INTERNATIONAL REVIEW OF MATHEMATICAL SCIENCES IN THE UNITED KINGDOM ...

International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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 Information for the Panel

PART I 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

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International Review of Mathematical Sciences 2010 Information for the Panel

PART I Evidence Prepared by EPSRC

List of Annexes Annex A Annex B Annex C Annex D Annex E Annex F Annex G Annex H

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

Table of Tables Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7 Table 8 Table 9 Table 10 Table 11 Table 12 Table 13 Table 14 Table 15 Table 16 Table 17 Table 18 Table 19

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

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International Review of Mathematical Sciences 2010 Information for the Panel

PART I Evidence Prepared by EPSRC

Table of Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Figure 24 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29

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

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International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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

1

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 1 information on how QR funding is spent is available , and care should be used in drawing

For an explanation of ‘QR funding’ see 2.1.4

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International Review of Mathematical Sciences 2010 Information for the Panel

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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

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PART I 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 2 context of a t en-year Science and Innovation Investment F ramework , 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 3 public body (the Technology Strategy Board ) 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 4

• seven Research Councils 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 ar e channelled v ia t he R esearch C ouncils ( mainly in the f orm of 5 research grants) and the four funding bodies 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 3 4 5

This was updated in 2006 and is available at http://www.bis.gov.uk/files/file29096.pdf , See also section 2.4 and http://www.innovateuk.org/aboutus.ashx For further information on the Research Councils se section 3 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.

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International Review of Mathematical Sciences 2010 Information for the Panel

Figure 1

Table 1

PART I Evidence Prepared by EPSRC

UK higher education institutions’ income from research grants & contracts and 6 funding council grants (Source: Research Information Network)

Research Councils and Funding Bodies actual research budget allocations for 2004/05 to 2010/11 (£000s) 2010/11

Funding Bodies

Research Councils

AHRC BBSRC EPSRC ESRC MRC NERC PPARC CCLRC STFC total HEFCE SFC HEFCW DELNI total Overall total

2.1.6

6

2004-05 67,746 287,571 497,318 105,252 455,279 314,256 274,037 127,940 n/a 2,129,399 1,078,750 174,220 66,480 37,158 1,356,608 3,486,007

2005-06 80,536 336,186 568,193 123,465 478,787 334,047 293,916 167,004 n/a 2,382,134 1,249,254 181,408 68,690 40,301 1,539,653 3,921,787

2006-07 91,379 371,644 636,294 142,468 503,461 359,367 306,540 182,256 n/a 2,593,409 1,340,843 201,958 70,257 44,155 1,657,212 4,250,621

2007-08 96,792 386,854 711,112 150,136 543,399 372,398 n/a n/a 573,464 2,834,155 1,413,014 217,444 75,016 47,264 1,752,738 4,586,893

2008-09 103,492 427,000 795,057 164,924 605,538 392,150 n/a n/a 623,641 3,111,802 1,458,447 228,890 76,965 49,251 1,813,554 4,925,356

2009-10 104,397 452,563 814,528 170,614 658,472 408,162 n/a n/a 630,337 3,239,073 1,582,636 240,543 84,738 53,932 1,961,849 5,200,922

2010-11 108,827 471,057 843,465 177,574 707,025 436,000 n/a n/a 651,636 3,395,584 1,603,000 242,696 84,905 56,994 1,987,595 5,383,179

increase against

2004/05 61% 64% 70% 69% 55% 39% n/a n/a 62% 59% 49% 39% 28% 53% 47% 54%

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

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-researchfunding-uk-higher-education

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the three financial years 2008/9-2010/11). The government has recently announced that the resource e lement of t he ‘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 ar e expected t o be published b y BIS i n 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 J une 20 09 by merging the Department f or Innovation, U niversities a nd S kills (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 S trategy Board are directly overseen by the Director General for Science & Research ( DGSR), currently Professor A drian Smith. P rofessor S mith w as 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 Information for the Panel

PART I 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 B IS t he DGSR and the S cience 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 7 balance allocated to the National Academies and other focussed areas of investment . 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. 8

2.2.4

An excellent overview 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-policyuk-introduction (also available in the ‘Research Council Overviews’ folder on the MSIR FTP site (login required).

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International Review of Mathematical Sciences 2010 Information for the Panel

Figure 3

PART I Evidence Prepared by EPSRC

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 9 co-funding of £180 m illion from t he R egional D evelopment A gencies . 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 r eplaced by t he Science R esearch I nvestment F und (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.

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(£40M per annum) and Cancer Research UK (£330M per annum) support research in a variety of areas. 2.6.3

The E U F ramework Programmes (e.g. FP6, FP7) are also a significant source of f unds 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 i s a lso available to r esearchers t hrough Marie C urie 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 en gagement. Through their investments Research Councils seek to ensure that the UK is one of the most attractive locations i n t he world f or s cience an d i nnovation. T he r elationship of R CUK with B IS 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 10 expertise. Through a Brussels office 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/

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International Review of Mathematical Sciences 2010 Information for the Panel

Table 2

PART I Evidence Prepared by EPSRC

The Research Councils

Research Council & Remit Investment Arts and Humanities Research Council (AHRC) Languages, law, archaeology, Approximately 700 research awards English literature, design, and around 1,500 postgraduate creative and performing arts. awards annually Biotechnology and Biological Sciences Research Council (BBSRC) Life sciences, agriculture, Supports around 1600 scientists and healthcare, food, chemical and 2000 research students in pharmaceutical research. universities and institutes in the UK Economic and Social Research Council (ESRC) The study of society and the Supports over 2,500 researchers and manner in which people behave more than 2,000 postgraduate and impact on the world students. including longitudinal studies, economics and development studies. Engineering and Physical Sciences Research Council (EPSRC) Chemistry, physics, Research and training, with a mathematical sciences, portfolio of around 5400 grants materials science, information supporting approximately 8000 and communications technology, researchers and 7,500 doctoral engineering and high students annually. performance computing. Medical Research Council (MRC) Mission to improve human Directly employs approximately 4000 health through excellent science staff and supports around 3,000 from molecular science to public researchers and 1400 postgraduate health research. students in universities, hospitals and its own institutes. Natural Environment Research Council (NERC) Increases knowledge and Directly employs approximately 2500 understanding of the natural staff and supports around 1200 world – including climate researchers and 1000 postgraduate change, biodiversity and natural students in universities and its own hazards institutes. Science and Technology Facilities Council (STFC) Responsible for the UK research Supports more than 700 current programme in astronomy, space research grants, over 600 students, science, particle and nuclear and UK access to world-wide physics, and for investment in facilities. large scientific facilities used across the research base. Figure 4

Diagram of Government – Research Council hierarchy

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2010/11 budget

Overview www.ahrc.ac.uk £110m AHRC (ftp site login required) £470m

www.bbsrc.ac.uk BBSRC (ftp site login required)

£180m

www.esrc.ac.uk ESRC (ftp site login required)

www.epsrc.ac.uk £850m EPSRC (ftp site login required)

£710m

www.mrc.ac.uk MRC (ftp site login required)

£440m

www.nerc.ac.uk NERC (ftp site login required)

£650m

www.stfc.ac.uk STFC (ftp site login required)

International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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 F ood S ecurity (BBSRC): a multi-agency programme br inging 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 interrelated 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 a nd 25% over 65. There ar e c onsiderable benefits to hav ing 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 through engineering to application ( EPSRC): a multidisciplinary programme that aims to evolve nanoscience and promote the responsible development and management of nanotechnology.

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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 i nvesting ar ound £850 m illion a year t o promote and enc ourage r esearch 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 11 following advice received from Advisory Bodies , 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 12 relevant to priority themes) . Table 3 outlines the coverage of the current core research programmes and their antecedents.

Figure 5

4.1.3

11 12

EPSRC - The Big Picture (2008-11)

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

See section 4.2 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.

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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

Table 3

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. EPSRC’s core programmes post April 2008 Current Programme

Current Programme Research Areas Include:

Information and Communications Technology (ICT)

Communications, computer science, people & interactivity, semiconductor materials for device applications including modelling, & electronic & photonic devices

Materials, Mechanical and Medical Engineering

Aero & hydrodynamics, control & instrumentation, generic manufacturing, mechanical engineering, medical 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 and Sustainability

Process engineering, Built environment & civil engineering, Water, 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 13 predominantly from the academic sector. UP represents the ‘user’ community , 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, gov ernment an d ed ucation. SIP aims t o hel p EPSRC t ake 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.

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• 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 14 Investment Framework 2004-14’ is articulated in the 2006 Strategic Plan and the 15 current Delivery Plan (2008-2011) . 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 go als for t he next three to five years. EPSRC i ntends 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

14 15

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

See http://www.epsrc.ac.uk/SiteCollectionDocuments/Publications/corporate/EPSRCSP06.pdf See http://www.epsrc.ac.uk/SiteCollectionDocuments/Publications/corporate/DeliveryPlanUpdatedFor201011.pdf

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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

and in t he Guidance on t he peer review pr ocess is available on the E PSRC w ebsite 17 EPSRC Funding Guide which can be downloaded from the website . 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.

16

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, 18 giving an overall f unding r ate of 26% by num ber and 27% b y v alue. 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

16 17 18 19

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 i nvolve 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 19 research areas .

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

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Figure 6

4.4.10

PART I Evidence Prepared by EPSRC

Distribution of requested support duration and value - Responsive mode applications 2009-10

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 19

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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 20 allow leading groups 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 r esearch c apacity t hrough 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 21 . Research quality of the training and other support has improved considerably” 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

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Table 4

PART I Evidence Prepared by EPSRC

Main EPSRC PhD Training Funding Mechanisms Approach

Main features

Doctoral Training Accounts (DTA)

• Flexible 4 year blocks of funding (Doctoral Training Grants (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 Training (CDT)

• Funded after a competition. • 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. 22 • EPSRC currently has more than 50 centres .

