Analysis: from theory to applications

Mathematical analysis is at the nexus of contemporary mathematics and its applications. It is the branch of mathematics most directly connected to the activity of describing aspects of the world in quantitative terms. Problems originating in science, engineering and industry typically reach mathematics by way of analysis, while new mathematical ideas, methodologies, techniques and algorithms, more often than not, enter application areas through analysis. Thus, the health of analysis, which provides the conceptual framework and technology and thereby underpins the majority of novel applications of mathematics in science, engineering and industry, and the provision of quality personnel in this subject are of key importance not just to the future of UK mathematics but to the UK science base in its entirety, indeed to the government strategy in fostering economic growth through scientific development and innovation.There is a critical need for knowledge transfer from specialized expert mathematical communities in analysis (partial differential equations, harmonic analysis, stochastic analysis) into the applied modelling community. This knowledge transfer is presently a recognized UK weakness. We believe that the reverse transfer, by which work on mathematical fundamentals is stimulated and focussed by modelling challenges in applications, is also a vital ingredient of a healthy mathematical community. Cambridge analysis is not confined to one specialized area but includes internationally strong individuals and groups in PDEs for mathematical physics, applications of PDEs, stochastic analysis, computational analysis, together with an unrivalled tradition in applied mathematical modelling. The ongoing realignment and interconnection of these groups gives an excellent environment in which to establish a Centre for Doctoral Training in analysis, which will enable doctoral students to experience the power and excitement of the mathematical modelling process from beginning to end, that is, from a physical, biological or industrial problem not yet formulated in mathematical terms, to a mathematical model, understood by rigorous theory and efficient computation, and then to see the results used for effective prediction, control or scientific understanding.The CDT will offer an enhanced graduate programme in pure and applied analysis. The aim is to create a distinctive team of young analysts who see the scope of their work as ranging from leading-edge theory to leading-edge applications. This is difficult within the standard three-year PhD framework. Through an initial period of richer training integrated with wider research experience, allowed by the CDT, students will develop a mixture of pure, stochastic, applied and computational skills, which will be a highly effective preparation for the specialized research required for a PhD. Continuing training activities throughout the duration of the CDT programme will encourage breadth of interest and approach. The CDT will engage with the Cambridge research environment, both within the University and outside in the research institutes and other enterprises which form the Cambridge phenomenon, building on existing links, to make a strong connection between the leading edge of core analysis and diverse and important applications areas.

The Centre for Doctoral Training will have impact in analysis, probability, computational mathematics and applied mathematical modelling in UK universities, through the distinctive orientation and training of its students. The key features of this orientation and training are the integration of expertise in modern theory and computation with experience of work in applications. We believe that this integrated approach will contribute significantly to the vitality and effectiveness of these academic communities, which will have a multiplier effect in the longer term on the downstream impact in industry and commercial sectors. The impact of the research outcomes themselves will also be significant and all steps will be taken to maximize this through dissemination in archives and academic journals, and where appropriate to the wider public. The Centre will also impact immediately those sectors of the UK economy and society that rely directly or indirectly on mathematical modelling and its applications, through the supply of appropriately trained individuals. Specific examples include areas of industry and government such as aerospace, medical imaging, weather and climate prediction and data mining, but there are many more potential examples. Long-term success in these areas depends crucially on recruitment of individuals who have acquired high-level mathematical skills and who have training and experience that enables them to contribute to applied projects, often team-based and requiring broad analytical expertise. Analysis has been selected as the mathematical focus of the Centre for Doctoral Training since it is the branch of mathematics most directly connected to the activity of describing aspects of the world in quantitative terms. The Centre for Doctoral Training will take on outstanding graduate-level mathematicians and train them in the full range of modern techniques of analysis for mathematical modelling, at the same time providing experience in the use of such techniques in scientific and industrial contexts. These students will then be in an excellent position to move on to contribute directly to important industry and government sectors, or in some cases reinforcing the aims of the Centre for Doctoral Training as new recruits to the mathematical analysis community who see the scope of the subject as extending from modern leading-edge theory to applications. The proposed programme of the Centre for Doctoral Training has been carefully designed to emphasise the importance of applications and of maintaining broad interests alongside the need for sustained periods of detailed research on challenging leading-edge theoretical problems. Opportunities for employment in non-academic sectors will be emphasised throughout and representatives of these sectors will provide input to the programme through lectures and workshops. Formal links to organisations such as the Smith Institute will ensure routes not only for knowledge transfer but also for transfer of personnel into applications areas at the end of the Centre for Doctoral Training programme.