Oxford, and challenging non-linear problems in materials science have been included in the research programme of this centre. A highly successful multi-million pound initiative has led to creation of Oxford Centre for Collaborative Applied Mathematics, which supports new exciting interdisciplinary projects, including those in fluid and solid mechanics and materials science and involving applied mathematicians, physicists and engineers. The Spencer Institute of Theoretical and Computational Mechanics at Nottingham University has been created to encompass novel interdisciplinary projects in continuum mechanics, and one of its exciting directions involves mechanics of multi-phase media. A major platform grant on Ultrasonic Waveforms was awarded to a group of researchers from Strathclyde University; as quoted in the abstract of this project, it is aimed at ``fundamental and broad reaching investigation into the design of new coded sequences, transducers and constituent materials, allied with the concept of electronic system and transducer design to maximise signal to noise ratio over extremely wide bandwidths’’. Very recently, a Mathematics Platform Grant (“MAPLE: MAthematics PLatform Engagement activity”) has been awarded to Manchester to underpin interdisciplinary activity, including in continuum mechanics. In addition to major funding on the scale of platform grants, individual applied mathematicians research groups have secured, on a regular basis, research funding for novel interdisciplinary projects in nonlinear analysis in materials science, phase transition problems, homogenization approximations and wave propagation in micro-structured solids, modelling of thin film structure and flows, and industrial and environmental fluid mechanics. Examples include Oxford, Cambridge, Bath, Warwick, Imperial, UCL, Keele, Manchester, Liverpool, Nottingham, Brunel, Glasgow, St Andrews and Birmingham, but successful applied mathematics groups working in interdisciplinary theoretical mechanics are to be found throughout the UK. In short, we feel that the theoretical mechanics community in UK mathematics is exceedingly effective in interdisciplinary research.

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77 [15] Chardard F., Dias, F. & Bridges, T.J. 2009 Proc Roy Soc Lond, 465, 2897-2910. [16] Korobkin AA, Ellis AS & Smith FT 2008 J. Fluid Mech. 611, 365-394. [17] Craster RV et al. 2010 Wave Motion 47(5), 299-317. [18] McIver P & McIver M 2006 J. Fluid Mech., 558, 53-67. [19] Stewart, P.S., Heil, M., Waters, S.L. & Jensen, O.E. 2010 J. Fluid Mech. 652, 405-426. [20] Goldstein RE et al 2009 Physical Review Letters 102, 168101 [21] Turner M.R., Bassom, A.P. & Gilbert A.D. J. Fluid Mech., 638, 49-72. [22] Dritschel DG et al 2009 J. Fluid Mech. 640, 215-233. [23] Gibbon J.D. 2008 Physica D, 237, 1894-1904. [24] Howison SD, Ockendon JR & Oliver JM. 2002 J. Engng Maths 42, 373-388. [25] Breward CJW & Howell PD 2002 J. Fluid Mech. 458, 379-406. [26] Villegas-Diaz M, Power H, Riley DS 2005 J. Fluid Mech. 541, 317-344 [27] Heap A. & Juel A. 2008 Phys. Fluids 20, 081702. [28] Kelmanson, MA et al 2006 J.Rheology 50(2), 207-235. [29] Chiu-Webster S & Lister JR 2006 J. Fluid Mech 569, 89-111. [30] Crowdy DG & Or Y 2010 Phys. Rev. E 81(3), 036313. [31] Staron L & Hinch EJ 2005 J. Fluid Mech., 545,1-27. [32] Healey JJ 2009 J. Fluid Mech 579, 29-61. [33] Shikhmurzaev, Y. D. 2007 Capillary Flows with Forming Interfaces. Taylor & Francis. [34] Wang, Q.X. & Blake, J.R. 2010 J. Fluid Mech. 659, 191-224. [35] Akhtar N, Decent SP & Kendall K. 2009 Int. Hydrogen Energy. 34, 7807-7810. [36] Abrahams ID 2007 Proceedings of the Royal Soc London A, 463, 613-639. [37] Parnell WJ 2007 IMA J. Applied Mathematics, 72(2), 223-244. [38] Zernov V, Kaplunov J 2008 Proceedings of the Royal Soc London A, 464, 301-318. [39] Fu YB, Brookes DW 2006 Journ. of the Mechanics and Physics of Solids, 54, 1-21. [40] Willis JR, Movchan NV 2007 Wave Motion, 44(6), 458-471. [41] Piccolroaz, A, Mishuris, G, Movchan, AB 2007 Journ. of the Mechanics and Physics of Solids, 55, 1575-1600. [42] Piccolroaz, A, Mishuris, G, Movchan, AB 2009 Journ. of the Mechanics and Physics of Solids, 57, 1657-1682. [43] Evans DV, Porter R 2007 J. Eng. Math. 58, 317–337. [44] Movchan AB, Movchan NV, McPhedran RC 2007 Proceedings of the Royal Soc London A, 463, 2505-2518. [45] Adams SDM, Craster RV, Guenneau S 2009 Waves in Random Complex Media, 19, 321-346. [46] Mazya V, Movchan A 2010 Math. Nachr. 283(1), 104 – 125 [47] Misuris GS, Movchan AB, Slepyan LI 2008 Quart J Mechanics Appl Math, 61(2), 151-160. [48] Misuris GS, Movchan AB, Slepyan LI 2009 Journ. of the Mechanics and Physics of Solids, 57, 1958-1979. [49] Ball JM, Mora-Corral C 2009 Communications on Pure and Applied Analysis, 8, 55-81. [50] Ball JM, Crooks ECM, 2010 Calculus of Variations and Part. Diff Equations. DOI: 10.1007/s00526-010-0349-8 [51] Friesecke G, James RD, Muller S 2006 Arch Rat Mech Analysis, 180, 183-236. [52] Sivaloganathan J 2006 SIAM J. Applied Mathematics, 66, 736-757. [53] Bevan J 2005 Calculus of variations and partial differential equations, 23, 347-372. [54] Parry G 2009 J Elasticity, 94, 147-166. [55] Ogden RW, Gasser TC, Holzapfel GA 2006, J Royal Soc Interface, 3(6), 15-35. [56] Fu YB, Pearce SP, Liu KK 2008 International Journal of Non-linear Mechanics, 43, 697-706. [57] Ockendon JR, Allwright DJ, Voskoboinikov R, Chapman SJ 2007, Journ. of the Mechanics and Physics of Solids, 55, 2007-2025. [58] Mikhailov S 2006 Engineering Analysis with Boundary Elements, 30, 218-226. [59] Warby M 2006 Computer Meth Appl. Mechanics and Eng., 195, 5220-5238. [60] Kenny SD 2007 Int J Materials Research, 2007(5), 430-437. [61] Ball JM, Zarnescu A 2008, Molecular crystals and liquid crystals, 495, 573-585. [62] Stewart I 2004 The Static and Dynamic Continuum Theory of Liquid Crystals, Taylor and Francis. [63] Osipov MA, Bedeaux D, Vlieger J 2004 J. Optical Soc America A, 21, 2431-2441. [64] Mottram N, Davidson A 2002 Phys. Rev E, 65(5), 051710. [65] Chapman CJ 2000 High Speed Flow, Cambridge University Press. [66] Chapman CJ, Sorokin SV 2005 J Sound and Vibration, 281, 719-741. [67] Chapman CJ 2001 J Fluid Mech., 426, 135-154. [68] Shearer, M., Gray, J.M.N.T. & Thornton, A.R. 2008 Eur. J. Appl. Math. 19, 61-86. [69] Harris D. 2009 Int. J. Eng. Sci., 47, 1208-1215

78 INTERNATIONAL REVIEW OF MATHEMATICS MATHEMATICS IN BIOLOGY & MEDICINE LANDSCAPE

Lead Authors: Philip K. Maini and Mark A.J. Chaplain Contributors/Consulted: The authors of this document were assisted by numerous contributions from the research community.

1. Statistical Overview: In compiling this document we focussed on advances in the field over the past 5 years. It proved impossible to collect reliable data in the time available. A particular problem in mathematical biology is that a significant number of UK mathematicians in mathematical biology also work in other areas of mathematics and this can result in misleading statistics. It appears that the number of mathematicians in the subject has grown substantially in the past 5 years, with several universities building large groups in the area, but is perhaps not keeping pace with the growth in the number of graduate students moving into mathematical biology. The subject attracts young people (most faculty being under 45) and the number of women in the area is higher than average.

2. Subject breakdown: Data taken from RAE 2008 modified by information received from colleagues suggest that there are 120 full time faculty in mathematical biology. Please note the unreliability of this datum as indicated in 1. Also, we have not broken down this number into different areas because many researchers span a number of research areas within mathematical biology as well as in other fields in mathematics.

3. Discussion of research areas: Mathematical Biology has grown enormously in the UK since the last IRM with the establishment by the BBSRC and EPSRC (funding around 8-9 million pounds each) of 6 systems biology centres in the UK (Edinburgh, Imperial, Manchester, Newcastle, Nottingham, Oxford). These centres are highly interdisciplinary and mathematicians are actively involved. The University of Warwick funded its own systems biology centre (10 million pounds) with the appointment of four new mathematics faculty. The Institute for Complex Systems and Mathematical Biology at Aberdeen has some 8 million pounds of outside funding in collaboration with their Institute of Medical Sciences. In addition there are several Doctoral Training Centres (over 10) funded by BBSRC and EPSRC designed to train graduate students in interdisciplinary mathematical biology (in its broadest terms). Recently, the EPSRC funded the continuation of some of these programs at a postdoctoral level (approx 10 million pounds) in which universities were required to collaborate with each other and industry.

The strength of mathematical biology in the UK has been recognised through a number of key elections and prizes. These include 2 recent presidents of the IMA (Grindrod (Reading), Pedley (Cambridge)), President of the Society for Mathematical Biology (Chaplain (Dundee)), Chair of the 2008 RAE Applied Mathematics sub-panel (Pedley), 2 ERC Advanced Investigator Awards (Chaplain (Dundee), Goldstein (Cambridge)), one ERC starting grant (Erban (Oxford)), 3 EPSRC Advanced Fellowships (Bees (Glasgow), Rand (Warwick), Twarock (York)), 2 Royal Society Research Fellows (Howard (Systems Biology, John Innes), Roose (Engineering, Southampton)), 2 Leverhulme Trust Fellowships (White (Bath), Chaplain), CBE (Grindrod), IMA Gold Medal (Pedley), 2

79 Junior Whitehead Prizes (Sherratt (Heriot-Watt), Owen (Nottingham)), Adams Prize (Sherratt), Julian Cole Prize (King (Nottingham)), Naylor Prize (Maini (Oxford)). It is also in evidence by the large number of invited conference plenaries (for example, Byrne (Nottingham) SMB 2007, Coombes, SIAM Annual meeting 2009, Jensen (Nottingham) Euromech Fluid Mechanics 2010, Chaplain ECMI 2010) talks and graduate school courses given. Moreover, UK mathematicians are increasingly involved in co-authoring papers in high impact journals such as Nature, Cell, Science and PNAS as well as being invited to write perspectives in journals read by biologists and medics.

Mathematical Biology is very popular at the undergraduate level and the majority of UK mathematics departments now have courses in the subject, although most of these are at the higher level and therefore students are exposed to the area late on in their undergraduate career. While the number of faculty appointments has grown in the area, it lags behind the growth in the subject area resulting in large supervision loads on faculty. The rather strict departmental structure of universities means that researchers are, by and large, judged either by their mathematical or their biological contribution, when it would be more appropriate for them to be judged by their mathematical biology advances. This can pose problems when young researchers are searching for academic positions within the UK or can harm promotion prospects of faculty members. Moreover, it pressurises doctoral students who are expected to finish their doctorates in the same time as their single discipline peers. The move towards the setting up of large centres funded by the BBSRC and EPSRC is going some way towards breaking down these departmental barriers.

UK mathematicians have made significant contributions to areas in mathematical biology too numerous to detail here, so we briefly highlight only some areas.

Bacterial Research: The recent appointment of Goldstein to Cambridge from Arizona has led to the establishment of a flourishing research group (5 post-doctoral fellows and 2 research students as well as an active laboratory). Experiments and theory advance together, focussing on physical aspects of multicellularity: individual and collective dynamics, mixing and transport (e.g. quorum sensing) in bacterial and algal suspensions. Ward (Loughborough) and King (Nottingham) have used a continuum framework and asymptotic methods to model and analyse bacterial infection and biofilm development (in collaboration with the Inst. of Infections and Immunity, and also the Dept of Genetics, Nottingham). Tindall (Reading) and Maini (Oxford), in collaboration with colleagues in Biochemistry at Oxford, used a reaction-diffusion model to identify a new signalling pathway in R. sphaeroides which was then identified experimentally. Howard (John Innes), in collaboration with cell biology colleagues at Newcastle, has uncovered the mechanism regulating low copy number plasmid segregation in bacteria. Grebogi (Aberdeen) and colleagues mapped the spatial distribution of bacterial mechanosensitive channels onto an Ising-like model to show that the aggregation of channels and the consequent interactions among them leads to a global cooperative gating behaviour with potentially dramatic consequences for the cell.

Biological Fluid Dynamics: Blake (Birmingham) and Gaffney (Oxford) have presented the first detailed explanation for the attraction behaviour of sperm near surfaces and found the first localised estimates of energy transport requirements in the sperm flagellum. This work is in collaboration with the Birmingham Women's NHS Fertility Centre and has attracted a lot of international media attention. The group at Cambridge under Pedley continues a very active program in biofluid dynamics. Luo (Glasgow) studies computational methods applied to fluid-structure interactions, flow in collapsible tubes, two-phase flow and biofilms. Bees and Hill (Glasgow) are carrying out theoretical and experimental investigations of pattern formation by suspensions of micro-organisms

80 (bioconvection). Please and Fitt (Southampton) and the Oxford group (funded by KAUST) are working on tear films and interior flows of eyes.

Cancer: Chaplain (Dundee) and Sherratt (Heriot-Watt), as well as Byrne and Owen (Nottingham), Alarcon (formerly Imperial) and Maini (Oxford) have developed the first models for tumour-induced angiogenesis that couple blood flow with capillary growth, include structural adaptation and link intercellular events at the molecular level through to tissue level events all in one framework. The latter group has established an experimental collaboration with the Moffitt Cancer Research Institute in Tampa to validate model predictions. Byrne, King. Jensen (Nottingham) and Chapman, Waters (Oxford) have been involved with colleagues in the Computing Lab in Oxford in developing Chaste, a general purpose simulation package aimed at multi-scale, computationally demanding problems arising in biology and physiology with particular application to colorectal cancer. This has also been used to address fundamental issues on how to derive continuum descriptions at the cell-level from individual-based models with a view to comparing different models at the cell level.

Webb (Strathclyde), in collaboration with Owen and Byrne and experimentalists at Sheffield, has developed novel PDE models that provide insights into the mechanisms controlling monocyte recruitment by tumours and their subsequent localisation in hypoxic regions. Gourley (Surrey) has analysed nonlinear equations for cancer invasion. Antal (Edinburgh), a recent appointment from Harvard, has developed quantitative models (Wright-Fisher process) for the progression to the malignant state. Ward (Loughborough) has been investigating regulation and growth of uterine myomas in collaboration with mathematical colleagues in Taiwan and the Kaohsiung Veterans Hospital (Taiwan).

Cardiac: Luo (Glasgow) and colleagues at NYU have applied novel computational immersed boundary methods to simulations of heart valves to show that the valve and chord bending stiffness is essential for complete valve closure. Simitev (Glasgow) and Biktashev (Liverpool) have developed a generic procedure for the asymptotic reduction of ionic cardiac models and tested it on several important applications. This is very significant as it has the potential to dramatically reduce the computational time for simulating large scale cardiac models. Chaste is being used by numerous cardiac modelling groups worldwide.

Ecology: McNamara (Bristol), in collaboration with Biology and the Vet School at Bristol, has extended game theoretical notions to include differences between individuals to show that this can fundamentally affect the outcome of a game, a result that has widespread application in behavioural ecology. Morozov and Petrovskii (Leicester), with colleagues in France and Germany, have studied reaction-diffusion waves and introduced a new conceptual mechanism for fat-tail formation. Speirs and Gurney (Strathclyde) have obtained a novel solution to the "drift paradox" in aquatic ecology and, in collaboration with Marine Scotland, developed the first spatially and physiologically structured model of an ecologically important marine zooplankton species (Calanus finmarchicus) throughout its geographic range. They also developed a new approach to modelling length-structured fish communities, demonstrating a commercially important trade-off between pelagic and demersal fisheries.

Epidemiology: Adams (Bath) and biologists in Hawaii have investigated the role of human movement in the epidemiology of vector-borne pathogens such as dengue. Hilker (Bath), with colleagues in France and Germany, has shown that a strong Allee effect may lead to complex dynamics in simple epidemic models, which may have profound implications for pest management and biological conservation. Britton (Bath) in collaboration with biologists at Bristol has developed the concept of strategy blocking to explain why certain organisms fail to employ a defence that seems obvious. White (Bath)

81 and biologists in Holland have modelled HIV strain heterogeneity obtaining new results on the transmission of drug-resistant strains to other implications suggesting changes in clinical practice. Greenhalgh (Strathcylde) has developed state-of-the-art models for the spread and control of infectious diseases such as HIV/AIDS, mumps and rubella, Bovine Respiratory Syncytial Virus (with colleagues in Holland), mumps and rubella, hepatitis C and pneumococcus. He collaborates closely with experimentalists and medics (Wyeth Pharmaceutics and Health Protection Scotland). Muldoon (Manchester) has played a long-term role in the development of the so-called mosaic vaccine antigens for HIV. Gog (Cambridge), in collaboration with experimentalists in Virology at Cambridge and the Institute Pasteur (Paris), has used bioinformatics techniques to predict locations of packaging signals in influenza A and then test the results experimentally.

The group at Heriot-Watt has developed a novel extension of adaptive dynamics that predicts the level and variation of host resistance to infectious disease and the mechanism used by parasites to counter host resistances. The results are in close agreement with laboratory findings on bacteria-phage co-evolution. They have also shown that a wave of disease can spread through a native population well in advance of any competition between species. This result has critical implications for the conservation of native populations. Bowers and Sharkey (Liverpool), in collaboration with biologists and vets, have modelled avian influenza in British poultry, and Salmonella and E. coli in British diary herds, as well as advancing the general theoretical aspects of epidemic modelling using network-theoretic approaches. Ward (Loughborough), with colleagues at the Inst of Animal Health, Pirbright, Surrey, has been modelling foot and mouth disease pathology and vaccination. Kiss and colleagues (Sussex) have used the interface of graph/network theory and modelling of livestock diseases in collaboration with DEFRA to obtain results which are now widely used for both practical and theoretical problems of disease transmission control.

The group at Stirling has made theoretical advances in the use and analysis of PDEs in structured population dynamics, the use of adaptive dynamics in studying evolution, the combination of epidemiological models and economics and the use of process algebra to model epidemiological systems. They have derived important results for tick borne diseases and plant pathogens and collaborate closely with the School of Biological and Environmental Science, the Inst of Aquaculture, Economics Division, Management School and Dept of Nursing and Midwifery in Stirling. Gourley (Surrey) has investigated the structured host population dynamics of various diseases and modelled migratory birds and their role in the spread of avian influenza. Recently he has been involved in modelling how to slow the evolution of insecticide resistance in mosquitoes by more effective use of existing insecticides which is currently a hot topic in malaria control.

Evolution: Beardmore and Gudelj (Imperial) have developed models of co-evolution. Bowers (Liverpool), Hoyle (Stirling) and White (Heriot-Watt), in collaboration with biologists, have developed a qualitative "geometric" method of analysis of the adaptive evolutionary dynamics using trade-off and invasion plots. Broom (Sussex) has developed kleptoparasitism and multi-player games models which are of pratical interest to empirical biologists as well as theoreticians. Grebogi (Aberdeen) used multi-layered network theory to obtain the surprising result that intraspecies infection can strongly promote biodiversity while interspecies spreading cannot.