Industrial CASE Studentships

• Enable companies to work with academic partners of their choice 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. • Currently awarded on research grants.

Project Students 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 t raining 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 23 and IT (20%) .

Masters 4.6.7

22 23

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.

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

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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, crossdisciplinary 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

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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 t he previous 10 years and who are w ithin 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 pr ovide 12 m onths f unding enabling r esearchers t o t ake t ime 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 24 . The following bespoke the chances of their research reaching those beneficiaries 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

24

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

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

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proof-of-concept, ent repreneurship t raining, networking, pe ople exchange (including t he 25 popular ‘Knowledge Transfer Partnerships’ (KTPs) ), 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 r esearchers t o devote time t o de veloping a higher media profile.

4.9.4

As part of t he forthcoming delivery plan, EPSRC h as been working towards embedding public en gagement as an integral element of t he ac tivity we support. From A pril 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

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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:

26

•

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 26 of which staff and other data to return against each cost centre. 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.

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/

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•

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 t o ot her d isciplines and r esearch ar eas, which are typically taken up by the application areas themselves. Research councils have 27 a mechanism in place to evaluate research projects where common interests exist.

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.

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28

different researchers , 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 29 (EPSRC data) Programme Cross-Discipline Interface Materials, Mechanical and Medical Eng Information & Communication Technology Process Environment and Sustainability Physical Sciences Other Energy Infrastructure & International Digital Economy Innovative Manufacturing Nanoscience to Engineering

5.4.3

Figure 8

Investigator number 144 140 138 119 96 63 35 33 24 12 1

% Mathematics Investigators 21.8% 21.2% 20.9% 18.0% 14.5% 9.5% 5.3% 5.0% 3.6% 1.8% 0.2%

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 i s not t rue f or t he s izes of t he ot her programmes’ rectangles, nor for their overlaps with each other. 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.

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5.4.4

Table 6

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. Number of common investigators between the EPSRC Mathematical Sciences programme and other research councils on fEC grants awarded since 2005 (Research Councils data) Research Council BBSRC NERC STFC

5.4.5

PART I Evidence Prepared by EPSRC

Number of Investigators also funded by EPSRC Mathematical Sciences programme 28 15 39

% of EPSRC Mathematical Sciences programme Investigators 4.3% 2.3% 6.0%

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 no t c ome as a surprise, s ince the mathematical physics portfolio of the E PSRC 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 30 themselves from the QR funding allocated to them by the Funding Councils . 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 c aveats ( 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 2 008/09 f or the Mathematics cost centre and all cost c entres r espectively, 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: 31

• In September 2005 full economic costing (fEC) 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 31

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

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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 32 new investments entered into during a particular period. • 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 33 funding for Science & Innovation (S&I) awards. • 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 34 core funding in this area is much lower. Table 7

UK university research funding - 'Mathematical Sciences' cost centres (£M) (excludes QR funding) (HESA data)

Research Councils 35 UK industry UK govt. & health auth. UK-based charities EU government EU other sources Other overseas sources Other sources Totals Table 8

2004-05 23.4 (62%) 2.9 (8%) 4.4 (12%) 2.4 (6%) 3.1 (8%) 0.1 (0%) 1.1 (3%) 0.5 (1%) 37.8

2005-06 28.7 (63%) 2.9 (6%) 3.5 (8%) 2.4 (5%) 4.6 (10%) 0.1 (0%) 2.4 (5%) 0.8 (2%) 45.4

2006-07 34.3 (68%) 2.3 (5%) 4.0 (8%) 2.9 (6%) 4.6 (9%) 0.2 (0%) 1.5 (3%) 0.6 (1%) 50.3

2007-08 41.7 (70%) 2.7 (5%) 4.4 (7%) 3.4 (6%) 4.6 (8%) 0.3 (1%) 1.8 (3%) 0.7 (1%) 59.6

2008-09 5-yr incr. 47.2(69%) 102% 3.3 (5%) 15% 4.4 (6%) 0% 3.6 (5%) 55% 4.5 (7%) 43% 0.7 (1%) 421% 3.7 (5%) 249% 0.8 (1%) 57% 68.2 80%

UK university research funding - all cost centres (£M) (excludes QR funding) (HESA data)

Research Councils 35 UK industry UK govt. & health auth. UK-based charities EU government EU other sources Other overseas sources Other sources Totals

2004-05 919.7 240.9 558.7 698.7 200.8 34.2 150.9 59.6 2,863.5

2005-06 1,064.4 254.5 568.7 720.4 217.4 40.2 171.2 55.8 3,092.5

32

2006-07 1,140.7 289.1 596.0 765.4 256.2 45.8 198.2 55.6 3,347.1

2007-08 1,349.3 295.5 629.3 824.1 277.5 51.7 216.5 51.8 3,695.7

2008-09 1,521.9 311.8 698.4 894.3 322.9 66.5 255.3 49.2 4,120.3

5-yr incr. 65% 29% 25% 28% 61% 95% 69% -17% 44%

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

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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 c ouncils are collectively t he largest s ingle 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 36 provided almost 70% of the funding from this source.

Table 9

Research Councils’ contribution to UK university research funding 37 'Mathematical Sciences' cost centres (HESA data) AHRC BBSRC EPSRC ESRC MRC NERC STFC Other

Totals

Funding (£ M) 0 2,894 32,295 1,770 708 1,335 5,759 2,407 47,168

% of total funding 0% 6% 68% 4% 2% 3% 12% 5% 100%

6.2

EPSRC support for Mathematical Sciences Research

6.2.1

The most significant individual component of EPSRC research funding to the 38 mathematical sciences comes from the Mathematical Sciences programme .

6.2.2

Table 10 below shows the budgets for EPSRC Programmes in the last 5 years

36 37 38

Recall that support drawn from QR funding is not included in this analysis – see section 7.1.1 This data is only available for 2009/10. The percentages are not expected to vary significantly in other years.

Details on the activities run in the areas of the mathematical sciences are included in Annex C 30

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Table 10 EPSRC research budget by programme: new investment (EPSRC data). Current Programme Cross-Disciplinary Interface Digital Economy Energy ICT Infrastructure & International Mathematical Sciences Materials, Mechanical & Medical Eng Process Environment & Sustainability Nanoscience to Engineering Next Generation Healthcare Physical Sciences User-Led Research

Programme 40 (pre-2008) Life Sciences Interface Energy

39

(£M, 2005/6 - 2009/10)

2005/06

2006/07

2007/08

2008/09

2009/10

26.9

31.2

22.5

36.8

41

26.3 88.8 16.4

41.2 87.2 21.5

56 83.3 41 24.2

Engineering

77.7

96.8

94.4

Chemistry Materials Physics -

48.2 47.4 38.2 -

51.7 52.6 49.2 -

42.8 56.3 33.1 -

46.2 57.9 75.6 21.3 15.3 60.7 27 21.1 3.7

19.9 43.8 72 8.2 42 14 63 30.6 5.6 7

100.4

88

82.5

68

-

6.2.3

Researchers i n the mathematical sciences have been successful throughout this per iod 43 in obtaining funding from other research programmes at E PSRC , 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 7 00 projects, and 12 3 of t hese received some el ement of c o-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)

Funding Programme Cross-Discipline Interface Energy Research Capacity Information & Communication Technology Materials, Mechanical and Medical Eng Physical Sciences Process Environment and Sustainability Grand Total 6.2.5

Total Value of Contribution (£M) 3.4 .2 3.3 2.2 2.5 1.2 12.7

Total Contribution as % of grants value 19% 50% 8% 7% 9% 6% 9%

Total Number of Grants 28 1 34 23 19 18 123

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 s hows t he contribution of other or ganisations to projects funded b y 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.

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Table 12 Co-Funding by other public sector organisations to Mathematical Sciences 44 programme grants awarded between 2005/06 and 2009/10 (EPSRC data)

6.2.6

Figure 9

Co-funding Organisation

Total Value Contribution (£k)

Number of Grants

BBSRC DSTL ESRC Inst. of Actuaries NERC Total

378 337 49 143 865 1,772

2 8 1 11 3 25

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 45 Mathematical Sciences) t o mathematics and statistics departments . T he 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 46 . The value of current EPSRC research mathematical sciences (including statistics) 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%. 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.

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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 c alls and activities. During t he f inancial 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.

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Figure 11 Mathematical Sciences programme funding (£M) by department (current grants) 47 (EPSRC data)

6.3.3

Table 13

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. 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) Lead Programme Cross-Discipline Interface ICT Materials, Mechanical & Medical Eng Physical Sciences Process Environment & Sustainability Grand Total

6.3.4

47

Value of Maths Contribution (£M) .9 4.2 1.2 .4 .5 7.2

Contribution as % of grants value 21% 26% 25% 23% 26% 25%

Number of Grants 18 61 18 5 7 109

Almost 60 universities throughout the UK currently receive grants from the Mathematical Sciences programme, however more t han 70% of the f unding has been awarded t o 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.

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.

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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 35

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Hautes Études Scientifique ( Orsay, France), which provides an important platform for collaboration of UK mathematicians with the French community and beyond. 48

6.4.2

is a national and international The Isaac Newton Institute for Mathematical Sciences 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 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 50 mathematical sciences .

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 di vided i nto 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 49 50

49

http://www.newton.ac.uk/ http://www.icms.org.uk/ 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.

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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 t o postgraduate training takes place through block gr ants t o 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

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allocation of EPSRC’s DTA budget by research programme over the last 5 years is 51 shown in Figure 15. Figure 15 Allocation of DTA budget by EPSRC programme over past 4 years (EPSRC data)

52

6.5.11

Data 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 r ecent information av ailable indicates t hat t he pr oportion of mathematics P hD 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 Information for the Panel

PART I 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 51 (2005/06 to 2009/10) (EPSRC data)

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 53 excess of £280M . 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 i n t he 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

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International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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 54 mathematical sciences. 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 Centres 6.5.16

55

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 56 TCCs is available in Annex A .

