EPSRC Centre for Doctoral Training in Analysis (Cambridge Centre for Analysis)

Our proposal builds on the successful start made by Cambridge Centre for Analysis (CCA), a current EPSRC Centre for Doctoral Training. We propose to develop further our activity in two important and rapidly evolving areas of analysis, namely mathematics of information and statistics of complex systems.

Beginning with Newton, for whom the development of calculus and the mathematical understanding of bodies in motion were closely intertwined, the mathematics used to describe real phenomena consistently involves notions of continuity, rate of change, average value, and basic challenges such as the relationship between discrete and continuum objects. This is the domain of analysis, encompassing modelling by partial differential equations and by random processes, and the mathematical theory which guides effective computation for such models.

The centrality of mathematical analysis in the relationship between mathematics and its applications has been acknowledged by successive International Reviews of Mathematics, as has the need to increase the capacity of UK PhD training in analysis. Mathematical Analysis and its Applications is an EPSRC Priority Area.

Beyond the established and important uses of analysis in modelling physical phenomena, digital technology has created new areas where mathematical analysis, in guiding the extraction of knowledge from massive discrete systems, plays an essential role. These include the fields of high-dimensional statistics and the mathematics of information, including compressed sensing. In each of these, one is looking for a reliable means to interpret massive high-dimensional data. Already several CCA students are working in these areas. Big Data is one of the Eight Great Technologies championed by the Minister for Universities and Science. Statistics and Data to Knowledge are EPSRC Priority Areas.

We propose a first year training programme based on our current successful model, now expanded by two further core courses, one in Statistics of Complex Systems and one in Mathematics of Information. These new courses will be paired with postgraduate level courses from the existing Cambridge Masters' (MASt), which students can use to consolidate their understanding. The core courses themselves are based on supervised student team assignments leading to student presentations. The other main components of the first year are research mini-projects (often the route to a PhD project) and an industry workshop. Years two to four are devoted mainly to the PhD thesis.

First year training establishes a collaborative ethos in the cohort and, by mixing students with different prior skills, encourages cross-fertilization of ideas across the different threads of analysis. This is sustained in later years through a programme of seminars, workshops and training in transferable skills. The students appreciate that their collective understanding of a given problem using different skills will often exceed each individual's understanding. This makes cohort-based training especially valuable in analysis.

We already expose all our students to the role of mathematics and the opportunities for mathematicians in industry and society, and we encourage first-hand engagement with applications through mini-projects, industrial seminars and study weeks, and, for some, PhD projects with industrial partners. The development of core skills and eventually the ability to generate new ideas is the hardest and crucial part of training as a research mathematician. This is necessarily our overriding task, in which we seek synergy and inspiration from user engagement. In the new CDT, our network of industrial connections will be further enhanced, along with our collaborations with Cambridge engineering colleagues, and our links with the Smith Institute for Industrial Mathematics.

The key impact of the CDT will be to produce a group of at least 50 outstanding PhD mathematical analysts, who will be equipped to tackle some of the most crucial research questions now facing industry, commerce and academia. A main feature of our program will be providing training in the mathematics of Big Data, including in the fields of non-parametric and high-dimensional statistics and the mathematics of information and compressed sensing. The analysis and interpretation of massive high-dimensional data sets is a fundamental challenge in an enormously broad range of applications, andthe UK Government has identified Big Data as being the first of The Eight Great Technologies upon which the future prosperity of the UK depends. The development of a new generation of mathematical analysts who will contribute to UK needs to tackle these challenges will have very significant economic impact.

The CDT will occupy a strategic position in a leading UK university, bridging between the competitive global community of mathematical research and established UK needs for highly trained mathematicians and statisticians in industry and society. We expect to achieve a shift in career path for many of our students, drawn from the top percentiles of mathematical ability, towards the applications of mathematics in the real world. Moreover, the nature of our training, emphasizing teamwork and cross-disciplinarity, at the same time as the highest international level of mathematical research, will produce highly skilled and effective personnel, for both industrial and university sectors.

The specific need for more UK PhD's in analysis and statistics has been identified twice by International Reviews of Mathematics, as has the lack of competitiveness of UK students in the highly competitive global market for young mathematical researchers. By providing breadth of mathematical education, highest-level original research in a specialist area and exposure to industrial challenges, our CDT students will benefit from an exceptional training in a world-class research environment.

The reputation and vigour of the UK academic sector will benefit from our focus on areas of mathematics which are strongly motivated by applications and which are rapidly developing globally. Significant new knowledge and insight will arise from the PhD programme, enhancing the knowledge economy and delivering the next generation of academics in the crucial areas of analysis, statistics and mathematics of information, disciplines in which the UK is dangerously under-represented. Our engagement with colleagues in engineering will ensure that these academic benefits will not be restricted to mathematics; our CDT will provide a bridge via which the very latest mathematical developments can be brought effectively into application areas, and via which application areas can in turn stimulate and inform mathematical research.