Immunology: The Leeds' group has developed the first model of T~cell homeostasis to provide quantitative estimates for the times to clonal extinction and this work has stimulated experimental immunologists from UCL and the Pasteur Inst to test and

82 validate the model. Maree (John Innes) has made important breakthroughs in understanding immune cell migration. Twarock's group (York) has shown, in iterative interdisciplinary research with colleagues at the Astbury Centre for Structural Molecular Biology (Leeds), that the viral genomic RNA plays important cooperative roles during virus self-assembly that have been neglected in the previous protein-centric approaches. They have also shown that there is a previously unrecognised scaling principle in virology that constrains the shapes and sizes of all viral components collectively. There is a large group in Warwick addressing many issues in immunology.

Molecular Level: Classification of biological function of proteins according to their translational dynamics (Grebogi and colleagues (Aberdeen)). They also formulated a mathematical model of the dynamics of DNA damage and repair which predicted a counter-intuitive relationship between survival and repair. It allowed the discrimination between two phases, which were tested experimentally with E. coli cells. Buck (Imperial) in collaboration with molecular biologists in the US and UK has developed a model of site-specific recombination, and successfully tested it in vitro. In collaboration with biologists in Montreal, Shahrezaei (Imperial) has found the molecular mechanisms of decision making in yeast cells. Together with the structural bio-informatics group in Copenhagen, the group at Leeds has published the first probabilistic model of local protein structure based on a new statistical distribution for conformational angles (TorusDBN) which will be used in Bayesian Protein Design. The Computational and Systems Biology department at the John Innes Centre has contributed to breakthroughs involving novel geometrical approaches for global genetic mapping (Dicks), and a greater understanding of the evolutionary signatures of secondary metabolic gene clusters (Dicks). Broomhead and Muldoon (Manchester) collaborate with one of the world leading experimental groups (White (Liverpool)) on the NfkappaB cell signalling pathway, an area in which the Warwick group also works. There has also been the establishment of the StoMP network, which comprises experimentalists and theoreticians interested in stochastic dynamical modelling of gene regulation.

Networks: The Aberdeen group used the theory of modular networks for the description of eukaryotic adaptive cell cycle control to unify various experimental paradigms in molecular cell biology. Popovic (Edinburgh), in collaboration with the Centre for Systems Biology (CSBE) at Edinburgh, is investigating noisy intracellular metabolic networks. Millar (CSBE) has used mathematical models to investigate why the genetic networks generating circadian rhythms are so complex and not simply composed of one negative feedback. Using in silico evolution with fluctuating light signal his group has shown that the complexity of the clocks can potentially be explained by the need to generate rhythms with a robust period despite seasonal (deterministic) and daily (stochastic) variations in light intensity. MacArthur (Southampton) works on gene networks in stem cell fate and cellular reprogramming while the pure mathematicians in Southampton are using graph theory on DNA problems in collaboration with biologists. Higham (Strathcylde), Spence (Bath) and the Beatson Institute for Cancer Research are developing models and algorithms for complex biological networks. Higham is analysing data sets obtained by neuroscientists at the Oxford Centre for Functional Magnetic Resonance Imaging of the Brain and has produced some of the first quantitative evidence in favour of the hypothesis that the adult brain can adaptively and globally re- wire itself after trauma. Bristol hosts the Synthetic Components Network, jointly led by Rossiter (Eng. Maths) and Woolfson (Biochemistry) and have been very successful at the IGEM annual synthetic biology competitions at MIT, last year winning a gold medal and a prize for the best model; the team of advisors included faculty from Eng. Maths, Biochemistry and Biological Sciences.

83 Plants: The group at Cambridge has strong collaboration with Plant Sciences on cytoplasmic streaming in plant cells. The analysis of properties of genes in various plant species by Tatarinova (Glamorgan) and colleagues has provided major insight for tissue specific and stress response in plants and resulted in patents in areas of plant biotechnology. Grieneisen/Maree (John Innes) have made significant breakthroughs in explaining the dynamics of the auxin morphogen gradient in roots and uncovering the link between mechanical stress and auxin flow. King and Owen (Nottingham) have made predictions on auxin redistribution in the root meristem during gravitropism.

Pattern Formation: Madzvamuse (Sussex) and Gaffney (Oxford) have shown, using a non-standard linear analysis on a non-constant coefficient system, that domain growth can drive spatial patterning without the restriction of short-range activation, long-range inhibition. Winter (Brunel), in collaboration with colleagues at the Chinese Univ of Hong, has used the theory of nonlinear PDEs, functional analysis and numerical simulation to derive novel results for the existence and stability of multiple spike solutions for classes of reaction-diffusion systems. Painter and Sherratt (Heriot-Watt) have developed a new integrodifferential equation model for cell adhesion. Painter, with colleagues in Alberta, has developed the first continuum model for chemotaxis that incorporates tissue volume effects, and the first demonstration of spatiotemporal chaos in chemotactic equations. The team at Heriot-Watt has also come up with a new combined experimental/theoretical approach to elucidate the synthesis of macroscopic and microscopic patterning during embryonic development. Sherratt is the first to apply the theory of absolute instability to biological models, providing important new insights into spatiotemporal patterning in biology. He has also undertaken the first detailed study of a model for vegetation patterns in arid environments, leading to the novel prediction of hysteresis in pattern wavelength as rainfall varies. Chapman and Erban (Oxford) have shown the profound effects that stochasticity can have on the behaviour of traditional reaction-diffusion models. Howard (John Innes) has investigated the effects of noise in gradient formation. Monk (Nottingham) has demonstrated how multiple levels of feedback enhance robustness of patterning in Drosophila.

Other: There is also major activity in neuroscience (for example, Bressloff (Oxford), Coombes (Nottingham), Broomhead (Manchester), Higham (Strathcylde), Warwick group) and biomechanics and tissue engineering (Heil (Manchester), Cambridge, Nottingham, Oxford, Southampton and Strathcylde groups)

4. Discussion of research community: Please see answer to question 1.

5. Cross-disciplinary/Outreach activities: By its very nature, this subject includes many mathematical areas being covered in other landscape documents, and as illustrated above, a vast amount of the activity is intimately linked with other departments outside mathematics. Some universities, for example, Southampton, are making full-time joint appointments such as between mathematics and medicine. It is important to note that a significant amount of the UK's activity in mathematical biology takes place in other departments (Biochemistry, Biology, Engineering, Medicine, Physics, Plants, Zoology) but we have, in the main, restricted ourselves in section 3 to research advances which have actively involved mathematicians from mathematics departments. This does not do full justice to the subject area as a whole. For example, much of government policy is informed by mathematical biology carried out by researchers who are not in mathematics departments; insights into the cell cycle, arguably to date the most successful study in systems biology, is based on mathematical modelling.

84 INTERNATIONAL REVIEW OF MATHEMATICS INDUSTRIAL MATHEMATICS LANDSCAPE

Lead Author: Prof Colin P Please Contributors/Consulted: Details at the end of the document

1. Introduction

Industrial Mathematics is defined in a European context (ref[1]) as the mathematical sciences considered in their broadest sense, when applied to any activity of economic or social value regardless of whether it is in the public or private sector. Hence research in this area overlaps with and interacts with many more tightly focussed interdisciplinary research areas including statistics, operational research, scientific computing, computer science, as well as math-biology math- medicine and math-finance. However, the borders of the area are necessarily fuzzy and highly porous to ideas and researchers. This document attempts to take a broad view of the topic, but emphasises those aspects not covered by other landscape documents (particularly Statistics, Mathematical Aspects of Operational Research, Mathematics in Biology and Medicine, and Financial Mathematics).

2. Statistical Overview

Data from Universities on the amount of industrial mathematics being done is difficult to extract due to the fuzzy borders described above and the differences between mathematics departments, particularly whether they include statistics, operational research or computer science groups. Using direct data from a relatively small sample of a dozen Universities with significant industrial mathematical activity I conclude that approximately 20% of the academic staff in UK mathematics departments are involved in Industrial Mathematics. Looking at the individual subjects that occur in the International Review of Mathematics and additionally guided by interests shown in the activities of the KTN in Industrial Mathematics, the largest percentage of staff involved with industry is in Operational Research and Statistics, then Applied Mathematics and Computer Science and only small fraction in Pure Mathematics.

3. Discussion of research areas

Industrial Mathematics has experienced a huge growth in activity both in the UK and internationally since the previous MSIR in 2004. This expansion of mathematical research, industrial interaction and economic impact has been chronicled in a number of influential documents. The international perspective is clearly given in the extensive review of Industrial Mathematics by the OECD (this review is contained in ref[2], which gives a detailed overview, and ref[3] that outlines routes for managing these activities). The final details of the extensive European assessment of Industrial Mathematics for the ESF being undertaken by the Applied Mathematics Committee of EMS is expected in December 2010 (ref [1]). A review focussed on UK activity is given in ref[4] and demonstrates the breadth of interaction between mathematics and industry. In all these documents the UK industrial mathematics community is identified as leading the way through its style of interaction, its mechanisms of engagement as well as the quality of its research outcomes. The role of industrial mathematics in giving clear, accessible and quantifiable examples of the impact of mathematical research on problems of practical importance to society is clearly demonstrated by the compiled sets of case studies being collected by the ESF (ref[5]) and the IMA(ref[6]). The UK style of industrial mathematics, outside the other landscape topics, is interdisciplinary and problem-driven and it has developed a skill base and an approach to problem solving that has made the UK the world leader in such activities (ref[4]). This style of Industrial mathematics has a large element of model building. Developing such mathematical modelling engenders a holistic approach to problems, seeking to identify and apply the most appropriate modelling methodologies. As a result industrial mathematics draws on the full breadth of mathematical techniques from continuum

85 to discrete, from deterministic to stochastic, and from building simple models to sifting large data sets. Its contribution to motivating and analysing new mathematical ideas is significant while it also has direct impact as a result of being industry-driven. The training of postgraduates and postdocs in this style has led to significant influence on a large spectrum of disciplines, as industrial mathematicians are naturally attracted to interdisciplinary research both in academia and industry.

The developments in industrial mathematics rely on four pillars namely i) leadership ii) academic intellectual capacity, iii) academic-industrial connections, iv) effective and efficient interaction mechanisms. The UK academic base in this area has great strength in all these pillars and has the capability to retain its internationally leading position.

Leadership: The international and national growth of this area has been led by key researchers from many universities commonly acting through centres funded by various mechanisms such as Universities or EPSRC (eg: OCIAM (Oxford), CICADA (Manchester), BCANM (Bristol), SIMM (Southampton), CMMB(Nottingham), BICS (Bath), RCMM (Liverpool) ) and by regional initiatives (eg: GWR (Great Western Research)). The common philosophy to using industrial problems and interaction to motivate mathematical research is seen as central to their success. Continuing support for these focal points of leadership is critical to growing and sustaining the current level of research activity. Several key leaders in the area are retiring but there is a depth of senior academics in the area who are ready to step up to keep the momentum of the expansion in the subject area.

Recent evidence of the international status of UK industrial mathematics is the central role this played in convincing KAUST to set up OCCAM, a $25million research centre in the UK in 2008. The rapid expansion of this centre has created a unique opportunity for industrial mathematics drawing together an array of international talent, exploiting new industrial connections, and creating a critical mass of activity. Identifying methods for the long-term support of these new initiatives will be key to consolidating this enhanced activity.

Academic-industrial connections: There are major connections between industry and academia, many of which have been informally generated through personal contacts. The great achievement of all the previously mentioned centres has been to facilitate the marriage of problems and mathematicians across the vast range of industrial research. In addition to these local initiatives the role of the KTN in Industrial Mathematics in creating such interactions has grown significantly and provided a highly effective national focus to industrial mathematics (see more details in ref [7]). This KTN activity was recently cited by the President of ECMI (Wil Schilder) as a paradigm for all the countries in Europe.

Academic intellectual capacity: Central to ensuring that industrial engagement occurs and that good mathematical ideas are produced from these interactions is the quality of the academics who participate in the process (faculty, graduate students but especially postdocs). For traditional mathematicians to become industrial mathematicians requires they become, to a certain extent, polymaths. Evidence for the UK strength in this respect is the extensive international requests for these industrial mathematicians to participate in workshops and study groups, and the interest shown by industry in being involved. There is a need to grow the academic capacity to respond to these demands.

Effective and efficient interaction mechanisms: Creating and nurturing interdisciplinary collaboration requires time to develop relationships, skill to interpret the different styles and languages, and for both industrialists and academics to be motivated and resourced to bridge such gaps. A major hurdle is that companies and particularly SMEs have insufficient trust or experience of academic research, particularly in mathematics. Use of academic effort has previously been considered by much of the mathematical community as a low quality aspect of research activity. However, with the increase in the need to demonstrate impact and the desire to exploit the success of research resulting from these interactions there is increasing interest in how these can be better

86 managed and promoted. Mechanisms that have been proven to be very effective include focussed workshops, MSc projects, PhD projects, study groups and internships.

MSc and PhD students are a very effective way to create highly collaborative research teams between industry and academia but these mechanisms have changed significantly in the review period. The general removal of EPSRC support for MSc courses, including those at Bath, Oxford, Loughborough, Southampton and Reading, has reduced the opportunities for diverse and opportunistic connections to be explored with industry. In addition the decline and restructuring of CASE awards, which previously funded about 30 mathematics PhD students annually, has reduced the opportunities for longer term collaborations. In contrast the ECMI initiative that is extending the number of European Masters degrees in industrial mathematics demonstrates how Europe has recognised the need to train students to realise the societal value of their mathematical skills and to do this across country boundaries. The UK needs to address how to retains its strength at the MSc level and ensure prospective UK and European students can gain some or all of their qualifications here in the UK.

The great success of the Mathematics internship programme, supported by EPSRC and managed by the KTN for Industrial Mathematics, has shown the depth of interest from industrialists, academic and graduate students for such interactions. The Study Group has proved to be one of the most successful mechanisms for generating interactions and use of these has grown from its UK base to a major international activity as well as being adopted as a successful mechanism by other interdisciplinary mathematical areas including Biology (eg: Mathematics in Plant Science) and Medicine (eg: the very successful MMSG). Since the 2004 besides annual UK meetings there have been a total of 65 study groups worldwide with central involvement of UK researchers in each of these. Their impact can be appreciated by noting that in 2010 UK industrial mathematicians will be key in doing research at study groups in 14 countries overseas each connecting about 50 local industrial mathematics academics with approximately 6 local industrial companies. Internationally study groups have proved a very effective mechanism for training postgraduate students and this process has been enhanced by specific graduate training study groups with UK academics sought to mentor at many of these meetings. There are increasing numbers of European universities where participation in workshops or study groups is part of a Masters degree in industrial mathematics.

Creating effective collaborative research interactions with industry requires full engagement between the different groups and this is best enabled by movement of people. Getting senior mathematical staff to transfer across is very difficult and almost impossible from industry into academia, although there are some successful examples through industrial fellowships and industrially funded positions. The mechanisms that use the more mobile postgraduates are however highly effective and relatively inexpensive. Support for these, including the mechanisms outlined above, should continue but there is a need to consider how to enable processes that can provide fast-track support to extend those engagements that reveal the possibility of long term collaboration.

4. Discussion of research community

There has been good support for industrial mathematicians at senior levels with fully-funded academic positions. The main issue is the small numbers of high-quality candidates available for such posts. Many of these positions are in Mathematics departments but some industrial mathematicians have moved into departments of other disciplines as their interests specialise. At postdoctoral level the funding is much more disperse with direct contribution from EPSRC mathematics being a small but growing fraction. A significant fraction of industrial mathematicians gain their postdoctoral experience within other disciplines. From limited direct data it appears that approximately half of graduating PhD industrial mathematical students pursue postdoctoral research and follow an academic career while most of the other half go into a very wide range of industries At PhD level student numbers are good. There is a significant hazard that as funding for collaboration between academia and industry is tightened the role of mathematics may be squeezed and

87 subsumed into other disciplines. This could greatly reduce the unique benefits gained by engagements that draw on the breadth of mathematical approaches, ideas and conceptual frameworks.

Central to ensuring that such industrial collaborations thrive is to ensure that new bright mathematicians, at an early stage in their career, are inspired by the collaborative atmosphere of the area and trained in effective interaction mechanisms. Such training requires experience of a vast range of interdisciplinary engagements. Recognition and appreciation from the wider community of the quality of the research resulting from these activities has been difficult as it spans so many different topics.

5. Cross-disciplinary/Outreach activities 5.1 Scientific Issues UK industrial mathematics is at the cutting edge of dynamic and efficient quantitative cross- disciplinary research. It has explored and identified unique mechanisms that are particularly effective at the interface of mathematics with other disciplines. This activity has a forty-year track record of uncovering and then pursuing new interfaces where mathematical research can have a major impact through more general cross-disciplinary collaborations. It has shown how to bring a focus where there was previously no coherence within the mathematical community or to the impact it could make. Examples of such previously identified new interfaces include mathematical finance, environmental mathematics and mathematical medicine. In such areas many mathematical methodologies, such as stochastic differential equations, exponential asymptotics, methods associated with free boundary problems and thin film models have transferred across. There have also been a significant number of researchers who have moved from industrial mathematics to concentrate their efforts in these areas.

There has been engagement across the EPSRC themes with mathematics providing underpinning theoretical frameworks, concepts and insights. There were some initial difficulties in identifying how best mathematics could fully participate in the very large grants that were considered in these areas. However, there is now much greater connection with these themes. For example one half of the recent EPSRC grants for mathematics underpinning energy and the digital economy exploit close links with industrial partners.

Examples of the ways in which industry can value industrial mathematics are cited in [7]. They include: applying a new approach which gives confidence in toolset selection (Frazer-Nash Consultancy); providing low cost opportunities to investigate new technology and achieve significant savings (National Grid); bringing together the environmental science community and the insurance industry (NERC); addressing SME problems within a short timeframe (Lein Diagnostic) and, considering more simulations to tip the balance for winning contracts (Wills). A further measure of industrial benefit is that many industrial partners return to identify further new collaborations by exploiting the engagement mechanisms. This is primarily seen with large industries such as Thales, BA Systems, DSTL, Schlumberger, Unilever and NATS while successful SME collaborations usually result in single long-term research interactions.

The academic community has primarily benefited from the boost that industrial mathematics has given to mathematical research. Examples include radio channel assignment in buildings, which has identified a new class of combinatorial problems; weapons research, which has motivated new theories in extreme plasticity; risk in the environmental, climate and resource allocation studies, which have stimulated a surge in stochastic control theory; nanoscale effects in photovoltaic devices that require new modelling ideas; behaviour of vehicular and electronic traffic flows that have posed many new challenges by the need to analyse vast amounts of data and network behaviour; and the increasing use of game theoretical epidemiological analogies in social science applications such a criminality and homelessness.

88 Other areas where mathematical research has yet to develop a coherent community but where industrial mathematicians perceive nascent collaborations that may grow and become more coherent in the future with potential for significant impact from such research are: 1) Drug delivery. 2) Decision making and resource allocation. 3) Sport, for example optimal design of sports equipment. 4) Entertainment, particularly simulation for video.

The outlook for the impact of industrial mathematics on the economy is summarised in the ESF Forward Look document (ref [1]). It concludes that innovation is the main challenge for UK industry in the global market and that mathematics is a necessary tool for leading it, but that it requires an interdisciplinary approach. The large number of SMEs could benefit enormously from access to innovative research. The development of stronger interactions and networking in the community would be an opportunity for both industry and academia. The increasing awareness of the needs of mathematical modelling, illustrated recently for instance by the financial crisis or the global environmental changes, makes the timing right to create the necessary synergy.

5.2 Directions for the future The UK industrial mathematics community has identified its strengths and sees the opportunity to exploit these, while also addressing possible areas of weakness. It has identified that its young researchers are creating a novel outreach activity through their active participation in carefully chosen collaborative events especially graduate workshops and industrial study groups around the world. The opportunity is there to expand and coordinate the developments necessary to inspire and nurture early career researchers and postgraduates via a “UK Global Outreach Network in Industrial Mathematics”. This would have the mutually beneficial effect to academia and industry of: i) addressing some of the training difficulties by broadening the skill base and effectiveness of young UK researchers; ii) expanding the role of mathematics in innovation by boosting collaborative industrial mathematics worldwide; iii) increasing the academic intellectual capacity by making the UK the “country of choice” for talented young industrial mathematicians.

Outreach is crucial to bring in new ideas, new problems and new people. Mechanisms that allow substantive initial contacts to be made between industrial mathematicians and many different industrial partners need to be in place through MSc projects, PhDs, workshops, study groups and other short term activities. In addition there need to support additional methods that allow these nascent collaborations to be nurtured further and grown into productive and effective research activities. In this way industrial mathematics can continue to provide strength to both the mathematical community and to the UK economy.

