EPSRC Centre for Doctoral Training in Partial Differential Equations: Analysis and Applications

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.

The core purpose of this proposed project is to establish a world-class doctoral training centre of excellence in PDEs, with a vibrant and stimulating training/research environment and a dynamic and sustainable doctoral training base, to train a new generation of first-rate UK leaders in academic and industrial research in PDEs and their analysis/applications with a portfolio of transferable, interdisciplinary, organisational and ethical skills to interact with industry and society and to help drive scientific advances for the next fifty years.

The proposed CDT cohort training programme will equip students with a broad and deep expertise in the analysis and applications of PDEs and related areas of Core Mathematics and its Interfaces, and will engage with other areas of the mathematical sciences and academic/non-academic users. In current practice, students often narrow down in an abrupt, stepwise fashion to a specific research project, leaving them with an incomplete, imbalanced perception of the research purpose, context, and value. Such a cohort training, beyond the traditional training practice, will provide students with new opportunities not only to learn many diverse aspects of PDEs and related analysis/applications through group-work and mutual support, but also to develop their ability to engage with a range of communities and to understand and communicate with experts in diverse topics.

We envisage impacts of several different kinds resulting from the project:

1. Creation of a world-class doctoral training centre of excellence in PDEs, which, because of the prevalence of PDEs in science and engineering, impinges on the majority of the EPSRC CDT call priority areas;

2. Involvement of more than 50 DPhil students, who will be trained by the CDT in PDEs and other related CDT priority areas and will go on to have careers in academia or applied sectors in industry and government;

3. Advancement of the field of PDEs, Analysis, and related Core Mathematics and its Interfaces within the UK, as well as other fields such as the global challenge areas of energy and climate change, by delivering a new generation of home-grown research leaders in the priority areas, which will also give the UK an improved standing (reputation and influence) within the global mathematical research community;

4. Reinforcement of the UK's knowledge-based economy, by providing users in commerce, industry, and government agencies that rely on an understanding of PDEs and related analysis with a new generation of problem-solvers and innovators;

5. Creation of a lasting web-based archive of lectures on major contributions to the field and delivery of web-based teaching materials with benefit to the wider community;

6. Development and application of new mathematical research in PDEs and related Core Mathematics and its Interfaces, which will be published in journal articles, conference proceedings, and arXiv articles, and exposed in seminar and conference talks, so that mathematicians and the wider scientific/industrial community can learn about and use the results, and pursue further research.