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 U K P hD s tudents i n 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 55 56

http://www.epsrc.ac.uk/funding/students/centres/Pages/default.aspx http://www.epsrc.ac.uk/about/progs/maths/train/Pages/research.aspx 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 Information for the Panel

Table 14

PART I Evidence Prepared by EPSRC

Destination of UK PhD students by subject area (HESA data)

Destination Further study Higher education Research related employment Other employment Education Not employed Not known or not reported

Chemistry 5% 35%

Physics 1% 39%

Mathematics 5% 45%

Computer Science 3% 46%

Engineering 2% 32%

Materials 5% 26%

Total 3% 37%

24%

16%

8%

7%

9%

11%

13%

24% 2% 7%

33% 2% 7%

30% 2% 8%

30% 2% 7%

48% 1% 6%

45% 0% 13%

36% 2% 7%

2%

2%

3%

4%

3%

0%

3%

7. Support for People The support of world-class people and projects is an ongoing strategic priority for EPSRC. 57 ‘Supporting Excellence’ was an objective in the 2006 Strategic Plan and ‘Developing Leaders’ is 58 one of the three priorities of the 2011 Strategic Plan . 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 59 describes the mechanisms and schemes offered by EPSRC .

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

58

More information on the 2006 Strategic Plan can be found here:

http://www.epsrc.ac.uk/SiteCollectionDocuments/Publications/corporate/EPSRCSP06.pdf

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.

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International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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.

42

International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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

43

International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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 es tablish r esearch 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 60 . responsibilities; their employment conditions have been evolving over recent years 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 r elatively s mall number of PDRAs in the mathematical sciences compared t o 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

60

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

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

44

International Review of Mathematical Sciences 2010 Information for the Panel

PART I Evidence Prepared by EPSRC

Figure 21 Total number of PDRAs on grants funded by the Mathematical Sciences 61 programme by area (from 2005 to 2010) (EPSRC data)

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 per iod it is f ound that PDRAs ar e least common i n AGTN research pr ojects, w ith 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 F ellowships ( PDRFs) enable t he 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 t he br eadth of the EPSRC r emit; i n 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, w ith 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 [1] institution was introduced in 2008. In 2010/11 the cap was lowered to 5 applications .”

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 Information for the Panel

PART I 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

Table 15

Career A cceleration F ellowships (CAFs) pr ovide up to f ive years f unding t o 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 a pplicants 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%). Number of Career Acceleration Fellowships per year (2007/8 – 2009/10), all EPSRC vs. Mathematical Sciences programme (EPSRC data) Outlines Invited Awarded

7.2.13

2007/08 437 61 22

2008/09 53 415 9 70 4 24

50 4 1

2009/10 61 449 11 72 5 30

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.

46

International Review of Mathematical Sciences 2010 Information for the Panel

Table 16

PART I Evidence Prepared by EPSRC

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 r esearchers at more advanced s tages of t heir 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) Outlines Invited Awarded

2007/08 296 59 23

2008/09 253 49 17

47 11 7

52 7 2

2009/10 206 38 16

37 10 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 no ted that there is a single b udget f or 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

62

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 62 London.

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

47

International Review of Mathematical Sciences 2010 Information for the Panel

PART I Evidence Prepared by EPSRC

7.3.7

Applicants must discuss their suitability for programme grants with EPSRC before submitting an out line; 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 i ndependent r esearcher 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

Table 18

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. A wards 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. Science & Innovation Awards since 2005 in areas of Mathematical Science (EPSRC Data)

Title

Date

Numerical Algorithms and Intelligent Software for the Evolving HPC Platform

Aug-09

The LANCS Initiative in Foundational Operational Research: Building Theory for Practice

Sep-08

Analysis of Nonlinear Partial Differential Equations Centre for Analysis and Nonlinear Partial Differential Equations Cambridge Statistics Initiative (CSI) The Centre for Discrete Mathematics and its Applications (DIMAP) SuSTaIn - Statistics underpinning Science, Technology and Industry The Centre for Research in Statistical Methodology (CRISM)

Value (£ million)

Mar-07

Location Edinburgh, Heriot-Watt, Strathclyde Lancaster, Nottingham, Cardiff, Southampton Oxford Edinburgh, Heriot-Watt Cambridge

Mar-07

Warwick

3.8

Aug-06

Bristol

3.5

Oct-05

Warwick

4.1

Oct-07 Aug-07

Total

4.5

5.4 2.8 2.8 2.3

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).

48

International Review of Mathematical Sciences 2010 Information for the Panel

Table 19

PART I Evidence Prepared by EPSRC

Multidisciplinary Critical Mass Centres funded bythe EPSRC Mathematical Sciences programme

Title The Bristol Centre for Applied Non-linear Mathematics Interdisciplinary Programme for Cellular Regulation: Mathematical Architecture of Biological Regulation Bath Centre for Complex Systems National Centre for Statistical Ecology Multidisciplinary Critical Mass in Computational Algebra and Applications New Frontiers in the Mathematics of Solids

Date Oct-02 Renewed Oct-07 Oct-03

Location Bristol

Value (£ million) 1.07 1.77

Warwick

1.26

Nov-04 Oct-05 Renewed May10

Bath Kent, Cambridge, St. Andrews

1.01 1.10

Sep-05

St. Andrews

1.09

Oct-06

Oxford

1.16 Total

1.04

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 t ime t hey 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 2005/06M

400

2004/05F 2004/05M

200

O th er G ra de s

Re se ar ch er s

Le ct ur er s

Se ni or L

ec tu re rs

&

R

es ea rc he rs

Pr of es so rs

0

7.5.3

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

49

International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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 oinvestigators who by definition will have gained some experience already.

50

International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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)

Proportion of Principal and CoInvestigators

20% 18% 16% 14% Maths Unknow n

12%

Maths Female

10%

Maths Male

8%

All EPSRC

6% 4% 2%

n

U

nk no w

>6 5

61 -6 5

56 -6 0

51 -5 5

46 -5 0

41 -4 5

36 -4 0

31 -3 5

25 -3 0

65 61-65 56-60 51-55 46-50 41-45 36-40 31-35 25-30

120 100 80 60 40 20 0 2005/2006

2006/2007

2007/2008 PI Age Range

51

2008/2009

2009/2010

International Review of Mathematical Sciences 2010 Information for the Panel

PART I Evidence Prepared by EPSRC

8. International Engagement 8.1

Overview

8.1.1

Numerous s ources of f unding are available t o s upport the flow of peo ple t hrough 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 Kbased researchers wanting to t ravel abroad, and f or overseas researchers who want t o 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 pr ogramme. 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)

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International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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 hi gher t han the average f or EPSRC (just below 45% ); t he 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 63 collaboration with CNRS . 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 i mpact 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). O btaining r eliable qu antitative evidence on K T 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 i n t he mathematical sciences and users in a variety of sectors. An important part in this is played by the 65 Industrial Mathematics KTN 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 The ‘Warry’ report is available to panel members on the review ftp site. 65 http://www.industrialmaths.net 64

53

International Review of Mathematical Sciences 2010 Information for the Panel

PART I 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 66

10.1.1

Although bibliometric indicators must be used with caution they are commonly used as a source of information when seeking to compare national research quality in fields such as mathematical s ciences where the journal c overage i s good and where pu blication 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: th

• Globally the UK is ranked 5 behind the USA, France, China and Germany in terms of the total numbers of citations; th

• the UK is ranked 5 in the number of papers globally; rd

• the UK is ranked 3 , behind the USA and Australia in terms of Citations per 67 paper ; • 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:

67

Limited to countries publishing at least 500 papers a year

http://www.mathunion.org/fileadmin/IMU/Report/CitationStatistics.pdf 54

International Review of Mathematical Sciences 2010 Information for the Panel

Figure 29

68

Citation impact of Mathematics papers

Papers here refers to journal articles and review articles

55

68

PART I Evidence Prepared by EPSRC

(Thomson Reuters ‘InCites’)

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 primarily by the Government, with additional support from charities (particularly for biomedical research), international sources and the private sector. Public funding comes from a range of Government Departments with research budgets at their disposal, but the bulk comes from the Department for Business, Innovation and Science (BIS), which funds research from its science and higher education budgets. Most of the funding BIS provides for research in higher education institutions (HEIs) is delivered either through non-departmental public bodies such as Research Councils, or through the Higher Education Funding Council for England (HEFCE). The devolved administrations support equivalent funding bodies in other parts of the UK (see boxes 1 and 2). The dual support system This division of labour between funding bodies and Research Councils constitutes an organising principle for research funding known as ‘dual support.’ With its origins in the 19th Century, it rests on an idea (known as the ‘Haldane Principle’) that Research Councils and universities should be able

to choose which research to support themselves, at arm’s length from any political control. It has evolved into a system that provides multiple points of decision-making about what research should be supported and where resources should be concentrated. • Higher Education funding bodies provide core funding as block grants to HEIs for research infrastructure and to support their strategic research priorities. The bulk of their recurrent quality-related (QR) funding is allocated through a formula that takes account both of institutions’ volume of research activity and an assessment of the quality of their research over the previous period. • Research Councils provide grants to specific research projects and programmes of research. Awards are made, after expert peer review of applications submitted by individual researchers and teams, on the basis of research excellence and potential, and on the importance of the proposed topic. • Each Research Council seeks to achieve a balance between ‘responsive mode’ awards made in response to proposals submitted by researchers 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 Directly-allocated and indirect costs are the 1980s and 1990s as core funding declined calculated using standard rates established by markedly in relation to the amount provided as the university through the Transparent Approach project grants. The two funding streams were not to Costing (TRAC) methodology that underpins meeting the full costs of the research and the UK 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 headings to set the price for the research projects model and concluded that changes were required to address under-investment in infrastructure. In its 10-year Science and Innovation Investment Box 2: Research councils Framework (2004), the Government reiterated The UK’s Research Council system is supported its commitment to a dual support system, but by the Government’s Science Budget. BIS has signalled that HEIs would be asked to recover statutory control of the councils, supported by more of the full economic costs (FEC) of research the Director General of Science and Innovation. and to manage their research portfolios in 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 • Engineering and Physical Sciences Research subsistence; 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 Executive