References ref 1). 2010 ESF forward look www.esf.org/activities/forward-looks/physical-and-engineering- sciences-pesc/mathematicand-industry.html) ref 2) 2008 OECD review www.oecd.org/dataoecd/47/1/41019441.pdf ref 3) 2008 OECD management methods www.oecd.org/dataoecd/31/19/42617645.pdf ref 4) 2004 EPSRC International Review of Mathematics ref 5) ESF Brochure of case studies – to appear December 2010 ref 6) IMA Brochure of case studies - in preparation ref 7) 2010, EPSRC International Review of Mathematical Sciences, Submission by the Smith Institute for Industrial Mathematics and System Engineering ref 8) KTN annual report 2007-8 www.industrialmath.net/web/?url=Documents/AnnualReport2008- 09/AnnualReport/AnnualReport09.pdf

89 People who were consulted or contributed to this document In writing this document the authors consulted numerous members of the research community.

90 Landscape document: mathematical ﬁnance and actuarial mathematics

Sam Howison & Andrew Cairns October 25, 2010

Authorship The lead authors are Sam Howison (Oxford) for Mathematical Finance and Andrew Cairns (Heriot–Watt) for Actuarial Mathematics. Both authors are happy to be identiﬁed. For practical reasons, the two topics are treated separately and in parallel although, as noted below, there are many overlaps between them.

Research areas: subject breakdown and statistical overview Mathematical finance. Mathematical finance is a quintessentially interdisciplinary subject and it is not easy (nor necessarily fruitful) to break it down into disjoint areas. Nevertheless, one can take various perspectives on the subject. One such is to think of the time scales involved. These range from minutes to decades. Much of the work of an investment bank or hedge fund deals with day-to-day modelling and managing of the risk involved in a huge range of assets, with short time horizons and usually within one of a class of fairly well documented standard models. On a longer timescale, the focus shifts to asset allocation and management, often with an insurance or pensions perspective, and the interface with actuarial mathematics comes into play. Although this distinction is somewhat blurred, it has been used as a starting point to divide activity in the area of mathematical finance into the following five areas, shown with an estimate of the volume of activity in each:1

1. Derivatives pricing in more-or-less standard market models or with a fairly short time horizon (the Black-Scholes model for equities and FX, models for ﬁxed income (interest rates); generalisations to include stochastic volatility or Levy processes; etc). Also robust modelling frameworks; risk measures and capital adequacy. 54 researchers, 49% of the total (probably an overstatement to the extent that their work extends into the other areas). 2. Mathematical approaches to asset pricing and allocation, to include preference theory, utility and stochastic control, behavioural ﬁnance, interface with economics. 10 researchers, 9%.

3. Modelling for less standard markets or longer-term perspectives e.g. energy, commodities, carbon markets; very illiquid assets, real options, interface with actuarial problems (also part of category 1). 24 researchers, 22%. 4. Numerical methods. 10 researchers, 9%.

5. Other approaches including data analysis, econophysics and OR. 12 researchers, 11%.

Actuarial mathematics The main actuarial research departments (with estimated numbers of research active actuarial staﬀ) are Heriot-Watt University (7), City University (Cass Business School) (8), LSE (2). A small number of other universities also have small numbers of actuarial researchers that are associated with what are primarily teaching-focused actuarial groups: Kent (2), Manchester (1). This implies 17 to 20 under the research area “Actuarial Mathematics”.

1It should be borne in mind that many researchers in the subject also have substantial interests in other areas (eg, probability) and no attempt has been made to account for the overlap. The headline figures therefore overstate the total research effort in the subject. Some researchers are based in business schools but nonetheless are counted here because their work is substantially mathematical. However, researchers in econometrics have mostly not been counted, nor have those in the financial services industry (of whom there are many, often publishing in both the academic and the trade literature) been included. Data has been sourced by the lead author from the websites of all relevant HEI’s and supplemented with personal knowledge, enquiries of the relevant departments, and responses from consultation with the community.

91 Discussion of research areas Mathematical finance. There is no doubt that the use of mathematics in (or by) the finance sector has become a controversial subject in the wake of the financial crisis (the ‘pensions timebomb’ is not unrelated). There is also no doubt that the finance industry remains a vital part of the UK economy, and it is most unlikely to become less complex or less quantitative in its practice. This is not the place to rehearse the arguments around the use of mathematics in finance. Nevertheless, one might point out that mathematical finance is a relatively new subject and that (often complex) product innovation in the industry has frequently occurred in advance of any sizeable body of academic research; credit derivatives such as the now infamous CDOs are a case in point. One might also note that it is vital to use good models and to be aware of their limitations. Unlike, say, fluid mechanics the models to use are not always clear. The danger of investing too much effort in a ‘standard’ approach, and relying on it too much, at the expense of variety, have become very apparent. The subject faces a considerable challenge if it is to prove its worth in the future; of course, this is also a huge opportunity. The UK’s research base in finance is world leading in both quality and quantity, albeit largely concentrated in a few leading institutions. We now discuss it in more detail. Returning to the headings introduced above, the first category, ‘near-market’ research, is long standing and well represented in the UK with much good work. It deals with the analysis of the models that are in day-to-day use by banks, and might be called ‘classical’ in so far as that is a good term less than 40 years since Black, Scholes & Merton kick-started the analysis of derivatives. It also includes the so-called ‘robust’ and related modelling approaches, which offer promise in replacing parametric models with less tightly specified methods yielding bounds based on a variety of market information, and in which the UK is strong. This (and other) areas of research benefits from close connections with strong groups in stochastic analysis and PDE. The second category attacks fundamental issues to do with the theory of preferences and their evolution, connecting with both classical and behavioural themes in economics. It leads to hard mathematical questions and is represented by some world leading researchers in the UK. The third area is more practically focused, including research into a wide variety of markets such as energy or carbon where there is still no general agreement as to the models that should be used; furthermore, the contracts may be very complex in reflection of market realities. In general the modelling issues take precedence over the sophistication of the mathematics but, although these markets are enormous and very important, the mathematical research effort is thinly spread (as indeed it is worldwide). The UK has a small but high-quality research effort in the fourth area, that of numerical methods, where the principal challenges are those of size and dimensionality; this group has made some very valuable contributions but it is vulnerable because of its small numbers. Lastly, the fifth category above includes a variety of attacks on (in particular) the vital question of connectedness in financial systems and networks; indeed, understanding network dynamics and their stability has become a key problem. This work emanates from mathematics, physics, statistics and econometrics. A prime issue is the analysis of the vast quantities of data that are gradually becoming available; the area of ‘econophysics’, which is only lightly represented in the UK, has produced many studies (some of them excellent but many with methodological weaknesses) which exemplify some of the gaps between reality and standard mathematical finance models. Econometrics, which is not covered here, provides others within a tighter theoretical framework. Engagement with data is sure to become an ever more prominent theme of all branches of quantitative finance. It would be helpful to all if the industry were to make a concerted effort to make currently proprietary data available for academic research and training.

Actuarial mathematics. From a mathematical perspective academic actuarial research typically develops ahead of implementation in practice. Much of the research in the UK is, nevertheless, strongly motivated by problems that are emerging in the practice of insurance and pensions. A significant proportion of published actuarial research finds its way over time into actuarial practice, in many cases affecting transactions or liability valuations amounting to billions of pounds. The UK is seen as a world leader in actuarial mathematics. This reflects its strengths in being able to develop innovative new statistical models to help understand the increasingly complex range of risks and financial products that face insurers, pension funds and banks. This is achieved through the original application of a wide range of methods and theories developed in various branches of mathematics: probability and statistics in particular. Some branches of actuarial mathematics require the ability to make the best and most-creative use of medical and other data collected for other purposes in order to answer financial questions being asked by insurers: especially modelling of genetic disorders and health insurance. One measure of the UK’s standing internatially is through the award twice in the last 10 years of the

92 Halmstad Prize - internationally the top annual prize in actuarial mathematics - to academics in the UK. Key areas of strength are as follows: • Modelling and management of longevity risk: a number of UK academics are seen by both international academics and practitioners and as the leaders in the develop of innovative new stochastic models for forecasting future improvements in mortality rates. This work is supplemented by development of mathematical techniques to help pension plans and insurers manage the ﬁnancial risks associated with uncertain future mortality. • Modelling of health insurance and the impact of genetic information on insurance business: the group at Heriot-Watt University is widely recognised as the world leader in this rapidly evolving ﬁeld. The group are expert at surveying the relevant medical literature and distilling the limited available statistical information in a way that allows construction of robust stochastic models that link potentially-complex insurance contracts with the progress of medical problems.

• The interface between financial mathematics and insurance, and quantitative risk management: at least 50% of actuarial research involves the use of financial mathematics and the development of new, arbitrage-free financial models to tackle complex problems arising in insurance. These models are used for the purposes of pricing insurance liabilities using arbitrage-free models, assessing the risk associated with these liabilities, and developing hedging strategies, including optimal securitisation structures to help manage these risks. The UK is amongst the world leaders in this field.

• Stochastic reserving methods in non-life insurance: this type of business generally involves greater short-term risks than, say, life insurance. The last decade has seen sigiﬁcant developments in statistical modelling to help understand and measure these risks. The UK is amongst the world leaders in this ﬁeld.

The research community Mathematical finance. There has been growth since the last IRM. As then, the bulk of the academic research community is concentrated in half a dozen institutions. There are also many talented researchers in banks, hedge funds and the like. Academic research is often underpinned financially by a Masters course.2 These courses are a distinctive feature of the subject, and although they have obvious benefits they also take effort to run. The best compete on a world stage and attract excellent students from around the globe. They also provide a source of doctoral students, who are again concentrated in a few centres. Some of these centres are able to provide good doctoral training on their own, and another noteworthy development is the London Graduate School in Mathematical Finance, a consortium between Birkbeck, Imperial, King’s and LSE. The number of postdocs is relatively small. Age distribution is good with a number of excellent younger faculty (the research already existed in a fairly coherent form when they moved into it, which was not the case for their more senior colleagues). Recruitment is not easy at any level, partly no doubt because of the disparity in salaries between the industry and academia; competition for the best talent from universities around the world is also fierce, as to both salaries and research support. It is perhaps also a concern that a very large proportion (probably a substantial majority) of new appointees come outside the UK. Whether for reasons of competition from the City, or otherwise, few talented UK early career researchers stay in academia.

Actuarial mathematics. Age distribution is reasonably uniform although 3 out of the c.20 are at or close to retirement. 2There are courses at the following [number of students, where known, in square brackets]: Cambridge (Finance, joint with Judge BS [10, + 10 Part III]), Cass Business School (Mathematical Trading & Finance), Essex (Mathematics & Finance [4]), Exeter (Financial Mathematics [19]), Heriot-Watt (Financial Mathematics [30]), Imperial (Mathematics & Finance [50]), King’s (Financial Mathematics [51 full time, 4 part-time]), Leeds (Financial Mathematics, joint with Leeds BS [32]), LSE (Financial Mathematics [25], Risk and Stochastics [25]), Manchester (Mathematical Finance), Oxford (Math- ematical & Computational Finance [30], Mathematical Finance (part-time) [25]), Reading (ICMA: Financial Engineering and Financial Risk Management), Swansea (Mathematics & Computing for Finance) [17], Sussex (Financial Mathematics [9], Corporate and Financial Risk Management [64]), UCL (Financial Computing), Warwick (Financial Mathematics, joint Maths/Stats/WBS/Economics [45]), York. However, not all of these institutions host a substantial research eﬀort. The numbers here total 450, indicating that there are more than 500 MSc students on these courses. At an average fee of at least £10K, they bring more than £5M annually into the university system, and probably the same again into their local economies (and courses in Actuarial Mathematics add yet more). Much of this cash ﬂow stems from overseas and it would be foolish to imperil it.

93 There are probably 20+ actuarial PhD students at present, but there are very few UK or European PhD students. Most come from the Far East, the Middle East and Africa. Some are sponsored by overseas universities and must return home after completion of their PhD and so are unlikely to join a UK university as a lecturer in the near future. The best UK bachelors and masters graduates are much more likely to be tempted into business careers in insurance, consulting or banking where salaries are much higher than in academia. There is no shortage of applications from well-qualified students for PhD places. The pipeline is mainly restricted by the limited availability of scholarships etc. Probably around 50% of new PhD’s continue with an academic career. Of the remainder, many will take up a research-oriented career in insurance or banking that exploits the student’s PhD. About 2/3 of current permanent actuarial researchers are native to the UK (although the proportion is declining), as the best home-grown graduates get tempted off into business careers. Further strains are placed on the staffing situation because of the proliferation of actuarial undergraduate and masters degrees in previously non-actuarial maths departments and business schools. As with mathematical finance, there is a risk, therefore, in a reduction in research time available for even the leading actuarial researchers.

Cross-disciplinary activities/Outreach Mathematical finance. Mathematical finance draws on and contributes to many areas of mathematics, economics and finance, and the range of its interactions is apparent from the description of research areas above. Interaction with the industry to which the subject owes its raison d’etre is widespread, as one would expect. The great majority of doctoral students move into the industry, whence a very few return to academia, and from where rather more maintain contact. There is a Knowledge Transfer Network, run by UCL; it is relatively new and its impact is only beginning to be felt. The Scottish Financial Risk Academy (SFRA) based at the Maxwell Institute and Heriot–Watt University runs a series of Knowledge Exchange activities designed to improve the interaction between the academic sector and the financial services industry. These include biannual Risk Colloquia, Knowledge Transfer Workshops, placements for MSc and PhD students, and postgraduate course modules taught by industry professionals. Some institutions have Centres with explicit industry involvement; a notable development is the establishment of the Oxford–Man Institute of Quantitative Finance, funded by Man Group plc (the support has just been renewed until 2015). This interdisciplinary research institute has a substantial mathematical component and, interestingly, is co-located with Man’s own research laboratory. It offers a new model for industry-academic collaboration and makes an interesting comparison with Risklab at ETH, Zurich and the Quantitative Products Laboratory in Berlin.

Actuarial mathematics. UK actuarial research has a strong track record of interdisciplinary research. There is strong engagement with other areas of the mathematical sciences, especially (financial mathematics and) probability and statistics. This reflects the fact that actuarial mathematic primarily concerns the development of innovative models to address challenging problems in the financial services using components that have their theoretical roots in other branches of the mathematical sciences. There is also considerable engagement (either formal or informal) with researchers in economics, finance and the medical sciences. Lastly here is a very strong track record of engagement with the UK and international financial communities. Several significant areas of university research have either found their way directly or indirectly into practice in insurance and actuarial consulting. For example, recent advances over the last 5 years in the modelling of longevity risk are already used in industry to both measure and manage the risks that face both life insurers and pension plans.

Annexe: academic researchers Categories 1–5 are as above; category 6 is Actuarial Mathematics. The classiﬁcation is necessarily very approximate and the boundaries are porous. Student and postdoc numbers are unknown unless stated otherwise. Bath: Cox (1), Kyprianou (1), Evans (1), 3 PhD students, no postdoc. Birkbeck: Brummelhuis (4), Baxter (4), Geman (3), Hubbert (4). Brunel: CARISMA centre, Hand (5) Mitra (2) Roman (1,2). Cambridge: Dempster (2,4), Medova (2), Rogers (1,2), Tehranchi (1) One postdoc.

94 Cass Business School: Ballotta (1,3, 6), Cerny (1), Gerrard (2,6), Hatgioannides (1,3), Heberman (3,6), Hodges (1), Kadam (1), Kaishev (6), Rickayzen (6), Spreeeuw (5), Tsanakas (6), Verrall (6). Also Iori (Economics, City University, 5). Edinburgh: Rasonyi (1). Sabanis (1). Essex: No MF research listed on website. Exeter: No researchers. Greenwich: Parrott (4) Heriot-Watt: Cairns (3,6), Chan (1), Donnelly (1,6), Johnson (2,3), Kleinow (3,6), MacDonald (6), McNeil (1,6), Waters (6), Wiese (1), Yuen (1,6). Imperial: Atkinson (1,2), Bingham (5), Brody (1), Davis (1,2), Hughston (1), Pistorius (1), Zheng (1). In Bus school, Buraschi (1), Perraudin (1, in Bus School). 30 PhD students (but some of these are in the Business School and the author is uncertain how many are in Mathematical Finance). Kent: Sweeting (3,6), Tapader (6). King’s: Shaw (1,4), Brigo (1), di Matteo (5), Macrina (3), Buescu (1,2), Jack (3), Emms (3). 8 PhD students. 0 postdocs. Lancaster: Financial Statistics group, Tawn (5), Whittaker (5), Muhkerjee (5). Also Widdicks (1). Leeds: Palczewski (1), Schenk-Hoppe (3), 4 students, 0 postdocs. Leicester: Grigoriev (1), Tretyakov (4). Liverpool: Chen (1), Bujar (2,3). LSE: Barrieu (3,6), Baurdoux (1,5), Cetin (1,3), Danilova (1), Gapeev (1), Lokka (1), Norberg (3,6), Ostaszewski (3), Rheinlander (3), Xing (1), Veraart (1), Zervos (3), ? students Manchester: Asimit (6), Duck (1), Peskir (1), Fedotov (1), Johnson (3), Evatt (3, business school), Howell (3, business school). 5 students. Oxford: Giles (4), Gyurko (1), Hambly (1), Hauser (4), Henderson (1,3), Howison (1,3), Jin (2), Lyons (1), Monoyios (1), Obloj (1), Reisinger (4), Winkel(3), Zariphopoulou (2), Zhou (2). 5 postdocs, 18 students. Reading (ICMA centre):, Alexander (1), Nogueira (1), Prokopczuk (5), students ? Strathclyde: Mao (4) Swansea: Wu (1), 4 PhD students, no postdocs. Sussex: No MF research listed on website. UCL: Currently no researchers on website but Shaw moves there in January from KCL. Warwick: Alili (1,5), Dias (WBS, 5), Hobson (1,2), Jacka (1), Jin (1), Kennedy (1), Kozlovski (5), Mijatovic (1), Webber (1, Bus school). Neuberger (3, WBS), Whalley (1, BS). 12 students. 1 postdoc. York: Roux (1), Zastawniak (1), 2 students.

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International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation

INTERNATIONAL REVIEW OF MATHEMATICAL SCIENCES IN THE UNITED KINGDOM

INFORMATION for the PANEL

PART 3

CONSULTATION RESPONSES

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation

EVIDENCE SUBMITTED TO THE ENERGY REVIEW

Arranged in sections by the Evidence Framework high level questions A-H.

Each section is preceded by a summary of the key points made (with the exception of section I). These summaries are reproduced in full below.

This is followed by two pages with links to each response submitted to facilitate easy navigation through the document.

INDEX

A. What is the standing on a global scale of the UK Mathematical Sciences research community both in terms of research quality and the profile of researchers?

B. What evidence is there to indicate the existence of creativity and adventure in UK Mathematical Sciences research?

C. To what extent are the best UK-based researchers in the Mathematical Sciences engaged in collaborations with world-leading researchers based in other countries?

D. Is the UK Mathematical Sciences community actively engaging in new research opportunities to address key technological/societal challenges?

E. Is the Mathematical Sciences research base interacting with other disciplines and participating in multidisciplinary research?

F. What is the level of interaction between the research base and industry?

G. How is the UK Mathematical Sciences research activity benefitting the UK economy and global competitiveness?

H. How successful is the UK in attracting and developing talented Mathematical Sciences researchers? How well are they nurtured and supported at each stage of their career?

I. Other comments

Response Key Point Summaries

A B C D E F G H

(Late submission by IoP - not included in summaries)

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summaries

RESPONSE KEY POINT SUMMARIES

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summary A

A. What is the standing on a global scale of the UK Mathematical Sciences research community both in terms of research quality and the profile of researchers?

(return to Main Index) The following key points were highlighted:

. The UK is considered to be one of the world leaders in mathematics research, the RAE demonstrated that excellent research is found right across the UK.

