Partial differential equations (PDEs) are at the heart of many scientific advances. The behaviour of every material object in nature, with time scales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modelled by deterministic and stochastic PDEs or by equations with similar features. The role of PDEs within mathematics (especially nonlinear analysis, geometry, topology, stochastic analysis, numerical analysis, and applied mathematics) and in other sciences (such as physics, chemistry, life sciences, climate modelling/prediction, materials science, engineering, and finance) is fundamental and is becoming increasingly significant. PDEs have consequently become one of the largest and most diverse research fields of present-day mathematics. There is a serious shortage of UK researchers and specialists in the Analysis of PDEs and related areas of Core Mathematics and its Interfaces, both in academia and industry, particularly compared to other G8 nations. More generally, several EPSRC reports and the 2010 International Review of UK Mathematics have drawn attention to the under-representation of analysis in the UK, compared to the rest of the world. It is therefore important that resources are invested in this area to remedy this deficiency. The central aim of the new Centre for Doctoral Training (CDT) is to produce cohorts of highly trained, outstanding mathematicians with deep expertise and interdisciplinary skills in the analysis/applications of PDEs and related areas of Core Mathematics and its Interfaces. A sizeable yearly cohort will allow the CDT to create new training mechanisms so that the students will learn theory, analysis, and applications in a variety of fields in a coherent manner with a natural progression, by-passing a traditionally separate `pure' or `applied' approach to learning. The training will be fundamentally connected to all aspects of PDEs and their analysis/applications which, because of the prevalence of PDEs in science and engineering, impinge on a majority of the EPSRC CDT call priority areas. Oxford is well placed to play a leading role, building on UK strengths in PDEs and their analysis/applications. The Oxford Centre for Nonlinear PDE (OxPDE) was created in 2007, jointly by EPSRC under a major Science & Innovation Award and the University of Oxford by significant matching funding. OxPDE has attracted a number of outstanding researchers in PDEs and Analysis, forming the largest research group that there has ever been in PDEs in the UK. The proposed CDT is based on this core group, along with a multidisciplinary cluster of high quality researchers with PDEs as a core connection spread across the Mathematical Institute and the Departments of Physics, Computer Science, Statistics, and Engineering Science within Oxford. The supervisors in our team have extensive experience of providing a high-quality research training environment for supporting doctoral level education/research. The University of Oxford is committed to the formation of the new CDT and will provide a significant contribution, in particular funding up to 3 students per year. One of the key partners, BNP Paribas, will undertake to fund 2 DPhil students commencing in 2014/15 and sponsor 2-6 internships per year for the CDT students. The CDT will have an international dimension with Partners from leading academic and research institutions in the US, China, France, Germany, Italy, Norway, Russia, and Switzerland; these partners have offered a variety of support for our CDT including attendance at their courses and funded visits by our students who will be equipped with a different research/education culture and will gain additional expertise which is absent in the UK.

Building upon our existing flagship industry-linked EPSRC & MRC CDT in Systems Approaches to Biomedical Science (SABS), the new EPSRC CDT in Sustainable Approaches to Biomedical Science: Responsible and Reproducible Research - SABS:R^3 - will train a further five cohorts, each of 15 students, in cutting-edge systems approaches to biomedical research and, uniquely within the UK, in advanced practices in software engineering. Our renewed goal is to bring about a transformation of the research culture in computational biomedical science. Computational methods are now at the heart of biomedical research. From the simulation of the behaviour of complex systems, through the design and automation of laboratory experiments, to the analysis of both small and large-scale data, well-engineered software has proved capable of transforming biomedical science. Biomedical science is therefore dependent as never before on research software. Industries reliant on this continued innovation in biomedical science play a critical role in the UK economy. The biopharmaceutical and medical technology industrial sectors alone generate an annual turnover of over £63 billion and employ 233,000 scientists and staff. In his foreword to the 2017 Life Sciences Industrial Strategy, Sir John Bell noted that, "The global life sciences industry is expected to reach >$2 trillion in gross value by 2023... there are few, if any, sectors more important to support as part of the industrial strategy." The report identifies the need to provide training in skills in "informatics, computational, mathematical and statistics areas" as being of major concern for the life sciences industry. Over the last 9 years, the existing SABS CDT has been working with its consortium of now 22 industrial and institutional partners to meet these training needs. Over this same period, continued advances in information technology have accelerated the shift in the biomedical research landscape in an increasingly quantitative and predictive direction. As a result, computational and hence software-driven approaches now underpin all aspects of the research pipeline. In spite of this central importance, the development of research software is typically a by-product of the research process, with the research publication being the primary output. Research software is typically not made available to the research community, or even to peer reviewers, and therefore cannot be verified. Vast amounts of research time is lost (usually by PhD students with no formal training in software development) in re-implementing already-existing solutions from the literature. Even if successful, the re-implemented software is again not released to the community, and the cycle repeats. No consideration is made of the huge benefits of model verification, re-use, extension, and maintainability, nor of the implications for the reproducibility of the published research. Progress in biomedical science is thus impeded, with knock-on effects into clinical translation and knowledge transfer into industry. There is therefore an urgent need for a radically different approach. The SABS:R^3 CDT will build on the existing SABS Programme to equip a new generation of biomedical research scientists with not only the knowledge and methods necessary to take a quantitative and interdisciplinary approach, but also with advanced software engineering skills. By embedding this strong focus on sustainable and open computational methods, together with responsible and reproducible approaches, into all aspects of the new programme, our computationally-literate scientists will be equipped to act as ambassadors to bring about a transformation of biomedical research.