Welsh Assembly Government

(£3.1bn from Science Budget)

EPSRC STFC

BBSRC

ESRC

NERC

AHRC

HEFCE

SFC

HEFCW

DELNI

Government Departments other than BIS, local authorities, health & hospital authorities

MRC

Research councils £1,892m

Funding bodies £2,266m

£706m

UK higher education institutions

UK-based charities £896m

UK industry commerce & public corporations £312m

EU and other overseas sources £648m

Other £51m

* 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

Box 3: TRAC sustainable management The guidance document An overview of TRAC defines sustainable management as follows: ‘An institution is being managed on a sustainable basis if, taking one year with another, it is recovering its full economic costs across its activities as a whole, and is investing in its infrastructure (physical, human and intellectual) at a rate adequate to maintain its future productive capacity appropriate to the needs of its strategic plan and students, sponsors and other customers’ requirements.’

they wish to undertake. The proportion of the FEC that universities recover will depend on the price that they set. Research Councils currently pay 80% of the FEC of the research projects they fund, which is intended to rise to 100% in the next few years. Government Departments, however, are now expected to pay 100% of the FEC. There is a broad consensus in the sector that the move towards the FEC approach is beneficial, but some concerns persist as to whether Research Councils’ increased contributions are causing them to fund less research, about the use of funding body 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 Department for Business, Innovation BIS reform. The purpose of the Research Assessment and Science Exercise (RAE), first conducted in 1989, was to enable funding bodies to derive a formula through FEC Full economic costs which to allocate funds to institutions selectively HEFCE Higher Education Funding Council and for extensive periods to promote stability. for England The system’s focus on rewarding performance is claimed to have had a positive effect on the volume HEIs Higher education institutions and quality of research outputs. However, so QR Quality-related [funding] profound has been the influence of RAE scores on RAE Research Assessment Exercise universities’ reputation and ranking that immense 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 introduced in 2014. In 2006 the Government announced its intention 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 worlduse 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 2004 – 2014 (July 2004) www.hm-treasury.gov. uk/spending_review/spend_sr04/associated_ documents/spending_sr04_science.cfm • An overview of TRAC (June 2005) www.jcpsg. ac.uk/guidance/downloads/Overview.pdf • Research Assessment Exercise website www.rae.ac.uk

• Research Excellence Framework website www.hefce.ac.uk/Research/ref • Research Councils UK www.rcuk.ac.uk • The financial information in this document is derived from a variety of sources, which are detailed at www.rin.ac.uk/makingsense-funding

Annex A - 4

International Review of Mathematical Sciences 2010 Information for the Panel

Annex B

PART I - Annex B Previous Review Recommendations

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 worldleaders 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 Information for the Panel

PART I - Annex B 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 themebased 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 topicbased 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: £M

2006/7 21.6

2007/8 18

2008/9 16

2009/10 14

2010/11 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 wellfunded 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

spend (£m)

3.0 2.5

1.93

2.0

2.18

1.6

1.5 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 Information for the Panel

Annex E

PART I - Annex E Research Assessment Exercise (RAE)

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 E2, 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 4* 3* 2* 1* Unclassified

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

Annex E - 1

International Review of Mathematical Sciences 2010 Information for the Panel

PART I - Annex E Research Assessment Exercise (RAE)

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

†

Imperial College London † University of Warwick † University of Oxford † University of Cambridge † University of Bristol Joint submission: Universities of † † Edinburgh and Heriot-Watt † University of Bath † King's College London † University of Aberdeen † University College London † University of Durham † University of Manchester † University of East Anglia † University of Sheffield † Queen Mary, University of London † University of Birmingham † University of Glasgow † University of Nottingham † Loughborough University † University of Exeter † University of Leeds † University of Leicester † Lancaster University † University of York † University of Liverpool † University of Southampton London School of Economics and Political Science † Queen's University Belfast London Metropolitan University Swansea University Aberystwyth University Cardiff University † University of Newcastle upon Tyne † University of St Andrews † University of Kent Open University Royal Holloway, University of London

Overall quality profile (percentage of research activity at each quality level)

FTE Category A staff submitted 21.8 32 55.16 55 34.53

4* 40 35 35 30 30

3* 45 45 40 45 40

2* 15 20 25 25 25

1* 0 0 0 0 5

unclassified 0 0 0 0 0

Quality Index 61.4 57.1 55.7 52.9 50.7

41 10 13 14 15.25 15 27 7 17.25 20.2 18 16.32 15 11.4 5 23.2 10 10 12.34 15 15.75

25 25 20 20 20 20 20 15 15 10 15 15 15 10 10 10 10 10 10 10 5

45 35 50 45 40 40 40 45 40 50 40 40 35 45 45 45 40 40 35 35 45

30 40 25 35 35 35 35 35 45 40 35 35 45 45 40 40 50 35 50 45 40

0 0 0 0 5 0 5 5 0 0 5 10 5 0 5 5 0 15 5 10 10

0 0 5 0 0 5 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0

48.6 45.7 45.0 44.3 42.1 42.1 42.1 39.3 38.6 37.1 37.1 37.1 36.4 35.7 35.0 35.0 34.3 32.1 32.1 31.4 30.0

12.5 8.2 4 20.5 8.3 30.45 10 12 6 16.5

5 5 10 5 5 5 5 5 0 5

40 40 25 35 35 35 30 30 35 25

50 50 50 50 45 45 60 55 55 40

5 5 15 10 15 15 5 10 10 30

0 0 0 0 0 0 0 0 0 0

29.3 29.3 27.9 27.1 26.4 26.4 26.4 25.7 22.9 21.4

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 Information for the Panel

PART I - Annex E Research Assessment Exercise (RAE)

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

†

University of Cambridge † University of Oxford † University of Bristol † University of Warwick † University of St Andrews † University of Bath † University of Manchester † Imperial College London † University of Portsmouth † University of Durham † University of Nottingham † University of Southampton † University of Surrey Joint submission: Universities of † † Edinburgh and Heriot-Watt † King's College London † Keele University † University of Liverpool † University of Newcastle upon Tyne † University of Leeds † University of Exeter † University of York † University of Strathclyde † Brunel University † University of Kent † Loughborough University † Queen Mary, University of London University of Sussex † University College London † University of Glasgow † University of Sheffield † University of Dundee † University of Birmingham † University of Leicester † University of East Anglia † University of Reading City University, London Coventry University University of Plymouth University of Glamorgan University of Brighton University of Chester Oxford Brookes University University of the West of England, Bristol Glasgow Caledonian University Staffordshire University

Overall quality profile (percentage of research activity at each quality level)

FTE Category A staff submitted 80.3 54.25 38 29.25 15 19.8 28.8 37.2 16 26 34.4 17 17.42

4* 30 30 25 30 25 20 25 20 15 15 20 15 15

3* 45 45 45 30 45 50 35 45 60 60 45 55 55

2* 25 25 30 35 25 30 40 35 25 20 30 30 25

1* 0 0 0 5 5 0 0 0 0 5 5 0 5

unclassified 0 0 0 0 0 0 0 0 0 0 0 0 0

Quality Index 52.9 52.9 48.6 47.9 47.9 45.7 45.7 44.3 44.3 43.6 43.6 42.9 42.1

35.46 18 9 23.33 11 24.9 18 22 24.33 19 6 20.4 15 13 16.5 19 17 10.7 25 12 7 21.3 8.2 7 4 4 3 5.8 5.71

15 15 15 15 15 15 10 10 10 5 10 10 10 10 10 10 10 10 10 5 5 5 5 0 0 0 0 0 0

50 50 45 45 45 40 50 40 40 55 35 40 40 40 40 40 35 35 35 40 40 35 15 20 25 20 15 15 10

30 25 35 35 35 40 40 45 45 30 55 40 40 40 40 40 50 45 35 50 45 45 65 70 55 45 45 45 55

5 10 5 5 5 5 0 5 5 10 0 10 10 10 10 10 5 10 20 5 10 15 15 10 20 30 35 35 35

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5 5 0

40.7 40.0 39.3 39.3 39.3 37.9 37.1 33.6 33.6 32.9 32.9 32.9 32.9 32.9 32.9 32.9 32.1 31.4 30.0 29.3 28.6 26.4 20.7 18.6 18.6 15.0 12.9 12.9 12.1

9 1 1

0 5 0

10 5 0

50 30 0

40 60 95

0 0 5

11.4 11.4 0.0

Annex E - 3

International Review of Mathematical Sciences 2010 Information for the Panel

PART I - Annex E Research Assessment Exercise (RAE)

Table E-4 Statistics and Operational Research unit of assessment 2008 RAE ratings

†

University of Oxford † University of Cambridge † Imperial College London † University of Bristol † University of Warwick † University of Leeds † University of Nottingham † University of Kent † University of Bath † University of Southampton † Lancaster University † University of Manchester London School of Economics and Political Science † University of St Andrews † Brunel University † University of Sheffield † University of Glasgow † University of Newcastle upon Tyne Open University † University College London Joint submission: Universities of † † Edinburgh and Heriot-Watt † University of Durham † Queen Mary, University of London † University of Strathclyde † University of Reading University of Greenwich University of Salford † University of Liverpool London Metropolitan University University of Plymouth

Overall quality profile (percentage of research activity at each quality level)