. Mathematical sciences sub-topics that were consistently rated as areas of strength include: - Algebra - Number theory - Mathematical physics - Mathematical biology - Fluid mechanics - Continuum mechanics - Applied dynamical systems - Industrial mathematics

. Research areas that the UK is thought to be weak in include: - mathematical analysis - approximation theory - mathematical modelling of real systems - the gaps between mathematics and theory and practice of computing

. There is strength in statistical methodology and applications in the UK however there is a gap in theoretical statisticians taking up academic posts. The quality of recruitment fields for lectureship positions has improved in recent years but there is concern about what will happen after S&I funds end. At senior levels recruitment in statistics and OR is very challenging as there is perceived to be a lack of mid-career researchers relative to the number of Chair positions.

. Several submissions considered the volume of funding for mathematical sciences to be inadequate to support such a large and diverse subject.

. Some members of the mathematical sciences community feel that the strategy of awarding large grants to individual institutions does not make sense in mathematics. They felt that the health of mathematical sciences requires breadth across the UK and that concentrating research in a small number of large centres is counter to the needs of the community.

. There is a general feeling that Centres for Doctoral Training concentrate a disproportionate quantity of training resource in a small number of geographical locations and subject areas, and that the country as a whole does not benefit from this approach.

. Many submissions commented that there are not enough funded PhD studentships for the UK to be competitive internationally. Several submissions also raised concerns that there were not enough postdoctoral or lecturer positions for early career researchers. There were also concerns that the recently imposed cap on non-EU workers would have a severe impact on the ability to appoint the best researchers.

. There are concerns that proposed government cuts could reverse recent progress in recruiting, supporting and retaining the best academics. Mathematics researchers are traditionally very mobile and consequently funding cuts may mean that the top people leave the UK. It is felt that the loss of one or two key researchers in certain fields would lead to significant loss of research quality.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summary B

B. What evidence is there to indicate the existence of creativity and adventure in UK Mathematical Sciences research?

(return to Main Index) The following key points were highlighted:

. Some contributors mentioned that research graded in the 3*/4* star band in the 2008 RAE, accounting for over half of the research outputs, is world leading and innovative and hence a measure of creativity and adventure.

. Some mentioned that Mathematical research by its very nature is highly creative and adventurous.

. Good indicators of adventure cited include: UK mathematicians are regularly invited to major international conferences, high level of publication in world leading journals, international prizes and high levels of collaboration with industry. Although it was noted that there are fewer UK-based winners of the Fields Medal compared to USA and France.

. Specific examples of research areas and collaborative initiatives were put forward as evidence of adventurous research.

. It was noted that adventurous research is encouraged by Research Councils however conservatism in peer review does not support high-risk research.

. Several contributors remarked that preparation for the Research Assessment Exercise/Research Excellence Framework can inhibit creativity as the emphasis on the number of publications encourages more research in already successful areas and short term projects.

. The heavy emphasis on publications as a means of establishing track record means that early career researchers cannot afford to take risks.

. The requirement of detailed work plans at the point of application can discourage adventurous research.

. The concentration of research funding, through large grants, does not guarantee creativity as it tends to encourage research with a clear outcome. There is a sense that the funding available for investigator-led research is shrinking and the suggestion that EPSRC should have dedicated funds for adventurous research.

. EPSRC’s funding structure and the perceived low levels of funding for responsive mode applications within the Mathematical Sciences Programme can in some cases impede adventurous research.

. Adventurous inter-disciplinary research which falls between the remit of different Research Councils can often be disadvantaged.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summary C

C. To what extent are the best UK-based researchers in the Mathematical Sciences engaged in collaborations with world-leading researchers based in other countries?

(return to Main Index) The following key points were highlighted:

. There are strong connections and collaborations between the british mathematical community and world-leading researchers in the United States and Europe, (specifically France and Germany), and to a lesser degree in Japan, China, India and the USSR. Evidence comes largely from joint publications and grant applications.

. The collaborations with countries such as Japan, China and India are on the rise, though travel costs and language barriers present continued difficulties.

. The importance of the Isaac Newton Institute (Cambridge) and the International Centre for Mathematical Sciences (Edinburgh) in supporting collaboration with world-leading researchers are highlighted. Maintaining these facilities is a matter of priority.

. International collaboration is hampered by a lack of travel money and a lack of travel grants.

. UK government visa regulations and employment law are seen as deterrents to international collaboration.

. Collaboration initiation needs to be face-to-face through workshops, conferences and research programmes.

. Much collaboration between world-leading researchers is made possible through small grant schemes such as those supplied by the EPSRC, LMS, IMA and EMS. These schemes are seen as extremely valuable, and funding for these must be maintained.

. Flexibility is the key to funds for collaboration as it is difficult to predict talks and conferences over an extended period of time.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summary D

D. Is the UK Mathematical Sciences community actively engaging in new research opportunities to address key technological/societal challenges? (return to Main Index) The following key points were highlighted:

. The majority of responses highlighted examples of mathematicians’ involvement in projects of relevance to key societal issues including: the environment, climate change and energy, the economy and financial services, health and well being, including biomedical research, future communications and electronics, manufacturing and industrial processes, defence, national security and crime prevention

. Technical challenges were referenced in an array of areas, particularly related to engineering applications and including: aeronautics, optimisation and queues, computing, large scale data analysis and mining, fluids, novel materials and networks.

. Many research areas were considered of relevance to societal/technical challenges including: mathematical physics, fluid dynamics, statistics, financial mathematics, operational research, numerical analysis and pure mathematics.

. Pure mathematicians were not perceived to engage as well as applied mathematicians with societal/technical challenges.

. More could be done to encourage mathematicians to work on large interdisciplinary projects as currently many challenges are not framed in ways that suggest mathematical approaches, however, disciplinary excellence does need to be maintained.

. Targeted initiatives led by the Research Councils were seen as essential in encouraging mathematicians to apply their research to large societal/technological challenges e.g. the recent ‘Mathematics underpinning Digital Economy and Energy’ call. However, it was warned that care should be taken to ensure that collaborations are not artificial and do produce meaningful and excellent results.

. There is a perceived difference between the type of research required for addressing key technological/societal challenges and that traditionally required by the RAE/REF.

. The impacts of mathematical research usually occur on very long timescales and attempts to encourage engagement with industry/other academic disciplines will fail to engage mathematicians if they perceive a short term focus. It is crucial that an excellent base of fundamental, curiosity-driven research be maintained alongside any more focussed funding schemes to address challenges.

. There were mixed views on the engagement of mathematicians with industry in the UK. For some sectors e.g. financial services the interactions appear to be well developed but other contacts depend very much on the interest of individuals. Some initiatives were mentioned to support mathematicians in their engagement with industry.

. Several areas were identified as under-supported in the UK but there was little consistency between submissions on this point. Ideas included mathematical analysis, discrete mathematics, applied mathematics, probability theory and dynamical systems.

. Most submissions considered the UK to have many research leaders of international stature in mathematical sciences but gaps were acknowledged in several areas including; geometric analysis, modelling extreme events, (non-linear) hyperbolic equations. Warnings were given that removal of funding by Research Councils from many areas may lead to more gaps in the future.

. Areas in which the UK is weak were suggested including; the mathematics/engineering interface, biostatistics, and quantitative training for social scientists.

. It was highlighted that good, broad based training is needed to enable researchers to apply their skills across application areas.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summary E

E. Is the Mathematical Sciences research base interacting with other disciplines and participating in multidisciplinary research? (return to Main Index)

The following key points were highlighted:

. The mathematical sciences, and particularly applied maths and statistics, lend themselves to multidisciplinary research, and there is much evidence of this in the UK. Maths provides underpinning understanding and tools that have application in a broad range of research areas. Across the UK, mathematics research contributes to physics, finance, computer science, geosciences, materials science, ecology telecommunications and biology. . Mathematical biology is a rapidly expanding area of strength in the UK. . There are numerous examples of multidisciplinary centres and institutes across UK institutions within which mathematics has significant representation. . Pure mathematics has been slower to engage in multidisciplinary research than other strands of mathematics; however this situation is improving and examples are emerging. . Often in multidisciplinary projects the research question is formulated by another discipline and mathematics contributes the tools to address it, as opposed to the research question being led by the maths component. . Mathematicians generally have to approach other disciplines to promote how mathematics can contribute to research questions. Other disciplines can be reluctant to engage with maths and will rarely seek out mathematicians for collaborations as they are often unaware of the role that maths could play in addressing their research questions. . Historically is has been difficult for mathematicians to engage with RCUK ‘Mission Programmes’ as it is not always obvious how maths would fit into their strategy. The manner in which these programmes operate (e.g. large consortia) is often at odds with the ways that mathematicians are used to working. Research councils could provide more guidance to mathematicians on the opportunities available to them. . A major barrier to multidisciplinary research is time. Progress requires a long-term collaboration with another discipline and this is often impossible due to the large number of external pressures on academics. . REF is a barrier to multidisciplinary research as it is discipline specific and multidisciplinary papers and projects may not be as well recognised. . Different language and terminology between disciplines takes a long time to break down. Effective translators that cross the disciplines are needed to break down these barriers. . It is difficult to attract the personnel to work on multidisciplinary projects. It is often more attractive to lead your own research project in your own discipline. Mathematicians get less recognition on joint papers in journals outside of mathematics disciplines. . Multidisciplinary proposals can struggle in responsive mode panels as they ‘fall between the gaps’ of panel and/or research council remits. Reviewers cannot always identify where the novelty lies if the whole project does not fall within their area of expertise. . There must be a balance between funding for multidisciplinary research in ‘thematic programmes’ and maintaining core mathematics research in responsive mode. It is important that funders recognise the importance of core research and do not force multidisciplinary research where it is not appropriate. . Multidisciplinary research in maths often arises in response to a specific call. EPSRC managed calls (e.g. Bridging the Gaps, Mathematics underpinning Digital Economy and Energy, Discipline Hopping Awards, Cross-disciplinary calls in mathematical biology, network theory and complexity science, Life Science interface/maths CDTs and Digital Economy Hubs) are very successful in promoting and funding multidisciplinary research in maths. . Isaac Newton Institute (INI) programmes have been successful in promoting multidisciplinary research in mathematics. 7

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summary E

. There is little opportunity for cross-national multidisciplinary research in maths, suitable mechanisms do not exist.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summary F

F. What is the level of interaction between the research base and industry? (return to Main Index) The following key points were highlighted:

. There is a good flow of people from academia to industry, but this is not reciprocated from industry to academia. The flow of people from industry into academia is rather small.

. It is felt that more potential exists for interaction between academia and industry, but that there is limited incentive for industry to become involved. Tax breaks for industry were suggested to try to encourage industry to interact more with academia.

. Academia and industry work on different time scales; academia works to long lead times whereas industry works to a more immediate time scale.

. Operational Research mathematicians are sought in the financial and statistical industries, while pure mathematicians through number theory and combinatorics are sought in the national security industry.

. There is widespread support for the KTN to continue.

. Industrial CASE awards are seen as a productive programme to involve academia with industry and should be continued, though there was mention of a lack of flexibility in these programmes.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summary G

G. How is the UK Mathematical Sciences research activity benefitting the UK economy and global competitiveness? (return to Main Index) The following key points were highlighted:

. The majority of submissions identified the training of students (BSc, MSc and PhD) in the mathematical sciences as a significant contribution to the UK economy with many being recruited by industry or government.

. Many submissions considered the economic benefit of mathematical science research to be long term with current research benefitting the UK in 10-50 years. However, it is felt that mathematical sciences underpins much of research in engineering and physical sciences with more immediate benefit to the economy.

. Areas of research which were considered to benefit the UK include: - Financial mathematics - Statistics - Operational research - Applied mathematics - Pure mathematics e.g. contributions to communications, coding and encryption technologies

. Some submissions felt that the mathematical science community needs to do better in presenting the case for the contribution made to the UK economy and increase public awareness of the benefits of mathematical sciences research.

. The Knowledge Transfer Network for Industrial Mathematics is seen as an effective agent in bringing together researchers and industry.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Key Point Summary H

H. How successful is the UK in attracting and developing talented Mathematical Sciences researchers? How well are they nurtured and supported at each stage of their career? (return to Main Index) The following key points were highlighted:

. Overall undergraduate supply is reported as sufficient and growing in the majority of cases. There is some concern that re-structuring of university sector threatens breadth of coverage and statistics in particular. . The majority of responses indicate that demand is sufficient for the number of PhD places available to be filled with good quality students. Despite growth in PhD numbers overall, several responses suggest there are too few PhD places being funded. . There are concerns that some of the best undergraduates are lost to alternative and more lucrative careers. There are also some concerns that UK students have only very limited support for Masters Degrees which are viewed as a prerequisite in some cases. . In comparision to other countries the UK undergraduate experience is favourable, although the longer cycle in other countries (and the lack of a separate Masters cycle in the UK) has impacts on the competitiveness of UK students later in research careers. . Generally the UK is producing a steady stream of researchers in the required fields. Specific shortage areas mentioned were in Statistics (multiple mentions), Pure Maths (one mention), OR (2 mentions) and Maths/Computing interface (one mention). Maths within the STFC remit was also mentioned as threatened by cuts. It was suggested that getting actuaries to consider academic careers was difficult. A strong supply of UK nationals throughout the career path generally is also important where security interests apply. . UK Fellowship schemes in general, and EPSRC’s in particular, are well regarded. However, the single strongest message repeated many times in responses is the lack of sufficient numbers of postdoctoral fellowship places to allow the individual to develop their own research post-PhD and prior to appointment. . Mid-to-late career support lacks specific schemes making Responsive Mode the main channel, this does not in the view of several submissions provide sufficient funds overall. There were also several inputs questioning the relative value of large vs. small grants and the trend towards concentration in the UK, although it was noted that departments in the UK are still generally small compared to the US. The RAE and REF were mentioned several times as contributing to increased conservatism in research and appointment choices and of reducing capability when combined with limited opportunity for winning grants. . Several submissions comment on the strength and quality of international researchers that the UK has attracted in recent years. Better salaries, particularly in the US, are seen as a threat to this along with cuts in public expenditure and visa restrictions. . There is a diverse range of provision described for postgraduates, including pooling of resources to provide courses across institutions. Once academic appointment is secured a common pattern involves mentoring and reduced teaching load in early years. . Mathematicians remain in high demand with employers. Submissions recognise this and the consequences for the mathematics community but do not comment on how they may respond to changes in the UK employment market.

. There are local reports of good progress in gender balance but broad acknowledgement that UK mathematics does not have as many female mathematicians as it should if the full talent pool is being accessed and this gets worse at more senior levels. There are suggestions that the situation is improving though.

. The ethnic balance in the mathematics community is not reported to be out of line with the general population due international recruitment.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Responses to A - H

RESPONSES TO INDIVIDUAL QUESTIONS

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Responses to A

A. What is the standing on a global scale of the UK Mathematical Sciences research community both in terms of research quality and the profile of researchers?

(return to Main Index) The following key points were highlighted:

. The UK is considered to be one of the world leaders in mathematics research, the RAE demonstrated that excellent research is found right across the UK.

. Mathematical sciences sub-topics that were consistently rated as areas of strength include: - Algebra - Number theory - Mathematical physics - Mathematical biology - Fluid Mechanic - Continuum mechanics - Applied dynamical systems - Industrial mathematics

. There is strength in statistical methodology and applications in the UK however there is a gap in theoretical statisticians taking up academic posts. The quality of recruitment fields for lectureship position has improved in recent years but there is concern about what will happen after S&I funds end. At senior levels recruitment in statistics and OR is very challenging as there is perceived to be a lack of mid-career researchers relative to the number of Chair positions.

. Several submissions considered the volume of funding for mathematical sciences to be inadequate to support such a large and diverse subject.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Responses to A

(return to Main Index) Follow the links below to view the responses

Sub-Panel A University of Durham visit: University of Leeds Newcastle University (no response against individual framework questions) University of York University of Edinburgh visit: University of Aberdeen (no response against individual framework questions) University of Dundee University of Glasgow Heriot-Watt University University of St Andrews University of Strathclyde ICMS University of Oxford visit: Rutherford Appleton Laboratory (no response against individual framework questions) WIMCS

Sub-Panel B The University of Manchester visit: Queen’s University Belfast (no response to question ‘A’) Keele University Lancaster University University of Liverpool University of Sheffield (no response against individual framework questions) University of Bristol visit: University of Bath University of Exeter University of Reading University of Southampton (late submission – not included in summaries) University College London visit: Brunel University University of East Anglia University of Kent (no response against individual framework questions) University of Surrey

Sub-Panel C University of Warwick visit: University of Birmingham University of Leicester Loughborough University University of Nottingham University of Cambridge visit: INI Imperial College, London visit: King’s College London University of London Queen Mary 14

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Responses to A

University of Portsmouth (no response against individual framework questions)

Others: Edinburgh Mathematical Society London School of Economics Institute of Animal Health Royal Astronomical Society Royal Holloway, University of London Professsor Peter Grindrod CBE CMath Professor Eduard Babulak, CORDIS Consulting European Commission Dr Andy Watham, University of Oxford K W Morton, Retired (no response against question ‘A’) University of Wales, Swansea Bangor University, Professor R Brown (1) Bangor University, Professor D Shepherd (2) (no response against question ‘A’) Cardiff University Aberystwyth University University of Salford Institute of Mathematics and its Applications University of Stirling Institute of Food Research Ben Aston, Gantry House Royal Statistical Society Defence Science and Technology Laboratory (DSTL) The Actuarial Profession The Operational Research Society The British Academy The London Mathematical Society Government Communications HQ (GCHQ) (no response against question ‘A’) Smith Institute/ Industrial Mathematics KTN (no response against question ‘A’) University of Plymouth (no response against question ‘A’) University of Sussex, Professor Robert Allison University of Sussex, Omar Lakkis University of Edinburgh, Professor Andrew Ranicki UK Computing Research Committee

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Responses to A

Response by University of Durham

. British mathematics has an extremely high international reputation. The UK produces research of the highest quality and its leading researchers are comparable with those produced by any other country. This quality is not uniformly distributed and a notable gap is the paucity of top quality research in Analysis.

End of response by University of Durham (return to ‘A’ response list)

Response by University of Leeds

. Generally, the standing of UK Mathematics is felt to be quite solid. There may be areas in science where the UK holds a stronger position internationally, but in comparison with many of our European peers the UK compares well in terms of the quality of the research community. Certain areas within Mathematics may be less representative than others (e.g. spectral analysis and PDE theory). But in other areas it is strong (e.g. mathematical physics, fluid dynamics, logic, algebra).

. Whilst the current position of the UK compares well internationally, there are worries that the long-term future may be less secure due to impending funding cuts, which will in particular affect the career prospects of early-stage researchers (which will have a potential knock-on effect on the future of mathematics education across the board).

End of response by University of Leeds (return to ‘A’ response list)

Response by University of York

. In areas in which the UK concentrates I think there is a good case that the UK research is world leading. I am thinking particularly of Algebraic Number Theory, parts of Algebra, some parts of Differential Geometry and Algebraic Geometry, Fluid Dynamics, Complex Systems. In some of these cases our impact owes a lot to spectacular contributions by outstanding individuals (for example Wiles in Number Theory and Donaldson in Differential Geometry, and taking a longer perspective Sir Michael Atiyah among others). In other areas, e.g. Complex Systems, it is more usual for outstanding work to be done in larger groups.

. A serious threat to opportunities for young mathematicians is a lack of postdoctoral research positions prior to taking up a full time position in a UK department. Also a threat is the lack of opportunities for permanent academics to have longish periods of research. It is true that most departments have research leave systems but often the leave is for a very limited time. The danger is that a mathematician may be appointed to a full time position and not be able to realize his or her potential because of departmental obligations.

. I think the last review concluded that we have a gap in Analysis, and I would say that is still the case.

End of response by University of York (return to ‘A’ response list)

Response by University of Edinburgh

. The overall research quality of the UK Mathematical Sciences is within the top 3 or 4 countries in the world overall. Perhaps because of the country’s size, the excellence is not uniform in all areas of mathematics. While there are truly world-class research pioneers in certain areas, other topics are not well covered.

. A feature of the research base is its fragility. It continues to be the case that the loss of one or two key researchers in certain fields would lead to a very significant loss of research quality. And because mathematicians, especially pure mathematicians, require only limited infrastructure, they are relatively easy to move between institutions and countries.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Responses to A

. The UK career structure for mathematicians is problematic. There are too few funded PhD places to be competitive internationally---we have the potential to attract the best in the world if only funding were more flexible. While the taught course centres which have been introduced since the last review have improved the breadth of PhD training, our programme probably remains uncompetitive when judged by the highest international standards. Moreover, in our view, the Centres for Doctoral Training have had the effect of concentrating a disproportionate quantity of training resource in a small number of geographical locations (and subject areas). The country as a whole does not benefit from this approach.

