Disordered networks are at the heart of a multitude of materials with functional properties where examples range from the glasses used in optical communications technology to the role of water in geological processes. Establishing the network structure, and its relation to a system's physico-chemical and opto-electronic properties, is a prerequisite for making new materials through the principle of rational design. Here we tackle this issue by using an integrated approach to investigate the fundamentals of basic networks, using pressure to manipulate the bonding and network topology. Oxide and chalcogenide glasses along with water will be investigated, the systems chosen to be exemplars of network forming materials with different bonding mechanisms. The contrasting bonding schemes confer the networks with different characteristics and have the potential for making modified materials with tailored functional and structural properties. Applications include the recovery to ambient conditions of materials with novel characteristics, sequestration of the green house gas CO2 by geological fluids, and the effect of rare-earth clustering on the photonic properties of glass. The inherent disorder of liquid and glassy network structures is a blessing, in delivering materials of unique scientific and technological importance, but is also a curse, in providing complexity on the atomic scale. The method of neutron diffraction with isotope substitution (NDIS) has played a pivotal role in unravelling the mysteries of disordered materials since it allows access to the so-called partial structure factors i.e. to the maximum information that can be extracted from a diffraction experiment. Over the last 3 years, Bath has led an initiative to develop the techniques for measuring accurate neutron diffraction patterns for glasses and liquids at high pressures using the Paris-Edinburgh press. Thus, the time is now ideal to exploit the NDIS method to make in situ high pressure and temperature investigations of structurally disordered materials. We intend to investigate the mechanisms of structural collapse in three classes of system with different bonding schemes and concomitant network properties, namely oxide glasses (GeO2), chalcogenide glasses (e.g. GeSe2, As2Se3, AsSe) and water. These particular systems are chosen because they are archetypical materials for the study of disordered networks e.g. they either show or are anticipated to show polyamorphic phase transformations in which there is an abrupt change in their structure and physical properties with change of pressure and/or temperature. In the case of the chalcogenide glasses, the large structural variability leads to the possibility of recovering new materials with novel functional properties to ambient conditions. The structure of two types of adapted networks will also be considered, namely salty water and rare-earth alumino silicate glasses. In the former, the experiments will be made under the high pressure and temperature conditions relevant for geological fluids where applications include the sequestration of CO2. In the latter, the phenomenon of rare-earth clustering will be investigated with a view to controlling the separation of nearest-neighbour ions and hence the optical properties of these materials. Complementary information will be provided, where applicable, by NMR (Warwick), high energy x-ray diffraction, EXAFS spectroscopy and other experimental techniques. The NMR work will include well established nuclei (27Al and 29Si for the alumino silicates) but will extend the boundaries of the method by using 17O, 73Ge and 77Se. A combination of isotopic enrichment and NMR enhancement schemes will maximise the amount of structural information that can be extracted by using these nuclei as probes. Importantly, the experimental work will be enriched and complemented by molecular dynamics simulations made in collaboration with groups in Oxford, Cambridge and Strasbourg.

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.