FTE Category A staff submitted 24.5 16 13.9 23 24 11 9 12 15 28 21.65 10.9

4* 40 30 25 25 25 25 20 20 20 15 15 20

3* 50 45 50 45 45 40 50 45 40 50 45 35

2* 10 25 25 30 30 30 30 30 35 30 35 30

1* 0 0 0 0 0 5 0 5 5 5 5 15

unclassified 0 0 0 0 0 0 0 0 0 0 0 0

Quality Index 62.9 52.9 50.0 48.6 48.6 46.4 45.7 43.6 42.1 40.7 39.3 39.3

13 7 10 10.7 13 13 7 13.5

15 10 15 10 15 10 10 10

40 50 35 50 35 45 40 40

35 35 40 30 40 40 45 40

5 5 10 10 10 5 5 10

5 0 0 0 0 0 0 0

37.1 36.4 35.7 35.7 35.7 35.0 33.6 32.9

30 11.6 8.2 10.33 7.7 2 9.8 5 4 4

10 5 10 10 5 0 0 0 5 0

35 45 30 30 30 40 35 35 20 30

45 45 45 45 55 40 55 50 40 45

10 5 15 15 10 20 10 15 35 25

0 0 0 0 0 0 0 0 0 0

31.4 30.7 29.3 29.3 25.7 22.9 22.9 22.1 19.3 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 Information for the Panel

PART I - Annex E 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

3b

1996 1992

5

5*

4

3a

5

3b/3a

2 0%

20%

5*

4

40%

60%

5/5* 80%

100%

Applied Mathematics Increasing Quality 2001

3b

1996

2

1992

3a

4 3b

1

5

3a

2

0%

4

5

3b/3a

20%

5*

40%

4 60%

5* 5/5*

80%

100%

Statistics and Operational Research Increasing Quality 3a

2001 1996

2

1992

4 3b

3a

2 0%

5 4

3b/3a 20%

5* 5

4 40%

Annex E - 5

60%

5*

5/5* 80%

100%

International Review of Mathematical Sciences 2010 Information for the Panel

PART I - Annex E Research Assessment Exercise (RAE)

Table E-5 Institutions included in RAE data

1

Institutions included in Pure Mathematics RAE Data

Goldsmith’s College Imperial College London King’s College London Open University 2 Queen Mary and Westfield College Queen’s University of Belfast 3 Royal Holloway and Bedford College 4 UMIST University College London 5 University College of North Wales 6 University College Swansea 7 University College of Wales University of Aberdeen

University of Bath University of Birmingham University of Bristol University of Cambridge University of Durham University of East Anglia University of Edinburgh University of Exeter University of Glasgow University of Hull University of Lancaster University of Leeds University of Leicester

University of Liverpool University of Manchester University of Newcastle University of Nottingham University of Oxford University of Reading University of Sheffield University of Southampton University of St. Andrews University of Sussex 8 University of Wales College Cardiff University of Warwick University of York

Institutions included in Applied Mathematics RAE Data Brunel University 9 Chester College 10 London Guildhall University City University Coventry University De Montfort University 11 Glasgow Polytechnic Heriot-Watt University Imperial College London King’s College London Loughborough University Nottingham Trent University Oxford Brookes University 2 Queen Mary and Westfield College Staffordshire University 4 UMIST University College London

University College of Wales University of Bath University of Birmingham University of Bristol University of Cambridge University of Dundee University of Durham University of East Anglia University of Edinburgh University of Exeter University of Glasgow University of Hull University of Keele University of Kent University of Leeds University of Leicester University of Liverpool

7

University of Manchester University of Newcastle University of Nottingham University of Plymouth University of Portsmouth University of Reading University of Sheffield University of Southampton University of St. Andrews University of Strathclyde University of Surrey University of Sussex University of Teesside University of the West of England University of York

Institutions included in Statistics and Operational Research RAE Data Brunel University City University Coventry University 1 Goldsmith’s College Heriot-Watt University Imperial College London London School of Economics and Political Science Nottingham Trent University Open University 2 Queen Mary and Westfield College 4 UMIST University College London University of Aberdeen

University of Bath University of Birmingham University of Bristol University of Cambridge University of Durham University of Edinburgh University of Exeter University of Glasgow University of Greenwich University of Keele University of Kent University of Lancaster University of Leeds University of Liverpool

1

9

2

10

Now Goldsmiths, University of London Now Queen Mary, University of London 3 Now Royal Holloway, University of London 4‘ ’University of Manchester Institute of science and Technology’ now merged with the University of Manchester 5 Now Bangor University 6 Now Swansea University 7 Now Aberystwyth University 8 Now Cardiff University

University of Manchester University of Newcastle 10 University of North London University of Nottingham University of Oxford University of Reading University of Salford University of Sheffield 12 University of Southampton University of St. Andrews University of Strathclyde University of Surrey University of Sussex University of Warwick

Now University of Chester London Guildhall, City of London Polytechnic and University of North London are now merged as London Metropolitan University 11 Now Glasgow Caledonian University 12 In 1996 Southampton entered two departments (‘Statistics’ and ‘Operational Research’) for the Statistics and OR UoA. The highest scoring (Operational Research) is used in Figure G-1. In 2001 both departments were awarded the same

Annex E - 6

International Review of Mathematical Sciences 2010 Information for the Panel grade, so this is used (still only counting one entry

PART I - Annex E Research Assessment Exercise (RAE) for the University of Southampton)

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

Number of Institutions Pure Applied Stats/OR 38 46 31 47 58 45 45 65 54 44 67 50

Number of Category A/A* Staff Submitted Pure Applied Stats/OR 685.25 850.05 388.78 509.5 734.71 386.64 468 721.5 423.7 506.4 770.6 407

Annex E - 7

International Review of Mathematical Sciences 2010 Information for the Panel

Annex F

PART I - Annex F Additional Bibliometric Evidence

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 years intervals

10.1.5

20022006

10.1.6

20032007

10.1.7

20042008

10.1.8

20052009

10.1.9

20062010

10.1.10 Number

110,541

115,585

124,552

131,286

134,498

10.1.11 Times

139,045

149,428

171,822

206,398

222,662

1.26

1.29

1.38

1.57

1.66

of papers cited

10.1.12 Citations per paper

Annex F - 1

International Review of Mathematical Sciences 2010 Information for the Panel

PART I - Annex F 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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Country USA France China Germany UK Italy Canada Spain Japan Australia Russia Israel The Netherlands Belgium Poland South Korea Brazil Switzerland Sweden Austria Taiwan India Czech Republic Denmark Hungary Greece Finland Romania Singapore Norway

Papers 66,764 23,071 27,458 18,795 14,876 13,287 11,994 11,038 12,644 5,632 12,556 4,919 3,498 3,283 5,533 5,338 4,220 2,793 2,856 2,553 3,492 4,745 2,507 1,382 2,636 2,062 1,597 2,464 1,376 1,351

Annex F - 2

Citations 305,546 87,984 78,992 71,813 63,653 44,910 43,312 36,252 32,918 24,379 19,523 17,986 14,869 13,166 13,045 12,862 12,478 12,109 11,844 10,502 10,163 8,935 7,488 6,691 6,685 6,347 6,286 5,967 5,860 5,845

Citations/paper 4.58 3.81 2.88 3.82 4.28 3.38 3.61 3.28 2.60 4.33 1.55 3.66 4.25 4.01 2.36 2.41 2.96 4.34 4.15 4.11 2.91 1.88 2.99 4.84 2.54 3.08 3.94 2.42 4.26 4.33

International Review of Mathematical Sciences 2010 Information for the Panel

PART I - Annex F 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 th rd 19 to 143 .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 nd nd st nd papers are from the UK; these are Imperial (22 ), Oxford (32 ), Bristol (51 ), Cambridge (52 ), nd th Warwick (92 ) and Nottingham (100 ). 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))

10.1.16

10.1.17

World ranking (papers)

10.1.18

World ranking (citations)

10.1.19

World ranking (citations 69 per paper)

10.1.14

10.1.15

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

10.1.13 Institution Oxford

papers

Citations

Citations per paper

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. 70

Their report showed that the majority of UK output in mathematical sciences-related fields 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 70

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

Annex F - 3

International Review of Mathematical Sciences 2010 Information for the Panel

PART I - Annex G 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 71 and Japan , 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 Information for the Panel

PART I - Annex G 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 Information for the Panel

Annex H

PART I - Annex H Evidence Framework

List of Acronyms

AGTN

Algebra, Geometry, Topology and Number Theory

AHRC

Arts and Humanities Research Council

BBSRC

Biotechnology and B Council

BERR

Department f or B usiness, E nterprise a nd R egulatory Reform

Created 2007 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 Councils

Merged with PPARC to become 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

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

DTA/DTG

Doctoral Training Account / Grant

DTC

Doctoral Training Centre

DTI

Department of Trade and Industry

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

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

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

iological S ciences R esearch

An EPSRC mission programme

Annex H - 1

Created 2007 Merged with BERR to form BIS

Abolished 2007, replaced by DIUS and BERR

An EPSRC core programme

An EPSRC core programme

International Review of Mathematical Sciences 2010 Information for the Panel

PART I - Annex H 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

TSB

Technology Strategy Board

UOA

Unit of Assessment

Annex H - 2

Created 2007 by merger of PPARC and CCLRC

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 MRCfunded 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

2

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

Return to Landscapes List

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.

Research Area Infinite Group Theory Finite Group Theory Semigroup Theory Representation Theory Noncommutative Algebra Other

Number of Researchers (IRM2003 total) 42 (43) 31 (18) 10 (5) 60 (21) 23 (23) 7 (13)

% of Total

24.3% 17.9% 5.8% 34.7% 13.3% 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 BridsonHowie-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 KacMoody 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-BreuillardTao (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

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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.

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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_2categorification, 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), 2137. [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.

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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 Algebraic Geometry (AG) Algebraic Topology (AT) Geometric Topology (GT) Differential Geometry (DG)

Number of researchers 44 48 36 50

% of Total in G&T 25 27 20 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 Poincar´e conjecture, finite generation of the canonical ring, the Mumford conjecture and the Kervaire invariant conjecture as particular highlights. The UK has worldleading 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.