. If the situation for PhD students is difficult then for early-career researchers it is even worse. There are far too few postdoctoral or tenure-track positions for early-career mathematicians. This can make it hard for institutions to make `risky’ appointments of unproven, but potentially brilliant early-career researchers.

End of response by University of Edinburgh (return to ‘A’ response list)

Response by University of Dundee

. The UK Mathematical Sciences research community clearly produces work that is internationally leading. The previous Research Assessment Exercise demonstrated this. It also demonstrated that this internationally excellent work takes place throughout the UK and not in a few select universities. The quality is definitely there. Whether the “appropriate quantity” is there, is another question. Specific research areas that the UK leads the world include: mathematical biology, fluid dynamics, theoretical physics, numerical analysis – personal opinion/perspective from an applied mathematics viewpoint.

End of response by University of Dundee (return to ‘A’ response list)

Response by University of Glasgow

. We believe that the standing of the UK is very high at international level. The results of RAE 2008 clearly indicate the large proportion of activity which takes place at international level. A high proportion of researchers have strong international connections and regularly participate in international events. A significant number of researchers are invited speakers at major international conferences and serve on international committees. UK academics are regular winners of international awards. All of these are indicative of high international standing.

. One of the trends of recent years has been a concentration of research activity into a smaller number of larger units. Some investment has occurred in response to previous IRM's and this has been very welcome. However, while the strategy of awarding large sums to individual institutions makes particular sense in disciplines requiring expensive facilities, the argument is less strong in the mathematical sciences. A problematic consequence of concentration is that there are areas of the UK where the presence of some types of mathematical science activity is sparse. To give one example, in Scotland there are no statistics groups in the mathematical sciences departments in the universities of Aberdeen, Dundee and Stirling. The effects of this on the exposure to undergraduates of central areas of the subject and insight into research opportunities, plus the missed opportunity of impact on other scientific areas represented in institutions, are regrettable. The health of the mathematical sciences requires breadth across the UK.

. Increases in the EPSRC budget for mathematical sciences research in recent years have been most welcome. However, the concentration effects plus the anticipated reduction in funding in response to government reductions are a significant concern. The mathematical sciences has a relatively small budget in EPSRC terms and the effects of income-based performance indicators, at both institutional and UK levels, have the potential to damage the perception of the discipline relative to other subjects, for those not fortunate enough to participate in the smaller number of larger grants available. The same issue applies to the doctoral training centres. There are clear potential advantages to the training experience

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Responses to A

which can be created by these but there are dangers to other institutions, and to the general health of the discipline, in the effects of funding and recruitment concentration.

End of response by University of Glasgow (return to ‘A’ response list)

Response by Heriot-Watt University

. The UK is a major player in a number of areas of mathematics, but not in all. Heriot-Watt has world-class research groups in Mathematical Biology, Mathematical Physics, Actuarial and Financial Mathematics, Probability and Statistics, Computational Mathematics, and in a wide area of Analysis centred around Partial Differential Equations.

End of response by Heriot-Watt University (return to ‘A’ response list)

Response by University of St Andrews

. Global standing: UK is amongst the leading countries in Mathematics research, but it would be difficult to argue it is the leader. It has definite pockets of leading expertise in all branches of Mathematical Sciences, but lags behind some of its competitors in ensuring a stable, well-funded environment, protected from the short-term policy changes, in which to develop breadth and diversity. Pretty much all. Opportunities: Use of computer technology in Mathematics research; world-wide famous universities able to attract staff and students from around the globe; geographical and linguistic compactness of the UK, making collaborative and interdisciplinary research easier; strong demand from the employers for graduates and researchers from all mathematical backgrounds (especially mathematical modelling, statistics and discrete mathematics). Threats: The UK system seems to have a general tendency to be disproportionately more careful about money than people. This manifests itself in two main forms: (1) Persistent feeling among staff about the lack and inadequacy of funding; (2) increasing pressures on academic staff. New research council initiatives to support research are often fairly nominal and, in trying to 'get more for less', tend to have the net effect of further stretching department resources. Another threat is the current policy of concentrating research in a small number of large centres. While this may be appropriate in the technologically based disciplines such as engineering or biology, it is running completely counter to the needs of Mathematical Sciences. The effects are currently particularly dramatic in Statistics where one third of stats groups that submitted to RAE2001 did not submit to RAE2008. Gaps and weaknesses: Although some support was given to Analysis (at least PDEs) and Statistics following the last review, the strength in these areas in the UK is rather patchily distributed by international standards. Similar comments apply on the interface between Mathematics and theory and practice of computing. Despite a funding drive from the Research Council, it does not appear that these have been fully embraced by the community. In particular, code development is not yet recognised as a valid form of research activity. Standing of researchers: Generally high, but perhaps some danger to be viewed as somewhat behind the times

End of response by University of St Andrews (return to ‘A’ response list)

Response by University of Strathclyde

. The previous International Review of Mathematics in 2004 indicated that the Mathematical Sciences are increasingly playing an important role in all other science and engineering disciplines and in wider society. However, this underpinning activity is often different from other sciences in one fundamental way; the individual nature of the mathematical sciences means that the large part of this research is based on significant contributions made by individuals or relatively small groups of people (rather than large research teams). This was true in 2004 (as it was true before then) and is remains true now.

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Responses to A

. The last Review highlighted the international excellence in the UK within fluid mechanics, numerical analysis, optimisation, mathematical biology, discrete probability, population modelling and epidemiology; which form the main areas of research within Strathclyde. These areas remain internationally important with the UK leading the way. Within the UK Math OR community, the previous International Review of OR recognised world class areas in, for instance, optimisation, stochastic modelling, heuristics, financial engineering, scheduling, performance analysis and simulation, as well as strengths in applied OR, but pointed out that “a gap remains between the output of a successful research project and what is needed for direct use by industry". This is partly because OR is fragmented, small scale and specialised in sub-disciplines, thereby hindering the exchange of ideas that could facilitate significant advances. To engage internationally larger networks of such expertise, providing breadth and diversity to the mathematical community, are required. While programmes such as the LANCS initiative have been positive in some respects we need to expand the network to be more inclusive towards all pockets of excellence across the UK rather than concentrated on a few.

. The previous Review acknowledged the role of UK OR as significant as a signboard of Mathematics to society. The strength in applied OR has resulted in research towards mixing modelling approaches, which include bringing together different mathematical modelling approaches in a coherent framework as well as integrating quantitative approaches with qualitative problem structuring. The UK is a world leader in this. This will be most effectively fostered through multi-discipline community, scaffolded by funding bodies such as EPSRC with initiatives like Bridging the Gap (BTG).

. Evidence of excellence comes from within the UK and outside.

. For instance, in the Research Assessment Exercise of 2008 50% of research activity was judged to be at 3 star (“Quality that is internationally excellent in terms of originality, significance and rigour but which nonetheless falls short of the highest standards of excellence”) or 4 star (“Quality that is world-leading in terms of originality, significance and rigour”). This level of performance was consistent across the various sub-disciplines within the Mathematical Sciences and it was acknowledged that excellence was widespread across UK.

. The fact that excellence was found right across the UK is evidence of the fact that world- leading research is often undertaken by individuals and small groups but may to some extent be undermined by recent developments in the way funding is distributed. Specifically, the concentration of funding into a few Departments in the UK (through large research grants and Doctoral Training Centres) may, of course, help these Departments to further develop their internationally leading research, but will also be detrimental to other areas in the UK, given that the total amount of funding distributed is limited. Of course if the funding levels for the Mathematical Sciences was at a level of other countries (see below) this may not be a problem.

. However, with funding consisting of such a small proportion of the overall budget, this can have catastrophic effects to small but world-leading groups of researchers outside the handful of the largest departments.

. External evidence of world-leading research can be found in the Thomson Reuters statistics. Recent citation indices (data from 1998-2008) reported in the Times Higher Education supplement showed that, in terms of citation indices, Scotland ranks 2nd and England 4th in the world in terms of Mathematics and Statistics research. (http://www.timeshighereducation.co.uk/story.asp?sectioncode=26&storycode=406463)

. From the same data (ISI Highly Cited list) the UK is only second to the USA in terms of the numbers of individual researchers.

. All of this has been achieved by a research community which is both inward in its ability to form networks within the UK and to foster new home-grown talent, and outward in its desire to collaborate with the best groups in the world (from all disciplines). The relatively low-cost nature of most areas of the Mathematical Sciences is perhaps the reason why only around 19

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4% of the EPSRC budget is allocated to the Mathematical Sciences Programme. However, given that other countries (i.e. USA or other European nations) fund Mathematics at a far greater level (proportionate to other sciences, say), it is miraculous that this level of international excellence has been achieved.

. Substantial and persuasive anecdotal evidence indicates that research staff in the Mathematical Sciences are becoming disillusioned by the EPSRC because they see the small proportion of funding allocated to their subject as indicating a lack of importance given to the Mathematical Sciences. Other measures, such as the concentration of funding in certain geographical and disciplinary areas, and the arbitrariness of blacklisting of staff from applying for research funding (the blacklisting cut-off is always set at 50% of applications, regardless of the quality of the proposed research) are also causing staff to become disillusioned.

End of response by University of Strathclyde (return to ‘A’ response list)

Response by ICMS

. The UK is internationally leading in the mathematical sciences. Britain is a medium-sized rather than a small country, but in mathematics it still punches above its weight by almost any measure of success: number of major international prizes won, citation levels per staff member and so on. This is despite the fact that, relative to other sciences, Britain spends much less on mathematics than is typical for developed countries. The ICMS plays a key role in British mathematical activity and its success illustrates the strength of the subject.

. Each ICMS workshop has at least one UK based organiser. They are funded only if they are deemed by international referees to be of an extremely high standard: to involve the world’s leading researchers on topics of fundamental importance. The workshops thus attract top scientists to the UK and demonstrate the UK’s central place in world mathematics. A list of workshop titles from the past 3 years is included in the data sheet submitted by ICMS. This demonstrates the breadth of UK mathematical research and provides a picture of areas in which the UK is particularly strong.

. After workshops, feedback from participants is collected in questionnaires. This is invariably very positive. Examples from recent meetings (including comments relevant to later questions) are:

. Painleve equations: “There was a lot of cross-disciplinary discussion which will generate important new research.” “This was an outstandingly successful workshop.”

. Hydroelasticity: “The online presentation from New Zealand executed without problems was characteristic of the workshop’s high technical and scientific level.” “[The highlight was] the interaction between researchers working in different fields (mathematics, engineering, industry) on the common theme of hydroelasticity.”

. Thin films: “There were several [highlights]: I found the organisation outstandingly efficient…, the talks were of consistently high quality, and the convivial atmosphere made it one of the most enjoyable meetings of my career.” “The younger generation were impressive, with some really good work being done in many countries.” “[The workshop] definitely helped [me]. I shall be visiting a number of colleagues in Europe in the coming months to foster research collaborations.” “The organisation of the meeting by the ICMS team was excellent: it was a joy to work with them.” “The best and most interesting scientific meeting I have ever seen!”

. The current financial climate poses a threat to scientific research everywhere. Dedicated mathematical conference facilities are recognised worldwide as being vital to a healthy mathematical community because of the fact that collaboration has become so important in mathematics and that it can be conducted without the use of laboratory equipment, but almost invariably begins with personal contact.

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. One of the major opportunities for mathematics everywhere is the growth of mathematical biology. The UK is well-placed to take advantage of this because of its historical strength in applied mathematics and the biological sciences. In recent years ICMS ran a programme of studentships in Biomathematics and Statistics in collaboration with the BioSS institute.

End of response by ICMS (return to ‘A’ response list)

Response by University of Oxford

. In general the UK's standing is very high. For example, the UK had 13 speakers at the recent ICM in Bangalore, and a similar profile may be observed in other arenas. A small number of institutions carry out world class research across the range of mathematics, while (as shown by the recent Research Assessment Exercise), smaller groups can also foster excellent work. There is strength in most mainstream areas, albeit not always in depth.

. In applied mathematics, there are traditional strengths in many institutions in the areas of fluid and solid mechanics, in mathematical physics and in mathematical biology. There is also more concentrated excellence in numerical analysis, statistics and applied probability, mathematical finance, dynamical systems and other more specialist topics such as liquid crystals. As commented below, the UK mathematics tradition is especially well suited to promote interdisciplinary research and this carries on in many institutions across a huge range of applications.

. The previous IRM commented on a perceived weakness in the analysis of PDEs and this has been addressed with the formation of new centres in Oxford and Edinburgh, as well as appointments elsewhere. However, there is still a perception that we lag behind the world in areas of analysis as well. This is at least as true in the general area of (sophisticated) discrete mathematics, which is having a widespread impact on many areas of pure mathematics while at the same time becoming increasingly prominent and important in immediately-applicable mathematics.

. The UK continues to be a leader in Geometry and Topology as well as aspects of algebra. Number theory, a traditional strength, has some outstanding people and is being a new lease of life by the expansion of the Oxford group following the arrival of Andrew Wiles.

. Excellence in research relies absolutely on recruiting, supporting and retaining the best academics, especially the younger ones. While the UK has fared relatively well in recent years, the outlook is much less certain in the light of proposed government cuts and this progress could easily be reversed. It cannot be too strongly emphasised that mathematics plays a crucial underpinning role in many other sciences and technologies, and that cuts to mathematics may have a leveraged impact elsewhere.

End of response by University of Oxford (return to ‘A’ response list)

Response by WIMCS

. Is the UK internationally leading in Mathematical Sciences research?

In some areas it is leading, and it is very competitive on a wider front although it would be naive to believe that the UK has the scale or depth of quality found at the best US institutions. We use evidence of strength originating in Wales to illustrate our replies.

. What are the opportunities/threats for the future?

The main opportunity is that the importance of mathematics has never been so clear. The growth of new problems, feeding new areas of mathematics and requiring rigorous answers is particularly exciting. The main threat to Mathematics in Wales is the availability of funding

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to respond to these challenges. Inadequate support for graduate training and for post-docs freezes out opportunities to contribute to the wider economy.

. Where are the gaps in the UK research base?

WIMCS has helped the Welsh research base to a reasonable balance, although parts of pure mathematics e.g. algebra, number theory, combinatorics, needs to be secured. Through the investment and strong pan-institutional co-ordination WIMCS has created 5 cross institutional clusters (analysis (32 members), computational modelling (82 members), mathematical physics (45 members), statistics and operational research (53 members), stochastics (24 members) (with significant cross membership as well as members from outside the mathematical community). It is enough to create a vibrant and challenging research environment; it has brought to the UK world class and prize winning researchers in topical areas of pure mathematics (e.g. Feng Yu Wang), and brought substantial new European Funding in applied areas (e.g. Gennady Mishuris). In collaboration with the INI, Cambridge it is organising and hosting very high quality international workshops and meetings that build on and consolidate its strengths (Ken Morgan, Computational PDE; David Evans, Non commutative geometry). WIMCS is a vehicle able to identify the strengths in Wales and work to maintain and develop the international standing through appointments, meetings, and by supporting early stage researchers. For a small community in a small country these developments in the mathematical and computational sciences seem exceptionally healthy.

. In which areas is the UK weak and what are the recommendations for …

One unique strength of WIMCS is the way that it both truly interdisciplinary and presents considerable disciplinary strength in each cluster area. This mix is absolutely essential if Mathematics is to make a sustainable and long term contribution to other areas. Prize winning work using random matrix theory to study reliability in large scale numerical models for engineering systems provides an example of science which has been transformed by WIMCS inter-cluster activity. WIMCS is exploiting its ability to co-ordinate mathematical strength to create a broader interface between the mathematical and health communities in Wales.

. What are the trends in terms of the standing of UK research and the profile…

The opportunities are there, the core talent is there. The trend depends on the funders and the attitude of the community. There are good and bad indicators.

End of response by WIMCS (return to ‘A’ response list)

Response by The University of Manchester

. In UK Pure Mathematics, the community contains a high proportion researchers at a high international level (> 50% 4* or 3* in RAE2008). However, there is a considerable block of researchers (40% 2* in RAE2008) who are active but do not contribute research that has a wide international impact. Areas of strength in Pure Mathematics in the UK are Analysis (Spectral Theory, PDEs), Arithmetic Combinatorics, Ergodic Theory and Dynamical Systems, Geometric Group Theory, Number Theory and Representation Theory. In all these areas the UK is one of the internationally leading countries. In Applied Mathematics, continuum mechanics continues to thrive, although funding opportunities for pure research in this area, without immediate industrial/ economic impact is now very difficult to obtain. Statistics is an area where it is very difficult to attract high-calibre staff (because of the attraction of considerably higher salaries in the commercial sector).

End of response by The University of Manchester (return to ‘A’ response list)

Response by Keele University

. The UK is internationally leading in many areas of Pure and Applied Mathematical research. For example, in Applied Mathematics, three strengths are fluid mechanics, 22

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elasticity theory, and asymptotic methods. Further details about this may be found in the various landscape documents, to which staff within mathematical at Keele have contributed separately.

. Particular opportunities are in biological and medical applications of Mathematics. The biggest threat is reduction in government funding for research in universities and education more generally.

. One gap is in the number of theoretical statisticians taking up established university posts.

. The international standing and profile of UK research have stayed approximately uniform in recent years, at a high level. There is no reason to expect a large change unless it is caused by external funding pressures.

End of response by Keele University (return to ‘A’ response list)

Response by Lancaster University

. We feel that views on the global standing of UK mathematical sciences research is best addressed elsewhere (e.g. by the Landscape documents).

. Our experience with recruiting gives some insight into the profile of researchers and how this has changed over the past 5 years. Throughout this period we have always had excellent fields for positions within Pure Maths. Within Statistics and OR, the picture is different. For lectureship positions, we have noticed a steady increase in the quality of fields over the past few years – though still not to the level we observe for positions in Pure Maths. The improvement appears to be as a result of investment in Statistics and OR, for example through the Science and Innovation Awards. We have concerns as to how this will develop once the funding from these awards ends.

. At senior levels, recruitment of quality researchers in Statistics and OR is very challenging. We are aware of many chair positions in Statistics and OR being unfilled (both at Lancaster and elsewhere). Even senior positions linked to Science and Innovation Awards in Statistics were unable to attract any overseas researchers to the UK; and there is a lack of mid-career researchers currently in the UK relative to the number of chair positions that need to be filled.

End of response by Lancaster University (return to ‘A’ response list)

Response by University of Liverpool

. According to the results of RAE2008, Applied Mathematics Research in the UK is extremely strong. In particular, such areas as theoretical continuum mechanics, numerical analysis, applied dynamical systems, non-linear analysis and industrial mathematics are very well developed in UK universities, and they attract world-class researchers with publications of exceptional quality. Among very successful research groups in Applied Mathematics are Oxford, Cambridge, Imperial College London, Bath, Bristol, Manchester, Nottingham and Warwick. These institutions have created an excellent research environment based around research institutes and graduate schools, funded either through the university infrastructure or via external platform grants from EPSRC or other research councils.

. As far as Pure Mathematics is concerned, research in areas such as Algebra, Number Theory, Algebraic and Complex Geometry, Geometry, Topology, Lie Theory and Generalizations, Analysis Dynamics and Combinatorics is very strong in the UK. In areas of theoretical probability, there is evidence that the UK has lost the leading role it enjoyed between the 1940s and 1970s, possibly since lack of funding has prompted some excellent researchers to leave the UK for the USA, Canada etc.

End of response by University of Liverpool (return to ‘A’ response list)

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Response by University of Bristol

. Is the UK internationally leading in Maths? In which areas? The UK is internationally leading in some areas of the Mathematical Sciences and internationally competitive in many areas. The IRM Panel will undoubtedly be aware of the RAE2008 results, particularly the subject overview reports, which give an overview report of each discipline.

. What contributes to the UK strength? What are the recommendations for continued strength? Top-rank mathematics can be achieved in both large clusters and individual researchers in small Departments. The key factors behind UK success in Mathematics are (i) recruiting the best people and (ii) providing them with the best research environment. We define `research environment’ to mean access to excellent graduate students and postdocs, funds and space for research interactions (meetings, workshops, conferences, travel) and equipment. Some subareas additionally require access to and control of high-performance computing. Most strong centres in the UK benefit from the strong proximity of a flourishing undergraduate programme enabling extensive two-way interaction.