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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 (Edinburgh). 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 hyperk¨aler 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, Gr¨ obner bases, 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 2

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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, highdimensional 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 3

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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 K¨ oln), 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 K¨ahler metrics, hyperk¨ahler metrics and other metrics with special holonomy). A distinctive feature of the UK is the variety of tools used: twistor theory, hyperk¨ ahler reduction, 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), Loughborough (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.

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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 Poincar´e conjecture. 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 (Cambridge) 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 LMSsupported 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

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Triangle (Leicester-Manchester-Sheffield). 5. Cross-disciplinary/Outreach activities There are numerous interactions between geometry, topology and other areas of mathematics. Commutative 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.

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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 Arithmetic 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 Algebraic Number Theory Arithmetic Algebraic Geometry Analytic number theory Representation theory Diophantine approximation Computational number theory

2008 24 23 15 9 7 8

% 34 32 21 13 8 8

2001 26 14 12 3 8 8

% 40 23 20 5 13 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. Despite 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.

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Return to Landscapes List 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 %

26 Real Functions 28 Measure Theory 30 Complex Analysis 31 Potential Theory 33 Special Functions 34 ODE 25 PDE 39 Functional Equations 40 Sequences 41 Approximation 42 Fourier Analysis 43 Harm Analysis (abstract) 44 Integral Transforms 45 Integral Equations 46 Functional Analysis 47 Operator Theory 49 Calculus of Variations

16 16 34 3 25 76 250 6 2 5 26 8

825 427 1553 167 677 4141 7727 847 165 674 1278 214

2 7 83 51 36

112 302 1941 2816 1455

1.9 3.7 2.2 1.8 3.7 1.8 3.2 0.7 1.2 0.7 2.0 3.7

UK Authors 16 8 21 3 20 77 200 7 3 5 26 8

1.8 2.3 4.2 1.8 2.5

2 15 61 44 34

UK Insts 12 7 14 3 13 36 41 7 2 4 13 5

RAE Nos. 3 30 11 0 1 17 145 1 3 2 22 1

UK/World% [ 2000-10 ] [ 1.5 ] [ 4.3 ] [ 2.4 ] [ 2.4 ] [ 3.7 ] [ 1.9 ] [ 2.6 ] [ 1.3 ] [ 1.4 ] [ 1.9 ] [ 1.7 ] [ 2.1 ]

2 8 27 20 18

0 1 43 74 10

[ 1.4 ] [ 1.8 ] [ 3.7 ] [ 2.4 ] [ 1.8 ]

The next table is an amalgamation of data from various MSC classifications and subclassifications rel ated 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.) 693

UK/World% 2009-10 7.1

UK authors 31

14

UK/World % [ 2000-10 ] [ 5.6 ]

22

430

5.1

17

9

[ 5.9 ]

17 14

300 377

5.7 3.7

13 20

11 11

[ 6.2 ] [ 4.5 ]

UK/World % [ 2000-10 ] [ 2.7 ]

MSC Subject

UK

World

46

49

Operator and Banach Algebras 37 Ergodic Theory & Complex Dynamics 28 Fractals, GMT 35 Spectral Theory

UK Insts

Data for the remaining part of functional analysis are MSC Subject

UK

World

UK/World% 2009-10

UK authors

UK Insts

46 Functional Analysis (without algebras)

34

1289

2.6

31

19

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 pa pers wa s roug hly the same a s in France, the main exception bei ng PDE (UK about 1/4 of France in the last 10 years, about 1/3 in the last 2 years).

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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 in cluded 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 1.Discussion of Research ar eas :

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 £7 M in the form of S&I award s 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 prize s in 20 05 and 2009, for work in the calculus of variation s, g eometric PDE s and general relativity, Maz’ya (Live rpool, half-time) was awarded the 2009 Senior Whitehead prize for lifetime achievements in linea r and nonli near PDEs, and Dafermo s and Stuart (bot h Cambridge) shared the 20 05 Adams Prize. Ball (Oxford) has received countless awards a nd 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 linea r elliptic operators including domain optimization problems. In a ddition 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 th eory, but relate d work i s 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 g eometric me asure th eory p opulation i s even smaller (unless o ne counts relate d work, for example i n geometric analysis) but highly signi ficant 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 term s of size and infl uence, this area necessarily follows the USA, France an d I srael, whe re classical erg odic theo ry 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

20

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

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 expa nded 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 oth er activity throughout the UK. OXPDE, wi th 1 1 faculty, 7 postdocs an d 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 que stions 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 lea dership 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 nonlin ear 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 rese archers a t Aber yswyth, Aberd een, B elfast, Ca rdiff, 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 le adership of Bonsall, Johnson and Ringrose, have retired since the l ast IRM. In spite of several new (young) a ppointments t o mat ch some of the retirements there has been a steady, thoug h slo w (till n ow) decline i n 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 ha s been excluded from re cent 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 predomi nance of eastern European trained researchers persists. (The o rganisers 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 th ere are distinguished individuals el sewhere, the subj ect ove rall 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 matt er fo r concern is the virtual no n-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 va rious sub-areas, o nly the theory of fractals has an app ropriately so und base with sufficiently well distributed a ctivity 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 peo ple thro ugh a ppointments and fello wships. The number of UK groups ha s 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 ari se 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 wo rkshop o n cell migration an d tissue mechanics) a nd risk analysis i n 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). A nalysis m ethods info rm all of numerical analysis an d 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 un iversities. Un dergraduates 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 numbe r the ory. Ho wever in the UK there i s ge neral 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 i n the Research-Council-identified centres of excellence i s p roving 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 3 away from fundamental aspects of analysis . 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 stoch astics, five P DE an d c omputational. T he a im 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 w hose ultimate direction is unclear.

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Return to Landscapes List 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 Model theory Proof theory, constructive mathematics, and category theory Recursion theory (aka Computability theory) Set theory Other

Number of Researchers

% of Total

17 7

49% 20%

3

9%

4

11%

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.

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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 ZilberPink 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 (HrushovskiLoeser). 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.

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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.

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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 wellknown 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

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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

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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).

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Return to Landscapes List 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.

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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 firstorder theories, sparse Ramsey theory, mathematical anal ysis of the AES and other ciphers, to mention just a few. The links to other parts of mathematics, and t he 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

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including Leader , Scott, Thomason, K uhn O sthus, R iordan, K eevash, a mong 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.

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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 U K or i ndeed t o any group o f co untries. T he st rategically i mportant 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.

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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 dem and. T his is also t rue for aca demic 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, C ombinatorics is a t hriving di scipline i n t he U K, but i ts health should be attributed to factors other than any specific initiatives taken after the last review: the points raised above remain valid today.

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Return to Landscapes List UK Landscape in Numerical Analysis 1

Statistical Overview

cases there is related UK-based work being con1065 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 worldysis, 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 oscilthat 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 interfaces that has made important contributions to the analysis activity from RAE2001. field of computational geometric evolution equa2 Subject Breakdown tions and PDEs on evolving surfaces. He also conThe greatest number of researchers are working tributed to the development and analysis of nuin differential equations, followed by optimization merical algorithms in a range of problems involvuli (Oxford) is active then numerical linear algebra then approximation ing nonlinear diffusion. S¨ in several areas of nonlinear PDEs including coutheory. pled 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 Ortital 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, Chapuli), and OxPDE: the “Oxford Centre for strength. Research in (partial) differential equa- man and S¨ 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 comsubarea. The UK is internationally competitive on putational nonlinear PDEs include Barrenechea (Strathclyde), Budd (Bath), Burman (Sussex), Kay all the major fronts within numerical analysis. Three worldwide trends in numerical analysis (Oxford), Jimack (Leeds), Lin (Dundee), Macurnberg (Imperial), Ramage and computational mathematics over the last 10– Donald (Oxford), N¨ 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 competitiveSecond, research straddling the different subareas ness with other leading countries. has grown significantly, for example at the linear alAdaptivity The construction and mathematgebra/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 adapformatics, data assimilation, engineering, materi- tivity and a posteriori methods. Houston (Nottingals science and networks. The hiring in the UK ham) has made important contributions to the deof 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 incompressentific research landscape than in other research- ible fluid flow problems. Other important contribuleading 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 1

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problems) and Lakkis (Sussex, norm-based a posteriori methods). This field is thus well-represented within the UK. Solution of Linear Systems Arising in PDEs The UK is strong in this important field, with research leadership provided by Wathen (Oxford) and Silvester (Manchester), with many contributions aimed at saddle-point problems and related systems rising in fluid mechanics and optimization, and the group led by Graham and Spence (Bath) studying discretization of boundary value problems. Other active researchers in this and related areas include Kay (Oxford), Loghin (Birmingham), Marletta (Cardiff), Powell (Manchester), Scheichl (Bath) and Duff (RAL, and partly through his team at CERFACS, Toulouse). An EPSRC network in the area would give the community a greater UK cohesiveness; an INI programme would enhance its international identity. Stochastic Problems This is an area of computational mathematics that has grown in importance significantly over the last decade, primarily because of the increasing use of stochastic modelling in many realms of science and technology. The UK is well-represented, with several internationally leading research teams. The group of Stuart (Warwick) has studied design and analysis of numerical methods for SDEs and SPDEs, focussing particularly on long-term behaviour, ergodicity and links with Markov chain-Monte Carlo. More recently this group has been active in the Bayesian approach to inverse problems. The multilevel Monte Carlo (MLMC) method of Giles (Oxford) has provided a step-change in the computational efficiency of certain calculations required in financial mathematics, and the basic idea appears to have far wider applicability: it is being extended by researchers in Germany, the UK and the USA to other stochastic settings. At Strathclyde the research groups of D. J. Higham and Mao are very active in the numerical approximation of SDEs, jump diffusions and the chemical master equation. There is also significant effort in the numerical solution of (primarily elliptic) PDEs with random coefficients, where the key issues are often related to linear algebra and preconditioning (see the references to Graham, Powell, Scheichl and Silvester in the previous item). Tretyakov (Leicester) is also very active in the numerical solution of SDEs and (with Milstein) has written an important research monograph in this area. Others active in this broad area include Lakkis (Sussex), Lord (Heriot-Watt), Lythe (Leeds), Shardlow (Manchester) and Voss (Leeds). The field is thus healthy and growing in tandem with broader trends in modelling. Geometric Integration and ODEs Geometric integration remains an area of strength within

the UK, with leading research groups at several institutions. Iserles (Cambridge) continues to be active in this area and his research agenda concerning Lie group methods, Magnus series and related topics has spawned a number of significant research problems, together with research groups in the UK and worldwide. Leimkuhler (Edinburgh) leads research at the interface of numerical integration, computational mechanics and statistical mechanics and has developed world-leading expertise in the construction of thermostats. In computational ODEs Cash (Imperial) is active in the development of code for singularly perturbed twopoint boundary-value problems and differential algebraic equations, emphasizing the importance of conditioning. Many of his codes are widely used, e.g., in Maple 14. Hill (Bath) has made significant contributions to the analysis of nonlinear stability of discretized ODEs, using control-theoretical techniques and within the framework of general linear methods. Iserles (Cambridge) has played a major role in establishing a theory for the integration of highly oscillatory ODEs and this too has led to new research questions and emerging junior researchers. This field has, to a large extent, been invigorated by links to broader trends in multiscale numerical analysis (see the next item), which provides a natural path for the future development of the area. Multiscale Problems There is a growing trend worldwide to study a variety of problems which possess two or more different scales under a single broad umbrella, and to look for common themes across different application areas, as evidenced, for example, by the successful SIAM journal Multiscale Modeling and Simulation introduced in 2003. This has led to an interest in developing and implementing numerical methods which account for, or exploit, these scales. The UK has developed a range of expertise in this broad area, and was recently host to an LMS-EPSRC Durham Symposium in the field (led by Graham and Scheichl from Bath, together with Hou). Some of the research in this area has appeared under previous items; in particular some of the work of Markowich, S¨ uli, Barrett, Graham, Stuart, Iserles and Leimkuhler has ramifications in the multiscale context. Additional work not yet mentioned includes the group of Chandler-Wilde in Reading, working on high-frequency wave-scattering (and including T. Betcke and Langdon) and that of Ainsworth (Strathclyde) who has worked on dispersion relations in (high-frequency) wave propagation, and on the adaptive multi-scale modelling of masonry structures. There is also research activity in the use of lattice Boltzmann models of fluids at Oxford (Dellar) and Leicester (Gorban, Levesley). The community of researchers engaged in multi2

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scale computation has thus emerged naturally from within existing research groups present in the UK. In particular the EPSRC funded network “Computation and Numerical Analysis for Multiscale and Multiphysics Modelling” played an important role in establishing this community. Inverse Problems On the applications side, interest in inverse problems is a significant growtharea, worldwide. This is driven by many factors including medical and military applications, design and optimization, and the integration of large data sets with mathematical models. Computational challenges within the field are significant. The British Inverse Problems Society provides a coherence to the community as a whole and has played a role in a number of international meetings held within the UK over the last decade. The inverse problems group of Lionheart and Dorn at Manchester plays a significant role within the UK and the innovative work of Dorn, using level set methods for inverse imaging problems, has strong links to research in computational nonlinear PDEs. Lesnic (Leeds) applies various computational techniques, including meshless and boundary element methods. The data assimilation group at Reading, led by Nichols, is involved in a range of geophysical inverse problems (mainly arising from weather forecasting) and drawing on a range of techniques from numerical linear algebra and control theory. The work of Stuart (see item on Stochastic Computation) is also directed at computational aspects of inverse problems, and in particular at using the Bayesian approach to quantify uncertainty. Others working in inverse problems include Arridge (UCL), Chen (Liverpool), Freitag (Bath), Johansson (Birmingham) and Potthast (Reading). The community engaged in the numerical analysis of inverse problems is thus a rather small one when compared to the fairly broad research goals represented and compared with international competitors. Integral Equations Research in this area is well-represented in the UK, especially in relation to previously mentioned work on multiscale and inverse problems, much of this activity developing from a 5-month programme at the Newton Institute in 2007 on Highly Oscillatory Problems (organised by Engquist, Fokas, Hairer and Iserles). In addition to the aforementioned work of Betcke, Chen, Chandler-Wilde, Langdon, Lesnic, Johansson and Potthast, there is significant research in high frequency wave propagation via the Bath group of Graham and Smyshlyaev (now at UCL) and on integral equation methods for the wave equation and time-dependent Maxwell equations by Davies (Strathclyde) and Duncan (Heriot-Watt). This latter work has motivated other work on the numerical analysis of first kind Volterra integral equations

(with Brunner, Newfoundland). Iserles (with Brunner and Nørsett, Trondheim) has developed methods for the computation of spectra of non-normal highly oscillatory integral operators.

3.2

Approximation Theory

For the past 20 years there has been a significant activity in UK approximation theory related to kernel based approximation methods in Euclidean space and on compact manifolds. Starting with Powell (Cambridge) and Light (Leicester) in the early 1990s, this has been continued by Baxter and Hubbert (Birkbeck), Levesley (Leicester), Wendland (Oxford) and Hesse (Sussex). These researchers are well-integrated with the international approximation theory community. More recently this has become more application based, with developments in practical algorithms for solving partial differential equations, and approximating difficult data sets. Davydov (Strathclyde) has worked in such kernel methods, but his work is more directed towards polynomial approximations such as are useful for the solution of PDEs. Goodman (Dundee) has also worked in multivariate approximation theory, in box splines, multiresolution and wavelet theory. A notable UK development on the computational side of approximation theory has been the growth of the Chebfun software system by Trefethen (Oxford) and collaborators since 2002. Chebfun exploits a mix of old and new algorithms of polynomial and rational interpolation and approximation and quadrature to produce a system that feels like MATLAB but computes with functions instead of numbers or vectors, including the solution of differential equations. Recently, the UK approximation theory and mathematical signal processing community has invested in “sparsity based” research, as part of the attempt to tackle the “data-deluge” of information. The EPSRC Network on multiScale Information RePresenation and Estimation (INSPIRE) has brought together members from the mathematics, computer science, statistics, and electrical engineering communities. Active in this area from the numerical analysis community within the UK are Cartis and Tanner (Edinburgh), who (along with M. Davies) have founded the Edinburgh Compressed Sensing Group http://ecos.maths.ed.ac.uk. This group is composed of eleven active researchers, the largest in the UK in this area and one of the largest in the EU. It is part of an EU FP7 FETOpen program “Sparse Models, Algorithms and Learning for Large-Scale Data” which has partners at Queen Mary London, INRIA, EPFL, and Technion. The research in this area is inherently interdisciplinary and partially addresses the concern stated in IRM2004 about insufficient connection be3

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tween the numerical analysis and computer science “algorithms” communities. There is also excellent activity in a number of other areas, for instance Sabin (Cambridge) and Goodman (Dundee) in computer aided geometric design, Iserles (Cambridge) in approximation on Lie groups, and Shadrin (Cambridge), famous for the solution of the de Boor Conjecture, in univariate spline theory.

3.3

mathematics and Network Science: for example, by Estrada and D. J. Higham (Strathclyde), Spence, and Grindrod (Reading). There is an excellent opportunity for the UK to build on existing computational strengths and compete with a highly active and data-driven international movement that is currently dealing with issues of high-dimensional, heterogeneous and time-dependent network data. An EPSRC Matrix and Operator Pencil Network (2009–2011) led by Marletta (Cardiff) and Levitin (Reading) is enhancing links between linear algebra, operator theory and engineering, and includes among its members a number of strong recent UK appointments in spectral theory. Although relatively small, the numerical linear algebra community provides international leadership across the spectrum of theory, algorithms and software.

Numerical Linear Algebra

The UK continues to have a small but world-leading effort in this area. The Manchester group of N. J. Higham and Tisseur has made significant advances in nonlinear eigenvalue problems, especially in analyzing conditioning and stability of linearizations of polynomial eigenvalue problems. Tisseur has derived tools and theory for eigenvalue and mapping problems with Lie and Jordan algebra structure. Higham’s work on theory and algorithms for matrix functions has had strong influence on software (e.g., in MATLAB) and has led to him writing the first research monograph on this topic. Dongarra (Manchester, part-time since 2007) has made major contributions to parallel numerical algorithms in linear algebra and to open source software packages and systems. At the Rutherford Appleton Laboratory, Arioli, Duff, Hogg, Reid, Scott and Thorne work on both direct and iterative methods (and are now looking at hybrid schemes) for large, sparse linear systems, developing theory as well as productionquality software made available through the HSL Library. Because this group is outside the University sector there is a consequent lack of graduate throughput. Research on preconditioning for iterative methods is represented at RAL and in several universities (see “Solution of Linear Systems Arising in PDEs” above). Spence (Bath) has obtained results on iterative methods for large-scale eigenvalue problems arising from discretizations of PDEs, notably having introduced (with Freitag) “tuned preconditioners”. Since completion of his monograph on pseudospectra (2005), Trefethen (Oxford) has developed methods for approximating eA and f (A)b by contour integration, exploiting connections with approximation theory. The emerging theme of Network Science offers a common framework in which to model, analyse and summarize data sets from a diverse range of areas in science, technology and even crime. Graph theory and linear algebra come together to offer tools for analyzing the type of sparse, unstructured but non-random connectivity patterns that arise. Currently there is a limited amount of UK activity at the intersection between applied/computational