. A key factor for success is the maintenance of a strong flow of people from school, through undergraduate, postgraduate, postdoc and through to Faculty positions, irrespective of nationality. Talented people can, and often do, switch into other career streams at each stage. Funding Bodies are to be commended for improving certain parts of this chain (for example, Postdoctoral and Career Acceleration Fellowships, undergraduate summer bursaries). However, there are still known obstacles to be removed which would benefit from funding body cooperation and action. For example, (i) the continuing debate about which funding body, HEFCE or EPSRC, should pay for strategic UK-important Masters courses; (ii) funding excellent international graduate students who, in many cases, would remain in the UK after their PhDs and contribute mightily to the UK’s future academic and economic wellbeing.

. Mathematics in the UK benefits from a number of organizations that are also active researchers of Mathematics (e.g. GCHQ, hedge funds) and a wide range of users of Mathematics (e.g. industry, government departments) who commission and stimulate research. These contribute considerably to the strength of UK Mathematics. We would propose that more, and more attractive, people-exchange opportunities be considered by Funding Bodies between industry, government and academia (although not at the expense of the existing precarious core Mathematics research funding).

. Opportunities Mathematics is almost universally recognised as a discipline that has huge impact on the world we live in. Indeed, some human endeavours would simply be impossible to pursue without Maths. Advances in Maths can generate dramatic benefits for those new disciplines that rely on Maths (for example, medical imaging, encryption, genetics) as well as many of the existing areas. Maths is a key pillar of STEM (Science, Technology, Engineering and Mathematics) and hence essential to the UK’s future prosperity and well-being. We predict that the importance and pervasiveness of Maths will continue to grow as the world continues to become more data-rich.

. Threats: A. Funding issues. 1. Targeted Government STEM funds for University Departments do not always reach their intended targets. 2. EPSRC’s Maths Programme funds are small, have been shrinking for a while, and experienced a large drop from £22M to £15M between 2007 and 2008 (EPSRC Annual Report). The shrinkage coincided with a large increase in Faculty into the UK prior to RAE2008. 3. We believe the Maths community is extremely worried about further diversion from basic research into `nearmarket’ application areas. Our competitors, and potential recruits, are acutely aware of these funding issues and they are already harming UK Mathematics. We are keen on extra funding for `interdisciplinarity’ and `applications’, but not at the expense of the core.

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B. Muddled national PhD options and limited funds: Unlike the strong, identifiable, US PhD brand, doctoral training options in the UK are confused to say the least. Even in the same institution there can be considerable diversity in programme requirements and duration. PhD programmes are often determined by the funding available rather than academic objectives, which is not satisfactory. Funding bodies have tried to help by the introduction of “Roberts money”, Taught Course Centres and Doctoral Training Centres (DTCs) and, although these have provided valuable advanced training options, they have essentially increased the confusion.

C. Departure of staff to other countries. Maths staff are traditionally very mobile. The looming crisis in funding will be much more damaging to those Depts that have successfully recruited research stars from abroad. With imminent cutbacks, these people are the ones most likely to leave the UK (and indeed some already have).

. Where is the UK weak? We believe mathematical (theoretical) statistics, applied analysis, analysis of pdes, complex networks, mathematical biology at the molecular scale, scientific computation, numerical analysis (compared to the US).

. Trends There is a feeling that the status of UK Research in Mathematics increased in the years up to RAE2008, with a net increase of Faculty and increasingly top international Faculty. Indeed, there is a feeling that we have done pretty well on the limited resources available to us. There is a general feeling that the recent trend is turning decidedly negative, for reasons we shall mention later.

End of response by University of Bristol (return to ‘A’ response list)

Response by University of Bath

. Toland elected to the executive committee of the IMU in Summer 2010

. Dawes is co-chair of the forthcoming SIAM Conference on Applications of Dynamical Systems, to be held in Snowbird, Utah, 22-26 May 2011. This conference, held every two years, is the major international applied dynamical systems meeting and attracts over 700 participants.

. Scheichl is the main organiser of a special semester in 2011 at the Radon Institute (RICAM) in Linz, on “Multi-scale simulation and analysis in energy and the environment”. Freitag and Graham (from Bath) are co-organising along with Bastian, Cullen, Engl, Langer, Melenk, Wheeler.

End of response by University of Bath (return to ‘A’ response list)

Response by University of Exeter

. I believe the standing of UK mathematical sciences is high on the global stage, and certainly well above any expectation based on the UK population. There is excellence both in particular areas (eg fluid dynamics, algebra, stochastic processes, nonlinear science) and a wide range of areas where it is competitive. Smaller research-intensive institutions (such as Exeter) tend to concentrate on niche areas while clearly the larger institutions (such as Oxford, Cambridge, Warwick) can cover larger areas. It is to the benefit of the UK that institutions should work together to maximise any benefits. This seems to be achieved in many cases, though there is room for improvement. For example, in Statistics, there seems to be somewhat of a division between those undertaking theoretical work (concentrated in a small number of institutions) and applied statistics researchers which are necessarily much more widely distributed.

End of response by University of Exeter (return to ‘A’ response list)

Response by University of Reading 25

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. In some subjects, UK's standing is extremely good. This includes many branches of pure mathematics, and some fields in applied, particularly fluid mechanics in its classical and more modern versions, and more recently some aspects of applied analysis and manay aspects of numerical analysis. In other branches of applied mathematics, the UK appears to lag behind the leading countries, especially France and the USA.

End of response by University of Reading (return to ‘A’ response list)

Response by University College London

. It is, of course, extremely difficult to make compelling statements at this level of generalisation, but our sense is that research quality in the UK is relatively healthy at present, but with some differences from subject to subject. In Pure, hard analysis and number theory are notable strengths. In Applied, industrial maths is performing well, aided by the KTN, and fluid mechanics is also strong. In Statistics, the strength is around methodology and application, particularly around Bayesian Statistics including Markov Chain Monte Carlo related areas and time series with environmental/ecological applications. We have some concerns that the interactions between Pure, Applied and Statistics are not as rich as they might be, however this could be considered to be a global phenomenon.

. The primary guide to assessing quality of research in the UK is of course the Research Assessment Exercise, and we recommend that the panel consult the sub-panel subject overview reports for mathematical sciences, which show compellingly that high quality research is distributed across a large number of institutions, although the number of institutions contributing has significantly declined since 2001.

. UCL’s own scoring in the 2001 and 2008 RAEs may be of interest. In 2001, all three submissions (Pure, Applied and Statistics and Operational Research) were judged to be of 5 rank quality. In 2008, Pure retained that level of strength with a 20/40/35/5 profile under the new system, although the panel did note: “Research income was slightly lower than would be found compatible with the overall profile. PhDs awarded were lower than the average for the UoA and significantly lower than would be found compatible with the overall profile.” However, in 2008 the Applied Mathematics and Statistics and Operational Research profiles were both 10/40/40/10, indicating a relative decline. However new appointments were made in all areas prior and/or subsequent to 2008, with 9 new members of staff joining Statistical Science alone.

. In terms of demographic profile, our impression is that the UK community is excessively top heavy and worryingly lacking in diversity (especially in Pure and Applied). The research community feels very “white and male” in comparison to wider society, the student body and comparable research communities elsewhere. We would especially welcome suggestions on how to improve this situation.

. As we note below in section H, we also have concerns that the future global standing of UK research may be adversely influenced by increasing concentration of PhD funding in too few departments.

End of response by University College London (return to ‘A’ response list)

Response by Brunel University

. The standing is very high and ranks alongside the best research centres in the world in all areas of mathematics. The researchers are of a standard equivalent to those of other first world countries. The most important contribution to the continued strength is the adequate funding of research students. In the present climate an opportunity is to recruit more PhD students in this economic downturn who would otherwise be unemployed. A threat to the 26

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future is to not provide this funding and to let the mathematics departments try to survive on none governmental support.

End of response by Brunel University (return to ‘A’ response list)

Response by University of East Anglia

. We believe that the UK’s standing is best assessed from the Landscape Documents and the individual Departmental Research Data responses. In the last few years, many mathematics departments have increased staff numbers with research quality one of the primary appointment criteria. This expansion, and the ability to draw on a pool of international talent has strengthened UK mathematics.

. The UK government has recently imposed a cap on the number of non-EU workers which universities can employ. This is already having a severe impact on our abilities to appoint the best postdocs and researchers and if it continues, it will restrict the flow of talent and ideas to the UK.

End of response by University of East Anglia (return to ‘A’ response list)

Response by University of Surrey

. One of the main contributions to the UK strength in mathematical research is our large undergraduate teaching activity. We believe that the UK probably has one of the highest percentages of university students specialising in the mathematical sciences in the world, possibly because this training is more highly valued by UK employers than those in other countries.

. A large undergraduate population provides both opportunities and threats to the research community. On the positive side it generates a large pool of potential mathematical science researchers and helps to maintain a large number of professional mathematicians in a diverse range of departments. From this point of view one might welcome the recent considerable rise in the number of students following degree programmes in the mathematical sciences. However the increasing undergraduate numbers have led to pressure to spend more time on teaching and less on research because staff numbers have not increased in proportion to student numbers.

. This shift can be seen in finances of the Surrey Department of Mathematics. In 2007-08 approximately half the Department’s income came from teaching (primarily HEFCE and undergraduate tuition fees) and the other half from research (HEFCE, grants and postgraduate fees). In 2010-11 the split will be 63% from teaching and 37% from research, a rise of over 250% in research grant income having been offset by a fall in the HEFCE research allocation.

. To ensure that the UK continues to be internationally leading in the Mathematical Sciences research the forthcoming review of university funding needs to redress the imbalance between funding for teaching and funding for research. Well funded teaching is essential to help realise the potential for training more high quality researchers offered by the large undergraduate population. But this must be matched by more funding for research to ensure that it is not overwhelmed by the teaching.

End of response by University of Surrey (return to ‘A’ response list)

Response by University of Warwick

. EPSRC has taken steps to strengthen certain areas through Science and Innovation Awards and other initiatives. In these areas, for example Discrete Mathematics and Statistics, continued steps are needed to sustain and add to the improvements.

. In Statistics the UK is currently known for its leadership and has a very high quality research profile. Nonetheless, this remains under threat because of the lack of resourcing 27

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of PhD's and post-doctoral positions in Statistics over the last 20 years. Despite the successes of recent efforts the threat of decline is still present.

End of response by University of Warwick (return to ‘A’ response list)

Response by University of Birmingham

. Research in the Mathematical Sciences in the UK has a strong and broad base within a substantial number of UK Universities, with developments having taken place over many years in response to contemporary industrial and sociological needs. In Applied Mathematics, strengths are traditionally in fluid mechanics, continuum mechanics and industrial mathematical modelling. More recently (over the last three decades) considerable strength has been developed in applied analysis, numerical analysis and computational methods, mathematics in the biological and medical sciences, mathematics in the social sciences and financial mathematics. The UK has developed firm foundations in these areas, producing internationally leading research which is being published in international journals of the highest rank. In Pure Mathematics the traditional strength in Algebra (especially Group Theory) and Combinatorics have been maintained. UK is also very strong in Representation Theory, Algebraic Geometry, Algebraic Topology, and many sub areas of Analysis. In optimization strength lies in computational optimization and operational research. In probability and statistics applications have strong links with the medical, social and economic sciences.

. Much of the UK strength lies in the diversity within individual schools and departments, and the collaborative opportunities this affords, both within mathematical disciplines, and with other disciplines which can be effectively enhanced with mathematical contributions. Opportunities for the future lie in the closer interaction of researchers in the Mathematical Sciences with biology, medicine, environmental science and the social sciences, whilst maintaining the strong links between the Mathematical Sciences and engineering and industry. Threats for the future come from a contraction of primary funding (which underpins such collaborations), increasing bureaucracy in obtaining funding and a contraction of diversity by a possible location of all significant funding in a minority of centres. The maintenance of funding at Postdoc level is essential for the recruitment and stable development of high quality researchers into the Mathematical Sciences. An area of weakness in the UK is the lack of establishment of long term links with industrial users (short term project funding is healthy). More formal arrangements with individual users and the Universities, both academic and financial, may encourage involvement in longer term research activities on the part of both partners.

. UK researchers in the Mathematical Sciences make internationally leading contributions in terms of publication profile. The profile of UK researchers in terms of major contributions at leading international conferences is less evident and needs improvement to properly reflect the quality of the research developed in the UK.

End of response by University of Birmingham (return to ‘A’ response list)

Response by University of Leicester

. The UK is in the forefront of many areas of modern mathematics, as indicated by citation data. Areas we are aware of are algebraic topology, topological dynamics, algebraic geometry, representation theory, differential geometry, number theory, fluid mechanics, computational solution of PDEs, numerical analysis, including numerical methods for stochastic systems and approximation theory, model reduction, financial mathematics and mathematical biology. We believe that the support for mathematics departments across the UK is important, and that the current popularity of mathematics as an undergraduate subject is key to securing the research base.

. The threats for the future: - are a collapse of the undergraduate population due to government policy in education (as happened at the turn of the millennium)

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- over emphasis of the research councils on interdiciplinarity at the expense of core funding for mathematics. Where there is natural cross-fertilisation then this should be supported, but social engineering here does not appear to work to well. The support from EPSRC for core funding in mathematics has taken a large hit in the name of this ideology, - the possibility of policy makers providing funding for short term populist activities, which is linked to the previous point, - over emphasis at EPSRC of concentrating funding in a small number of large universities. One outcome of the RAE was that increased input by way of research funding did not correlate strongly with research output. Biodiversity is what creates a rich environment, not concentration, which leads to stagnation and a maladjusted ecosystem.

. We believe there is a big problem in the UK research base with very applied mathematics, as UK culture does not seem to support links with industry in the way many other countries do. A net result of this is that applied modelling is not a well-valued activity, because the results of modelling could appear mathematically trivial from a puristic point of view subsequently. The RAE did not rate applied work highly (in our view), and so links with industry and other disciplines are being essentially inhibited. In order to produce the interdisciplinarity that EPSRC wants there has to be greater recognition of the challenges of this sort of work.

. There are gaps in approximation theory, numerical linear algebra, global analysis, mathematical modelling of real systems, CAGD, functional analysis.

. We believe that the standing of UK researchers and their research is very high.

End of response by University of Leicester (return to ‘A’ response list)

Response by Loughborough University

. Although the UK mathematics research community has a relatively small number of permanent faculty members compared to some countries like the USA, China etc., the UK is one of the leading countries in many areas of mathematics.

. As far as geometry and mathematical physics is concerned, the UK is a leading country. The example can be the world-leading research of Simon Donaldson and his geometry group at Imperial College. In integrable systems area, the UK is also deservedly considered as one of the leading countries. There are strong groups in Edinburgh, Imperial College, Leeds, Loughborough and Glasgow. Partial differential equations appear in a vast number of subjects such as the analysis of wave equations, the Navier-Stokes equation, but also Geometry and Topology. Global Analysis focuses on the interplay between Geometry and Topology with the Analysis of partial differential equations. Both fields are classical within pure mathematics and are developed intensively worldwide. The UK base in nonlinear PDE's and the Navier Stokes equation, curvature flows, some aspects of global analysis, spectral theory and scattering theory is well developed. High profiled international researchers have been hired in the last few decades, and a vibrant research atmosphere within the UK has been built. The UK is one of the world centres of Stochastic Analysis and in the international forefront of cutting edge research in the area. It has built a very strong research base in this field in the last few decades. Stochastic Analysis naturally lies between probability theory and analysis. It is one of the fundamentally important basic research areas of mathematics, having deep connections with other areas of research in mathematics such as analysis, topology, differential geometry, partial differential equations, and dynamical systems. It also plays a key role in understanding many random phenomena in many other fields such as physics, engineering, finance and economics. There are strong research groups in Edinburgh, Imperial College, Leeds, Loughborough, Manchester, Oxford, Swansea, Warwick and York. In the past ten years or so, UK researchers have done original work in a number of areas such as stochastic analysis on manifolds, rough path theory, stochastic dynamical systems and stochastic partial differential equations.

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. Prof Elworthy (Warwick) was invited to give a 45-minute lecture in ICM Madrid 2006 in recognition of his fundamental work on stochastic geometry and path space. Prof Lyons (Oxford) was elected as FRS and awarded the Polya prize in recognition of his groundbreaking work on rough path theory. Prof Truman (Swansea) had done seminal work on Feyman-path integrals and Prof Hudson (Loughborough) is one of the founders of quantum stochastic calculus. Though Elworthy, Truman and Hudson are or will be officially retired, they are still active in research. More importantly, a younger generation of stochastic analysts in the UK who have been nurtured, have become international leaders in this area (Z Brzezniak (York), B Hambley (Oxford), M Hairer (Warwick), X Li (Warwick), Z Qian (Oxford), F Wang (WIMCS), T Zhang (Manchester), H Zhao (Loughborough) et al).

. Traditionally the UK has been strong in fluid mechanics with pioneers like Lord Rayleigh and James Lighthill but with mathematical modelling applied to such a range of subjects in the modern era it is really only possible to say that the UK has a lot of applied mathematicians active across a whole range of subjects and almost certainly a wider range than many other countries because of the wide remit of applied mathematics in the UK. In many of these fields UK mathematicians are world leaders, even if many were not originally trained in the UK. The UK has been very active in appointing internationally leading researchers over the last decade so that the community is truly globally excellent. As far as Loughborough is concerned, we have a strong group in Mathematical Modelling in particular to materials and biology. There is a strong linear and nonlinear wave group in Loughborough. Roger Grimshaw is an international figure and has made fundamental contributions to the understanding of a number of aspects of fluid mechanics. The UK dynamical systems community is a noticeable part of global community in this field. In majority of universities there are dynamical systems groups. They represent all major directions in dynamical systems theory. There are high profile researchers in complex dynamics, one-dimensional dynamics, bifurcation theory, ergodic theory, perturbation theory, etc. Dmitry Turaev (Imperial College) was an invited sectional speaker on section "Dynamical systems and ODEs" at ICM 2010. International journals "Ergodic theory and dynamical systems" and "Nonlinearity" are published in UK. Prof. Neishtadt (Loughborough) is one of the Editors-in-Chief of “Nonlinearity”.

End of response by Loughborough University (return to ‘A’ response list)

Response by University of Nottingham

. UoN contributes to national strength in a number of areas, particularly in number theory, mathematical medicine and biology (including epidemic modelling) and operational research. The UoN groups in these areas have extensive national and international collaborations and enjoy outstanding levels of Research Council support. Other areas of strength at UoN, where the UK is less well represented, include scientific computation and statistical methodology (with applications). These areas offer numerous opportunities to engage with major technological and societal challenges, often arising in other disciplines and industry; however it can be difficult for universities to compete with industry and business to recruit the most talented experts in these fields. Further areas of strength at UoN include quadratic forms, quantum disordered systems, operator algebras, continuum mechanics and applied probability.

End of response by University of Nottingham (return to ‘A’ response list)

Response by University of Cambridge

. We believe UK Mathematics to be internationally leading in many areas despite lower effective funding for mathematics than in rival nations. Areas of particular strength in Cambridge are mathematical and theoretical physics, applications of fluid mechanics, geometry, combinatorics and number theory. Some weaknesses in UK Mathematics identified in the last International Review, e.g. in Analysis and in Statistics, are being addressed effectively in Cambridge (e.g. through new recruits at Lectureship to Professor level, through a joint DAMTP/DPMMS Centre for Doctoral Training in Analysis and through the Statistics Initiative in DPMMS).

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End of response by University of Cambridge (return to ‘A’ response list)

Response by INI

. Our role in facilitating collaboration means that we cannot disentangle the excellence of research done by UK and overseas participants at the Newton Institute. Rather our activities are directed to ensuring that the UK is integrally engaged at the forefront of world research. The evidence for that excellence is: - the scale and duration of the EPSRC grant award followed international peer review which recognised that the Newton Institute was amongst the leading institutions of its kind in the world; - international bodies such as the National Science Foundation of the USA and the Centre National de la Recherche Scientifique award funding to support participation; - the Scientific Steering Committee (SSC) http://www.newton.ac.uk/ssc.html draws on rigorous international peer review in selecting Programmes from the proposals submitted to it; - the stream of proposals to the Institute is very strong. All available slots are now committed through to the end of 2012, with a further 11 proposals under consideration at the October SSC meeting for 2013 onwards; - in exit questionnaires for 2008/9 (the last academic year for which we have full data), 72% of participants rated Programme quality as ‘Excellent’ and 24% rated it ‘Good’; . The Institute broadly meets its target of attracting on average 2 Fields medallist participants per year, and 1 other senior award winner (e.g. Nobel, Wolf, Abel), of course with considerable annual variation.

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Response by Imperial College London

. The UK is definitely one of the leading forces in international Mathematical Sciences particularly in such areas as Geometry and Topology, Algebraic Geometry, Number Theory, Combinatorics, and also within Applied Mathematics, Statistics, Mathematical Biology and Financial Mathematics. The latest UK investment in the area of nonlinear PDE’s has been very positive and UK is now sufficiently strong in this area.

. However, the international trend, which was clearly emphasised at the ICM 2010 in Hyderabad, shows that Mathematical Analysis has become an influential area of research. Two out of four of the latest Fields medal winners (S.Smirnov, C.Villani) work in different aspects of Analysis. Smirnov's area is the Percolation Theory where he applies his outstanding skills in Complex Analysis. Villani works is the area of the non-linear Bolzmann equation with methods rooted in classical transport problems and different types of Sobolev inequalities.

. The area of one other Fields medallist, E.Lindenstrauss is Analytic Number Theory. One of his main results is the proof of the fact that the notions of quantum and classical ergodicity coincide in the arithmetic case. Such a problem is still open for a class of linear pseudodifferential operators on manifolds. Moreover, the winners of the Chern and Gauss prizes are L.Nirenberg and Y.Meyer who are the most prominent specialists in Functional analysis.

. Y.Meyer is one of the creators of the theory of wavelets, which revolutionised the transmission of information and is an area in which UK does not have any expertise. Mobile telephones and Internet would simply not exist without this theory. . L.Nirenberg is the person whose influence in different aspects of linear and non-linear partial differential equation is crucial for future generations of mathematicians.

. Unfortunately the situation regarding Analysis in UK still remains unsatisfactory. At the moment it is unthinkable that UK would be able to deliver a Fields medallist in Analysis. Often the most talented students who have studied in UK choose to do their PhDs in Number Theory or Differential and Algebraic Geometry, which are the traditional areas of

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Mathematics in UK. This tendency is not easy to break due to the very long period during which Analysis was not properly acknowledged in the UK.

. Recently the EPSRC has decided to fund a Doctoral Training Centre in Analysis in Cambridge. I find it very strange because the Cambridge University does not have many specialists in Analysis.

. After this award a number of experts from different UK universities received letters from Cambridge inviting them to give intensive courses in Analysis. Of course it is very positive that the leaders of the DTC in Cambridge have understanding that they do not have enough their own specialists to run the centre. However, intensive courses do not contribute properly to the scientific environment required for PhD students. Lecturers come and go and the students remain remote from continuous contacts with leading specialists which is crucial for their development.

. Mathematical Analysis is a very rich and technical subject with a growing number of applications and is one of the most vital subjects for future technological development of the country.

. It is important to say that in order to prepare a high level specialist in Analysis takes longer time than a specialist in some other areas of Mathematics. Besides, it is necessary to have a right environment with regular seminars and courses given by specialists.

. The major threat to research in Mathematics is the unsatisfactory training of future researchers. After only three years of undergraduate University education, one year of graduate study and three years of PhD study, UK PhDs are usually not competitive in the job market. Most PhD students choose to leave Mathematics after receiving their doctoral degree because they do not feel confident enough to continue doing Mathematics.

. As a result a large number of new positions are given to foreigners who are better prepared for research in Mathematics.

End of response by Imperial College London (return to ‘A’ response list)

Response by King’s College London

. The UK mathematics research community is, in comparison with our international competitors, relatively small in size. Nevertheless, in key areas of Geometry, Topology, Number Theory, Statistical Mechanics and Theoretical Physics it has for many years had, and continues to have, a considerable international impact and is held in extremely high regard. In the last ten or so years the UK’s standing in key areas of Analysis, and more especially of Differential Operator Theory, has also risen substantially.

. Geometric Analysis is of fundamental importance in pure mathematics (it was, for example, pivotal to the recent proof of Poincare’s Conjecture by Perelman) and yet the level of research in this area remains relatively weak in the UK.

. The UK’s research strength is helped in part by the large proportion of immigrant research mathematicians, especially from Europe, and to a lesser extent from North America. Most research-led faculties are now of a truly international nature and this has dramatically improved the research profiles of many departments.

. A major threat for the future is the likely impact of decreases in Government funding on the continued attractiveness of UK departments for highly mobile international researchers.

End of response by King’s College London (return to ‘A’ response list)

Response by University London, Queen Mary

. UK statistics is highly regarded in terms of research quality (second only to USA) and, in particular, there is a distinctive image of UK research being strongly motivated by 32

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applications, but methodologically sound. Specific areas of excellence include Bayesian statistics and hierarchical modelling. The INI is a great resource, which allows us to show leadership.

. The greatest threats are the “demographic time-bomb”, which has probably already exploded and lack of funding for MSc and PhD students. While many excellent young researchers have been recruited in recent years, there is a vacuum where the research leaders should be. (SGG/RB).

. The volume of research funding is inadequate to support a large and diverse subject such as mathematics.

. Some areas of traditional UK strength, such as algebra, are wilting through lack of support, as more and more resources are poured into interdisciplinary areas, and other more fashionable subjects. (RW)

End of response by University of London, Queen Mary (return to ‘A’ response list)

Response by Edinburgh Mathematical Society

. Scotland comprises 8.4% of the total UK population, but submitted 12.8% of the mathematical sciences academic researchers in the 2008 Research Assessment Exercise (RAE) – 248 academic staff in the Units of Assessment for Pure Mathematics, Applied Mathematics, and Statistics & Operational Research, out of a UK total of 1940. The overall performance of Scottish mathematical sciences in RAE 2008 in comparison with the overall UK results are summarised in the table below (grade 4* in the RAE denoted "world leading" and 3* "internationally excellent").

weighted % in: 4* (Scotland) 4* (UK) 4*+3* (Scotland) 4*+3* (UK) Pure 19.3 18.2 61.2 58.1 Applied 13.8 16.7 57.0 59.1 Stats & OR 11.1 17.0 47.0 59.0 Total 15.0 17.3 56.0 58.7

. The results show the quality of Scottish research to be excellent, especially in pure mathematics, although the Scottish performance at the top, world-leading end of the range in applied mathematics and statistics & OR is slightly below the UK average. When the individual UK submissions are ordered by "quality rating", Q, (a weighted average of the quality profile), then the top Scottish submission is 6th in the UK in pure mathematics (out of 38), 4th in applied mathematics (out of 46), and 14th in statistics & OR (out of 31). [These are the rankings based on the Q-weightings used by SFC and in 2009-10 by HEFCE. HEFCE's 2010-11 weightings give the top Scottish positions as 6th, 5th, and 15th, respectively.]

. Scotland also performs well in bibliometric studies – e.g. the Times Higher (7 May 2009) published a table showing Scotland to be second in the world ranking of mathematical sciences citations per research paper, ahead of both the US and England (although care needs to be taken in any interpretation of citation counts).

. Scotland is host to the International Centre for Mathematical Sciences (ICMS), the sister organisation of the Isaac Newton Institute (INIMS). The two perform complementary roles: INIMS running 2-3 long programmes and ICMS running 15 or so workshops per year. The two centres are the UK members of ERCOM, the umbrella organisation for European Research Centres. The mathematical standard of ICMS workshops is extremely high as indicated by the comments of the international experts who are asked to referee proposals, while both the academic and organisational quality of the workshops is attested to by the exceptionally favourable responses of participants.

. Scottish departments have had considerable success in attracting large grants in the decade from 2000. These include awards for two EPSRC multidisciplinary critical mass 33

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centres (out of 6 funded); two EPSRC science and innovation awards (out of <10 funded in mathematical sciences); a significant (£1.7M) 'rolling grant' and share of £2.4M for high performance computing from STFC; and an €1.7 Million EU Advanced Investigator award. Many of these awards are collaborative, involving other disciplines or more than one institution. Even the largest Scottish mathematical sciences departments are dwarfed by the larger English departments, but collaboration enables Scottish departments to compete for research funding with institutions from England or overseas. It is thus profoundly damaging to the health of the discipline in Scotland that the Scottish Funding Council (SFC) is no longer in a position to fund a mathematical sciences "research pool", because of UK public funding cuts (for information on SFC's pooling initiative and a list of funded disciplines see www.sfc.ac.uk/research/researchpools/researchpools.aspx ).

. The UK spends much less on the mathematical sciences relative to other disciplines (such as physics) than is typical for a developed country, and future funding cuts threaten the (currently excellent) quality and amount of UK mathematical sciences research. In fact, Scottish departments are especially vulnerable, because SFC has already cut mathematical sciences research funding by about 25% over the past two years (this was due to a change in funding methodology; see Section I for more information). Further funding cuts pose a severe threat to the health of the discipline, especially when combined with concentration of funding into far fewer departments. The EPSRC Mathematical Sciences Programme's new invitation-only "platform grant" scheme was particularly disappointing in this regard: it was restricted to the top 5-6 UK departments as measured by grant income from that EPSRC Programme. (No Scottish departments were allowed to apply under this scheme.)

End of response by Edinburgh Mathematical Society (return to ‘A’ response list)

Response by London School of Economics

. The UK is at least as well-placed as any other country (including the USA) to conduct cutting-edge research in time series analysis - especially in nonlinear time series, econometric time series, and high-dimensional time series. Internationally leading figures are found among both the statistics and econometrics communities in the UK. The editors of the Journal of Time Series Analysis and the Journal of Time Series Econometrics, the only two major journals in time series, are UK-based.

. In social statistics, the UK has many centres within and outside universities that produce research and training in the areas of applied social surveys, demography, population changes, social exclusion, well-being, educational testing, psychometrics, sample survey methods, epidemiology and health. The research is conducted by leading researchers in the various fields but is generally concentrated in a small number of institutions. The journals published by the Royal Statistical Society - Series A: Statistics in Society and Series C: Applied Statistics - are highly rated.

. The UK has long had a strong base of research in probability and statistics, with internationally competitive academic centres across the country. This provides an ideal environment for research in financial and insurance mathematics, which relies on advanced methods from stochastics. In addition, due to a high demand for MSc programmes in Financial Mathematics, universities have rapidly expanded in this field.

. Discrete mathematics is well represented as a topic in the UK at a number of universities. The area is relevant to the design and analysis of algorithms and helps strengthen theoretical computer science.

. Mathematical game theory is a relatively research area, but the UK has notably attracted a number of excellent researchers in the field. This is particularly evident in the area of algorithmic game theory – a field which has experienced recent rapid growth not least because of the importance of the internet and its interactions.

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. In general, world-class expertise in these areas of mathematics is concentrated in the UK at a small number of institutions where it has grown up around groups of outstanding researchers who generate "critical mass".

End of response by London School of Economics (return to ‘A’ response list)

Response by The Institute of Animal Health

. UK researchers are at the leading edge in applying mathematical modelling to epidemiology, both in terms of developing novel modelling approaches and statistical methodologies for estimating parameters based on partially-observed processes. However, most of these researchers are based in biology departments and medical or veterinary schools, rather than mathematics departments and, with a few exceptions (e.g. University of Warwick), interactions with mathematics departments tend to be limited. The principal reasons for success in this area is the availability of good-quality data-sets and active engagement with policy makers.

. More recently, the application of mathematical modelling to problems in cellular and molecular biology has increased, with UK researchers playing a leading role, notably through the Research Councils’ Systems Biology centres. However, the continued successof this area will depend on future availability of funding and greater engagement of experimental scientists.

End of response by The Institute of Animal Health (return to ‘A’ response list)

Response by The Royal Astronomical Society

. The UK is certainly world-leading in many areas of theoretical astronomy, especially in magnetohydrodynamics (MHD) and cosmology. A great tradition has grown up in the UK to do internationally leading research in this area and the UK’s expertise complements the observational expertise in other parts of the world, notably North America. Input from UK researchers in theoretical astronomy is crucial to gaining a proper understanding of observations from a series of new space satellites launched by NASA, Japan and ESA.

. The Society therefore recommends that this work continues to receive the support it needs. In particularly, investment in postdoctoral researchers and research students in applied mathematics in MHD and particle astrophysics should be maintained at current levels.

. There are great opportunities for the future, for example for this work to support the analysis of data from international space missions. However, the Society believes that with the current uncertainty in funding, there is a real threat that the UK’s leaders emigrate and that their research groups will not be maintained and enhanced, putting our world-leading role at risk.

End of response by The Royal Astronomical Society (return to ‘A’ response list)

Response by Royal Holloway, University of London

. The UK community continues to be strong and dynamic. Progress in additive number theory has had an influence on group theory and combinatorics as well as within number theory itself. There have been exciting developments in random matrix theory and L- functions, relating to applications in physics. The interest in probabilistic methods and graph theory continues to rise, with progress both in pure problems and in areas motivated by various applications. The UK research community is central to these and many other exciting developments over the past few years.

. We believe that there is an opportunity to develop much stronger links with Computer Science and with Engineering than exist at present. A comparative lack of strong links between Discrete Mathematics and Computer Science was identified by the previous International Review in Mathematics; there has been significant investment in research at

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this interface at some institutions, but we feel that there is still much that can be done. Our perception is that the number of engineers with an interest in mathematics is small when compared to the United States; certainly, there is no direct UK equivalent of the IEEE, which caters (in particular) for researchers on the mathematics/engineering interface.

End of response by Royal Holloway, University of London (return to ‘A’ response list)

Response by Professsor Peter Grindrod CBE CMath

. Excellent.

End of response by Professsor Peter Grindrod CBE CMath (return to ‘A’ response list)

Response by Professor Eduard Babulak, CORDIS Consulting European Commission

. I believe that the standing of the current UK Mathematical Sciences research community is excellent. It is important to work together to create strategies to build a support infrastructure/learning foundation for the next generation of mathematicians in the UK.

End of response by Professor Eduard Babulak, CORDIS Consulting European Commission (return to ‘A’ response list)

Response by Dr Andy Watham, University of Oxford

. Good students have been attracted to the UK for such 1- or 2-year courses and many have either taken the skills learned into industry and commerce or have continued for doctoral research. . Many excellent home students, uncertain about taking on research directly after a first degree have found real interest in the applicability of mathematics and have gone on to industry or research. . MSc study provides a real bridge between undergraduate study when one is taught things that are known and graduate research when there is a large amount to understand to get to the forefront of research. Advanced courses with project work and a significant dissertation as usually found.

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Response by University of Wales, Swansea

. The U.K. is a net importer of research mathematicians of international standard. In some departments 70% of all research active staff has a non-U.K. background, in many mathematical subject areas a share of 50% or more is normal. This high portion of non- U.K. trained mathematicians has transformed the landscape dramatically. In particular smaller places are now recruiting often top researchers not anymore linked by their research or training to the traditional strong and large places such as Cambridge, Edinburgh, Imperial College London, Oxford or Warwick, and thus bringing exciting mathematics to the U.K. which was not represented here before. Consequently these places could built up own profile on the international scene. However, in some cases, especially when national peer review is crucial (EPSRC, RAE/REF), such departments might have clear disadvantages due to the fact that the traditional centres and their off- springs dominate the process.

. Non-commutative geometry and non-commutative (or free) probability and their relations to operator algebras and mathematical physics is a particular good example where in close collaboration smaller places such as Aberystwyth, Cardiff and Swansea have built up a strong, internationally highly regarded group which under the umbrella of WIMCS and with further support (including EU and RCUK funding) raised the profile of this area in the U.K. The strength of the pure mathematical work conducted in this group becomes evident for example by pointing to papers published in Acta Mathematica or Communications in

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Mathematical Physics. The international collaboration includes partners in Australia, Belgium, Canada, China, Germany, Italy, Japan, Poland, Russia or the U.S.A.

. Smaller institutions can achieve a lot when concentrating their resources. Traditionally stochastic analysis (in a wide sense) is very strong in the U.K., indeed here can the U.K. claim world leader status. Strong groups in Cambridge, Imperial College London, Oxford and Warwick are supplemented by equally strong groups in Bath, Manchester, Swansea or York. In this context, by looking at the probability group in Swansea, we can see how by concentration a smaller place may become a major player. This group has currently 11 research active members ( a bit more than 50% of all staff) including 4 professorial staff. Building on the tradition founded by D. Williams and J. Hawkes, this group is strongly interacting with many U.K. places as those mentioned above and on the international scene it has collaborations for example with colleagues in Beijing (CAS and BNU), Bielefeld, Bonn, Dresden, Kyoto, MIT, Moscow, Osaka, Paris, Seattle, Tokyo, Toulouse, Wroclaw or Zurich (ETH and Univ.) Three areas are of particular strength: 1. Probabilistic methods in mathematical physics, where the U.K. as a whole is leading, especially when it comes to probabilistic methods applied to fluids, often in the context of stochastic partial differential equations. 2. Micro-local analysis and jump processes, where Swansea together with Warwick host some of the most distinguished researchers. 3. Functional inequalities and spectral analysis, where the appointment of F.-Y. Wang as a WIMCS research professor was a coup much to the benefit of the whole U.K. community.

. We should add a further important feature of this group : the active knowledge transfer from pure probability theory to engineering. Probabilistic techniques are used rarely by U.K engineers (when compared with France, Japan or the U.S.A.). Meanwhile there are two groups led by world class engineers (R.Owen FRS and S.Adhikari, Leverhulme Prize winner) working with probabilists of this group and publishing joint papers in leading engineering journals. Again WIMCS plaid a crucial role in establishing this collaboration.

. As pointed out in the last IRM, the U.K. was losing out in parts of analysis, especially in nonlinear partial differential equations. The funding of two new centres by EPSRC (Oxford and Edinburgh) made a difference which was supported by some departments such as Bath or Swansea by strengthening the existing group or in the latter case by building up a new group. The Oxford centre is giving enormous support to these groups. The new, small group in Swansea (2 professors and 3 further staff) immediately integrated into the U.K. community, and on the other hand, due to its own strength and international connections it added value, for example in the field of nonlinear parabolic equations or the application of the calculus of variations. Part of this research is rather applied, for example a small team of mathematicians and engineers led by the WIMCS research professor K. Zhang have applied for a patent with several applications in imaging processing. The international collaboration includes groups in Australia, China, Germany, Israel, Russia, Spain or Ukraine.

. The U.K. has a strong tradition of rigorous spectral theory going back to E. Titchmarsh’s time. With many overseas researchers being appointed in this subject, especially in the London area, the U.K. continues to have a leading role. Recently surprising results on the spectral theory of non-symmetric operators were obtained, especially by E.B.Davies. Again, a small place such as Cardiff, by concentrating its resources, has build up a very active group in this area which is highly respected. The collaboration of this group with the nonlinear PDEs group in Swansea and colleagues in Aberystwyth (under the umbrella of WIMCS) has added to the U.K. landscape.

. In logic and theoretical computer science the UK track record is truly world class since the work of Boole, Russell and, in particular, Turing. This has been noted in the International Review of Computer Science. The hugely successful Computability in Europe (CiE) organisation is an example of a world-leading initiative started by UK pure mathematics at Leeds. Swansea plays an important role in CiE.

. There are further examples of small places with serious contributions such as Exeter in arithmetic geometry.

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. To summarize: The aim of our comments is to highlight by giving examples the important contributions made by smaller departments to the overall U.K. strength.

. Sad but true, in the current funding climate combined with a policy to strengthen the existing large places, smaller mathematics departments are, independently of their achievements, seriously under threat and by this the diversity of mathematical research in the U.K. will suffer. In particular we may see world class researchers located at some of these smaller places leaving the country, aiming for better working conditions, i.e. research funding or support for PhD-students. It is also apparent that in a situation where the performance and stability of larger places dependent on a transfer of resources so far allocated to smaller places might jeopardize the peer review system.

End of response by University of Wales, Swansea (return to ‘A’ response list)

Response by Bangor University (1)

. I have not the resources to investigate this question overall.

I am a founder member of the Editorial Board of 3 International Journals, all listed by ISI.

In some sense this is a sociological question on ‘standing’ and not the same question as research quality. I hate this deference to ‘global’!