3.4

Since IRM2004, UK research in optimization has continued to flourish. Three leading universities (Birmingham, Edinburgh and Southampton) now have sizable optimization groups, while interactions between individuals scattered elsewhere have grown. Most significantly, cross-area collaboration between optimization, linear algebra and differential equations, particularly in the area of simulation-based (or PDE-constrained) optimization, is commonplace—three recent international workshops (Durham, Oxford and Sussex) attest to this. There is also now an established biennial series of Birmingham conferences on optimization and linear algebra organised by the IMA. Current research in the UK has focused on theoretical and practical investigations into interiorpoint methods for linear, semi-definite, conic, convex and nonconvex problems (Birmingham, Cambridge, Edinburgh, Oxford, Southampton, RAL), the solution of (large) problems for which some or all unknowns are required to take integer values (Birmingham, Dundee, Edinburgh, RAL), methods for stochastic optimization (Edinburgh, Oxford, Southampton) problems involving “complementarity” constraints (Birmingham, Cambridge, Dundee), structural optimization (Birmingham, Southampton), second-derivative SQP methods (Dundee, RAL), the efficient approximate solution of (NP) hard combinatorial problems (Edinburgh, Imperial), multiobjective optimization (Birmingham, Southampton), support vector machines, (Dundee, Edinburgh), the complexity of nonconvex optimization (Edinburgh, RAL), derivative-free optimization (Cambridge, Southampton), steepest descent methods (Dundee), financial optimization (Imperial, Oxford), and algorithms for infinitedimensional problems for which (some of) the con4

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Optimization

straints are differential equations or variational inequalities (Birmingham, Cambridge, Dundee, Imperial, Oxford, RAL, Sussex, Warwick). Upcoming areas such as optimization over semi-algebraic sets and robust optimization are less well represented, but work is underway here too. A continuing trend in the UK is to match theoretical developments in optimization with the design and implementation of software. Generically this is available in the HSL and NAG libraries, and in general-purpose packages such as FilterSQP, HOPDM, LANCELOT/GALAHAD and PENOPT. Many of the key developments– the DFP and BFGS secant updates, augmented Lagrangian and SQP algorithms, and the nonlinear conjugate-gradient method—occurred either entirely or partly in the UK and as a direct consequence of the software needs of both commercial and academic sectors. More recent key research—the trust-region and cubic regularisation paradigms and their extensions, methods for largescale linear, quadratic and conic programming, large-scale interior-point and SQP methods, and the filter and funnel approaches—has continued this trend. Current computing developments are having profound implications for optimization, and UK researchers are evaluating and exploiting both massively-parallel HPC and more modest multicore and GPU systems.

4

Other key meetings include: LMS Durham symposia on Numerical Analysis of Multiscale Problems (2010) and Computational Linear Algebra for Partial Differential Equations (2008); EPSRCWarwick Symposium Challenges in Scientific Computing in 2008–2009 (seven workshops and one conference); and IMA Conference on Numerical Linear Algebra and Optimization (2007 and 2010). The Newton Institute has also supported a number of numerical-analysis related themes, including a 5 month program on highly oscillatory problems in 2007. The EPSRC funded network on Computation and Numerical Analysis for Multiscale and Multiphysics Modelling (see above) ran several workshops and seeded collaborations. There have also been several relevant LMS-EPSRC instructional courses aimed at graduate students, including a long running series in Numerical Analysis (now Computational Mathematics and Scientific Computing, most recently in Durham, 2010) and, in 2007, a stand-alone short course on Multiscale Methods. Two recently established EPSRC Centres for Doctoral Training include significant components in computational mathematics (at Cambridge and Warwick) and the EPSRC-funded Taught Course Centres have also involved many courses in numerical analysis. Individual Items of Esteem Higham and Trefethen were elected FRS in 2007 and 2005, respectively, and Trefethen was elected to the US National Academy of Engineering in 2007. Dongarra, Duff, Hammarling, Higham and Trefethen are ISI Highly Cited Researchers. In international prizes the 2005 SIAM Dahlquist Prize was awarded to D. J. Higham, the 2007 SIAM J. D. Crawford Prize to Stuart and the 2010 SIAM Richard C. DiPrima Prize to MacDonald (Oxford), and Elliott and Markowich were awarded a Humboldt Prize in 2009–2010. Duff, Gould, Fletcher, D. J. Higham, N. J. Higham, Stuart and Trefethen were elected SIAM Fellows. Recent UK awards are the LMS De Morgan Medal to Morton (2010), LMS Whitehead Prizes to Ainsworth (2004) and Tisseur (2010), the LMS Fr¨ohlich Prize to N. J. Higham (2008), and a Philip Leverhulme Prize to Tanner (2008). N. J. Higham held a Royal Society Wolfson Research Merit Award, 2003–2008. S¨ uli was elected Foreign Member of the Serbian Academy of Sciences and Arts (2009) and to a Fellowship of the European Academy of Sciences (2010). Research Funding EPSRC’s Grants on the Web lists 96 grants totalling £54 million as of 31-810 with “Numerical Analysis” as “Research Topic” (though, like the totals for most subareas of mathematics, this includes approximately £17 million of generic postgraduate training grants and the block grants for the Isaac Newton Institute and the Inter-

Strengths

Editorial and professional service UK numerical analysis punches well above its weight in this regard. 11 UK numerical analysts number among the boards of the SIAM journals SIMAX, SINUM, SIOPT, SIREV and SISC. The UK has major involvement in SIAM through (in 2010) Duff (SIAM Chairman of Board), Higham (SIAM Vice President at Large) and Trefethen (SIAM PresidentElect), and new SIAM Student Chapters in Edinburgh, Manchester and Oxford, while the SIAM UKIE Section remains very active. Editor-inChief positions are held by Iserles (Found. Comput. Math., Acta Numerica), Gould (SIAM J. Opt.), Ralph (Math. Prog. Series B), and N. J. Higham (SIAM Fundamentals of Algorithms book series). The UK-based IMA Journal of Numerical Analysis (est. 1981, with current EiCs Iserles and S¨ uli) continues to thrive, with a 2009 Impact Factor placing it in the top 10% of Applied Mathematics journals. Stuart and S¨ uli are co-editors of the Oxford University Press Monograph Series in Numerical Mathematics and Scientific Computation. Conferences, Networks and Summer Schools The highly successful Biennial Dundee meetings in Numerical Analysis have been taken over by Strathclyde and the Leslie Fox Prize continues biennially. 5

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national Centre for Mathematical Sciences). This total exceeds those for most other areas of the mathematical sciences, but it does include a number of grants held by researchers outside the numerical analysis community. Notable recent major individual grants include an EPSRC Career Acceleration Fellowship for T. Betcke (2009) and an EPSRC Leadership Acceleration Fellowship for Tisseur (2010), both of 5 years duration. Stuart was the first UK Mathematician to be awarded a European Research Council Advanced Investigator Award (2008–2013). IRM2004 noted the need for increased support for numerical analysis (“to remain at the level presently achieved in Numerical Analysis much more will be required”—page 27). The EPSRC initiatives that have impacted on the area are all directly related to high performance computing. The 2008 Science and Innovation call had as one of the 5 solicited areas “Numerical Analysis and High Performance Computing: Software Development”; of two proposals in this area that reached the full proposal stage, one was funded: “Numerical Algorithms and Intelligent Software for the Evolving HPC Platform” involving Edinburgh, HeriotWatt and Strathclyde (£4.5M). An EPSRC sandpit “Extreme Computing” in January 2010 led to two NA/HPC projects being funded at around £1.5M in total.

5

interfaces between different parts of numerical analysis. Research in PDE analysis has grown after the last IRM through a number of EPSRCfunded initiatives and developing this further to integrate with, and build upon, the potential in the Numerical Analysis PDE community is a key to competing internationally. The interface between computational PDEs and optimization is a rich research area, worldwide; both communities are strong within the UK but links between them are at an early stage. Research at the interface of statistics and numerical analysis, including uncertainty quantification, is an area of growing importance worldwide where the UK has world-leading strengths which could be further built upon, both on the theoretical side (for example the work of Tanner at the interface of approximation theory and statistics, and the integrated research groups of Roberts and Stuart at Warwick) and in applications such as data assimilation in the geophysical sciences, where the recently (NERC-funded) National Centre for Earth Observation provides a significant driver, as do the interests of oil companies. The UK has been very successful in building library software on top of research in numerical linear algebra and optimization over several decades (NAG, LAPACK, HSL, LANCELOT/GALAHAD etc.). There is also new UK expertise in finite element software (DUNE, via Dedner at Warwick) and further opportunities exists to build further on the substantial research output in differential equations. The numerical analysis group at RAL is funded directly by EPSRC (from the Mathematical Sciences and other programmes) and has recently received a further 4 years funding from September 2011. The long-term future of the group, and of development of the HSL Library is, however, uncertain. Support for MSc courses is no longer available from EPSRC. Yet MSc courses in Numerical Analysis (and more generally Applied Mathematics) have been of significant benefit to the health of the subject, PhD recruitment, industry and commerce. MSc funding needs to be considered within the contexts of the Bologna framework for European higher education and EPSRC support for Doctoral Training Centres. Finally, the age distribution of the community is healthy, thanks to a steady stream of lectureship appointments over the last decade. However, with future new academic appointments likely to be increasingly strategic it will be important for numerical analysts to exploit fully links with other disciplines and with other areas of applied mathematics.

Weaknesses, Threats and Opportunities

Whilst UK research in numerical analysis is strong, it does not play the central role in applied mathematics that it does in major research competitors. A strategy aimed at ensuring that the best numerical analysis informs large-scale scientific computing and mathematical modelling will have long-term benefits to the science and engineering communities as a whole. To realize these benefits a computational infrastructure is needed that allows for the use of state-of-the-art computer systems, including visualization, and training of researchers at this interface. While many universities do attempt to provide computational resources and training there is no national strategy or funding mechanism in this area. Individuals have to fight within their institutions to maintain minimal support. Funding to support software development has always been problematic. The 2006, 2008 and 2010 EPSRC HPC Software Development calls, are welcome exceptions, but these are open to applications from all areas and the majority of proposals do not involve a significant numerical analysis component. A significant opportunity exists to grow interfaces with other areas of mathematics, and to grow 6

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Return to Landscapes List

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 (