End of response by Professor R Brown, Bangor University (return to ‘A’ response list)

Response by Cardiff University

. UK research in mathematics is behind the USA, Russia, Germany and France, in general. Sweden and Denmark out-perform the UK in areas such as analysis of partial differential equations, complex analysis, and operator algebras. The UK is leading in areas such as geometry, algebraic geometry, some branches of analysis, industrial applied mathematics, computational mathematics and fluids.

. The main threats are the way that Mathematics is funded, inadequate support for PhD training and post-docs, particularly in the smaller departments where there are pockets of excellence which cannot fulfil their potential, and which may not achieve the impact of which this excellence is capable.

. Mathematics is a subject where small departments can make very significant contributions (despite our small size, at Cardiff, for example, we are coordinating a EU network in noncommutative geometry, contributing to the mathematical foundations of operational research thanks to a major EPSRC Science and Innovation award, and exploring innovative links between mathematics and health modelling). The trend towards concentration of research funding into a small number of larger centres, and the uneven distribution of funds resulting from the award of a small number of large grants, makes it more difficult for smaller departments to fulfil their potential.

End of response by Cardiff University (return to ‘A’ response list)

Response by Aberystwyth University

. The issue of size is one that affects all Welsh Universities. At present Cardiff, Swansea and Aberystwyth are involved in internationally leading research, as can be demonstrated by research outputs. The Quantum Systems, Information and Control group has produced publications is strong traditional journals such as Communications in Mathematical Physics, Journal of Mathematical Physics, Journal of Functional analysis, etc., while also conducting interdisciplinary research which has lead to publications in engineering journals (IEEE transaction in automatic control) and physics journals (Phys. Rev. A). International collaborators include Matthew James, Hendra Nurdin, Masahiro Yanagisawa (ANU, Canberra), Burkhardt Kuemmerer (Darmstadt, Germany), Oleg Smolyanov (Moscow,

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Russia). The Mathematical Modelling group likewise publishes in Applied Mathematics and appropriate interdisciplinary journals.

. While these research groups are active and have achieved international recognition for their research, and have attracted PDRAs and international visitors, their size has been a disadvantage when it comes to evaluation (especially in research assessments and research council funding). Nevertheless it should be emphasized that smaller groups can conduct world-leading research when their efforts are focussed, and supported.

. An example where the UK is internationally leading is Quantum Probability, which until recently had the largest European centre for this subject in Nottingham, however, the same researchers are now spread out amongst Nottingham, Aberystwyth and Lancaster. The Aberystwyth group was set up in 2007 on the adventurous theme of quantum feedback control, a rapidly developing subject involving collaborations with theoretical and experimental physicists and engineers, and is distinctive in applying quantum stochastic calculus to this field. It has attracted Dr Claus Koestler as EPSRC PDRA who is an acknowledged expert on non-commutative independence which has implications for pure mathematics topics such as braid and symmetric groups. Members of the group have visited India (Bangalore), Korea and Australia (Canberra). We would like to retain Dr Koestler on a permanent position, but this is difficult at present.

. Professor Simon Cox is supported by an EPSRC advanced fellowship and Professor Gennady Mishuris as WIMCS professor of computational modelling.

. Mathematical Physics has historically been a UK strength (Newton, Maxwell, Dirac, Penrose & Hawking), and there is a particularly promising body of research, which involves mathematicians with strong theoretical physics backgrounds, that is applicable to technological developments.

. At present, there is a concentration of graduate training in small number of centres. It would immensely benefit Welsh mathematics to see more graduate students ships available here.

End of response by Aberystwyth University (return to ‘A’ response list)

Response by University of Salford

. The UK contributes internationally to mathematical sciences research across the full range of subjects. Several universities in the UK have pure mathematics departments, while most have applied mathematics (physics, statistics, computing, operational research, etc.) departments. The UK publishes many journals, hosts many conferences and has many professional bodies that are concerned with research in the mathematical sciences. Perhaps the largest threat to research in this area is the doubtful availability of government funding for the foreseeable future, which is likely to harm recruitment plans and might close degree courses. As well as affecting research in the subject, both consequences might subsequently and adversely affect the uptake of mathematics in schools and colleges, as was previously evidenced when AS-levels were introduced.

End of response by University of Salford (return to ‘A’ response list)

Response by Institute of Mathematics and its Applications

. UK mathematics, both pure and applied, enjoys excellent international standing. As identified by the RAE08, `almost all applied mathematics research in the UK is recognised internationally’. Moreover, the REA08 panel identified the strength in depth of the community, and in particular noted that `world-leading research is done all over the UK’.

. The IMA understands, and supports, the need for the UK mathematics community to increase its engagement with other areas, including the cross-cutting themes identified by RCUK. This is particularly important in the current economic climate. However, we do wish to stress that the UK has an internationally leading research base in mathematics, and that core activity of fundamental curiosity-driven research in the subject needs to be 39

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protected as key underpinning for research of more immediate economic impact. We also note that undergraduate numbers in mathematics continue to be exceedingly buoyant, and that these students will be best taught by an active and thriving university mathematics research community.

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Response by University of Stirling

. UK Mathematical Sciences research community has a globally excellent standing. In particular, in our areas the UK has an internationally leading position, second only to the USA. For example, about 50% of participants in the recent top conference on mathematical epidemiology (Epidemics 2, Greece 2010) came from the UK.

End of response by University of Stirling (return to ‘A’ response list)

Response by Institute of Food Research

. UK produces some world-class mathematicians as the number of various international awards prove.

End of response by Institute of Food Research (return to ‘A’ response list)

Response by Ben Aston, Gantry House

. The profile of mathematics researchers suggests a weakness in communication.

End of response by Ben Aston,Gantry House (return to ‘A’ response list)

Response by Royal Statistical Society

. The UK is internationally leading in statistics. It has particular strengths in applications domains, developing new statistical methodology to meet the new challenges which arise. Recent examples of application areas which have presented new challenges, and to which UK statisticians have made central contributions, include genomics, proteomics, metabonomics, and bioinformatics in general, large data set problems in particle physics and astronomy, and banking, especially the retail banking sector. It is no exaggeration to say that the UK was (and remains) at the forefront of the Bayesian revolution, which is sweeping statistics itself as well as a huge range of application areas of statistics.

. There is a general concern that insufficient statisticians are being trained for sustainability. A report entitled "The teaching of statistics in UK universities", published in 2007 (J.Roy.Stat.Soc.A, 170, p581-622) concluded that "the number of [statistics] staff in core mathematical science groups peaked in 1996 and has declined since then", and projected a continuing decline. In particular, because statisticians are in demand in so many application domains, the report added "on present trends the UK academic system within statistics is unlikely to be self-sustaining; it will depend on recruiting qualified statisticians from overseas". It is true that attempts have been made to tackle this problem (e.g. taught course and doctoral training centres) but concerns remain.

. In particular, statistics has continued to suffer from non-integrated local decisions by individual vice-chancellors to cut or reduce statistics teaching at their universities. Two very recent examples of this are the termination of the Masters courses at Sheffield Hallam University and Edinburgh Napier University. There is a very real danger that such uncoordinated local decisions will leave the UK with insufficient ability to produced skilled personnel in an area which is increasingly important to the UK national economy and competitiveness. Some kind of integrated national strategy is needed.

. In terms of gaps, there is increasing concern that insufficient social statisticians are being trained. This may be partly because there is considerable, and increasing demand for 40

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statisticians in other areas (bioinformatics, for example) so that the small finite pool has shifted away from the social sciences, and partly because the world is becoming more quantitative.

. Collectively, these concerns and the diverse nature of statistics lead the Society to comment on the findings of the last IRM with the following: “Consultation with other Research Councils towards establishing a one-stop-shop for statistics research funding, both core and applied: Such an initiative would be able to take a strategic view of the discipline, and maintain a balance between applications and methodology. It would be a natural development from the existing systems of cooperation." This comment is still strongly patent in todays research landscape.

End of response Royal Statistical Society (return to ‘A’ response list)

Response by Defence Science and Technology Laboratory (DSTL)

. Unable to comment in general as most of our interaction is in niche areas that require applied mathematical support rather than with the full breadth of the UK and international community.

. We have a number of senior staff - typically Dstl Fellows or Senior Fellows - who are at the forefront of their fields internationally; some of these are in mathematical-related disciplines including Operational Research.

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Response by The Actuarial Profession

. The UK actuarial science research community has internationally recognised centres of excellence and is active in international actuarial research. Actuarial research has traditionally tended to be concentrated at a few universities which specialise in this area but, in the last 5 years, the Actuarial Profession has accredited more universities to teach its core syllabus. This has led to more actuarial teaching staff and now those universities are seeking to expand into actuarial research. Examples include Leicester, Manchester, Queen’s Belfast and UEA There is therefore an opportunity to grow the number of university departments active in actuarial research, so improving the diversity of the research that can be covered (including more specialism’s), the possibility of different interdisciplinary relationships developing, and the quality that comes with competition. The UK has a great opportunity to nurture this expanding research base and so enhance its international reputation both in breadth and depth.

End of response by The Actuarial Profession (return to ‘A’ response list)

Response by The Operational Research Society

. In 2004, EPSRC carried out an international review of Operational Research which was separate from the international review of Mathematics. The report can be found at http://www.epsrc.ac.uk/pubs/reports/Documents/OpResReview.pdf

. The OR community had expressed a strong preference for having a separate review again but we understand that EPSRC were unable to do this because of timing issues. The UK OR community has responded extremely positively to the observations and recommendations that were outlined in that 2004 report. For example, since 2004, EPSRC funding for OR has increased significantly. Indeed, there have been a number of high profile OR grants in the period. The LANCS programme was funded by EPSRC under the Science and Innovation initiative in 2008. This is a £13M consortium of four universities (Lancaster, Nottingham, Cardiff and Southampton). Over £5M was provided by EPSRC with the remainder being provided by the institutions. We believe that this represents the largest OR grant in the world. It is complementary to a £3.8M award to Warwick in 2007 (under the same initiative) in discrete mathematics and its applications (DIMAP) which has 41

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a significant OR contribution. In, 2010, EPSRC funded a Doctoral Training Centre at Lancaster University in Statistics and OR (STOR-i) to the value of £4.5M. In 2006, Nottingham was awarded a £2.6M grant to explore the automation of the heuristic design process and in 2010 the same group was awarded a prestigious £1M EPSRC platform renewal award. In 2006, a consortium of six universities was awarded an EPSRC grant of £240K to establish NATCOR, a national training centre for PhD students in OR. This brief list of grants is certainly not exhaustive. The OR Society’s view is that UK OR has strong international recognition and a high profile in the international community and that this portfolio of national funding is crucial in maintaining and enhancing this position.

End of response by The Operational Research Society (return to ‘A’ response list)

Response by The British Academy

. The UK is internationally leading in statistics. It is particularly strong in applications, where new challenges have stimulated the development of new methodology. In particular, electronic data capture and storage has facilitated the growth of large and complex data sets describing social structures and phenomena, which in turn require the development of new statistical methods and tools. Three recent examples of such developments which had their origins in the UK are (a) the Bayesian revolution, (b) multilevel models, and (c) generalised linear models.

. Since statistics might be defined as the science of quantitative evidence, it is clear that the discipline must be ubiquitous, underlying a great many other disciplines. This is particularly true of the social sciences.

. When we move to the formal and mathematical techniques that are increasingly used in linguistic and psychological research and theory construction, a similarly positive picture emerges. There are a number of world-renowned research centres and teams located in the UK. Salient contributions have been made and continue to be made in relation to natural language processing, deep and shallow text mining, and human computer interaction, as well as in the modelling of human choice and decision making.

. In terms of supply, however, there is a particular concern that an insufficient number of statisticians are being trained to ensure sustainability. Many statisticians, quite properly, go to work in applications areas and so are not available to train the next generation of statisticians. A report published in 2007 drew attention to this risk (J.Roy.Stat.Soc.A, 170, 581-622), noting (p. 605): "On present trends the UK academic system within statistics is unlikely to be self-sustaining; it will depend on recruiting qualified statisticians from overseas".

. Furthermore, the lack of coordinated decision-making by different universities is having an adverse impact on statistics training. In particular, in recent years a number of universities, driven by local financial considerations rather than national needs, have axed Masters courses in statistics without taking into account the wider needs of the UK, or what other universities are doing. The danger is that such unilateral actions will seriously damage UK capability in statistics, especially social statistics - an area one might justifiably describe as an important infrastructure technology.

. Since too few statisticians are being trained, and since the demands and opportunities for statisticians from competing areas continue to grow (e.g. bioinformatics, the financial services, etc, which are very attractive alternatives for statisticians) it is perhaps not surprising that there is concern that there are insufficient social statisticians being trained.

End of response by The British Academy (return to ‘A’ response list)

Response by The London Mathematical Society

. UK Research activity in the mathematical sciences was rated very highly in RAE2008, with 49% or more of research activity1 in each of the three relevant subject areas being rated as internationally excellent (3*) or world-leading (4*). Eminent international 42

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mathematical scientists were members of each of the three mathematical science sub- panels. Two additional features are worthy of note in the table below:

- Performance is very similar across all three disciplines. - Excellence is widespread throughout the research community, not being concentrated only in a small number of large submissions.

%4* %3* %2* %1* Pure: all submissions 14 39 39 6

Pure: biggest 4 submissions 30 43 26 1 Pure: biggest 8 submissions 22 39 30 6 Applied: all submissions 12 37 38 13 Applied: biggest 4 26 45 29 0 Applied: biggest 8 24 43 31 2 Statistics+OR: all submissions 14 40 37 9 Statistics+OR: biggest 4 22 45 29 4 Statistics+OR: biggest 8 22 44 30 3

. Evidence from the RAE also indicates that the volume of research activity in UK mathematical sciences increased in the period 1997-2007: 1783 FTE were submitted to the three mathematical sciences sub-panels in RAE2001, 1946 FTE in RAE 2008, an increase of 9%.

. Citation indices are widely recognised to be a rather unreliable means of judging performance in the mathematical sciences, at the level of individual researchers, or small groups of researchers. Nevertheless, when one is dealing with normalised indices at the scale of a country’s output, they offer a useful guide to research strength. As such, they provide a second indicator of the strength in depth of UK mathematical sciences in an international context at the start of the 21st century – the Thomson Reuters Citations Top 20, for mathematical sciences 1998-2008, shows Scotland in second place, England in fourth2.

. Notwithstanding the current high quality of research performance noted above, the UK spends much less on the mathematical sciences relative to other disciplines (such as physics) than is typical for a developed country, and future funding cuts threaten the (currently excellent) quality and amount of UK mathematical sciences research.

End of response by The London Mathematical Society (return to ‘A’ response list)

Response by University of Sussex, Professor Robert Allison

. The UK has several world leading centers in mathematical sciences as well as a number of pockets of excellence. In many cases UK mathematics is on the forefront of mathematics, whereas other fields with strong UK traditions that are no longer considered important research fields internationally tend to adapt only slowly to international trends.

. The research quality is in general high, but tend to be geared to quick payoff type projects as a consequence of the national and regional evaluation systems. Due to funding aspects on research all types of theoretical mathematics is under pressure and has to give way to more applied research. That is, in the sense of “applying mathematics to science” instead of “analyzing mathematical methods and mathematics from science”. This is also affecting the profile of researchers and in particular in smaller institutions the width of the subject is threatened.

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Response by University of Sussex, Omar Lakkis

. The UK, as a single entity, ranks in the top 7 in Mathematical Sciences in the world. In Europe it is, along with Germany, France and, possibly, Russia, one of the most prolific countries and a clear leader in the sector.

. I believe the UK has a margin of advantage at the individual researcher and small groups (2 to 4 persons) level, while Germany and France benefit from team work, which can be an advantage in very applied mathematical and computational sciences.

End of response by University of Sussex, Omar Lakkis (return to ‘A’ response list)

Response by University of Edinburgh, Professor Andrew Ranicki

. The UK has brilliant individual researchers rather than a "research community". The overall standing of the UK in the world of mathematics is excellent at the research level. In mathematics researchers, it is individual researchers who have a high profile, rather than individual universities.

End of response by University of Edinburgh, Professor Andrew Ranicki (return to ‘A’ response list)

Response by UK Computing Research Committee

. The UK's standing in research quality and profile of researchers is outstanding in many areas of mathematics of relevance to computer science. The UK computing community welcomes the increasing interest of leading mathematicians in problems inspired by computing, and is excited by the opportunities this offers for advances. Dynamical systems, topology, number theory, stochastic processes, statistics, and operations research all provide current examples, and Bayesian statistics, logic, and combinatorics, in particular, are areas that have been transformed over the past few decades by their importance to computing challenges, with UK scientists in academia and industry taking the lead. While it is not the purpose of this paper to delineate the boundary between mathematics, and other sciences such as computing, physics and biology, it is clear that to see the whole spectrum of mathematical activity in the UK, one needs to look beyond mathematics departments.

. Some examples give a flavour of the breadth and richness of engagement: - Bayesian statistics are becoming increasingly widely used in data mining, natural language understanding, image analysis and risk assessment, with strong groups at Edinburgh, Oxford, UCL, and Cambridge, and broad industry take up. - algorithms and discrete mathematics at Cambridge, Warwick, Oxford, Leeds, Durham and Queen Mary, in particular the award of the to Mark Jerrum of Queen Mary (with Sinclair of Berkeley and Vigoda of Georgia Tech) of the AMS Fulkerson Prize in 2006 - stochastic modelling of programs and processes at Oxford, Edinburgh and Glasgow, with a paper on biological applications of these techniques by Kwiatkowska (Oxford) and others receiving the Top Cited Article award in the journal Theoretical Computer Science for the period 2005-2010. - Web science at Southampton, Oxford, Cambridge, Edinburgh and IBM: Ian Horrocks (Manchester now Oxford) devised a logical technique for reasoning about web pages now widely used as an international standard. - Mathematics of networks and complex systems at Warwick, Cambridge and Imperial: Cambridge statisticians and computer scientists are at the forefront of international activity and standards for measuring internet performance. - Information security, both specialised cryptography and broader systems and approaches at Cambridge, Belfast, Bristol, Imperial, Royal Holloway and UCL, with strong interaction through government, industry and GCHQ’s Heilbronn Institute. - Logical foundations of programming at Oxford, Edinburgh, Cambridge, Imperial, Queen Mary and Microsoft Research: Sir Tony Hoare and the late Robin Milner both held Turing Awards, the computing "Nobel Prize". Abramsky (Oxford), Burstall (Edinburgh) and Plotkin 44

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(Edinburgh), have all recently been awarded major international retrospective prizes, and O’Hearn and Cook received widespread recognition for practical new techniques for proving programs terminate. - Work in quantum information science at Imperial, Oxford and Bristol, drawing together computing, mathematics, physics and electronic engineering. - Operations research at Lancaster, Nottingham, Cardiff and Southampton, stimulated by a recent major EPSRC Science and Innovation award: the group’s OR based software is about to be rolled out for runway scheduling at Heathrow. - Recent winners of the BCS Roger Needham Award, the UK's flagship award for computer science research conducted within ten years of a PhD, have included a number of international researchers, with a strong mathematical trend in nearly all recent awards: Joël Ouaknine (control theory), Byron Cook (logic), Wenfei Fan (logic and databases), Mark Handley (network modelling), Andrew Fitzgibbon (computer vision), Ian Horrocks (logic and the web) and Jane Hillston (stochastic processes). - It has been exciting to see the UK leading in the use of new computing technologies in mathematics – for example Gonthier (Microsoft Research Cambridge) made the first machine checked proof of the four colour theorem, Gowers (Cambridge) has helped initiate

PolyMath, a new web-based approach to collaborative problem solving, and Linton (St Andrews) leads the international GAP consortium for algebra software.

End of response by UK Computing Research Committee (return to ‘A’ response list)

International Review of Mathematical Sciences 2010 PART 3 Information for the Panel Responses to Stakeholder/Public Consultation Responses to B

B. What evidence is there to indicate the existence of creativity and adventure in UK Mathematical Sciences research?

(return to Main Index) The following key points were highlighted:

. Some contributors felt that research graded in the 3*/4* star band in the 2008 RAE, accounting for over half of the research outputs, is world leading and innovative and hence a measure of creativity and adventure.

. Some felt that Mathematical research by its very nature is highly creative and adventurous..