SAMBa aims to create a generation of interdisciplinary mathematicians at the interface of stochastics, numerical analysis, applied mathematics, data science and statistics, preparing them to work as research leaders in academia and in industry in the expanding world of big models and big data. This research spectrum includes rapidly developing areas of mathematical sciences such as machine learning, uncertainty quantification, compressed sensing, Bayesian networks and stochastic modelling. The research and training engagement also encompasses modern industrially facing mathematics, with a key strength of our CDT being meaningful and long term relationships with industrial, government and other non-academic partners. A substantial proportion of our doctoral research will continue to be developed collaboratively through these partnerships. The urgency and awareness of the UK's need for deep quantitative analytical talent with expert modelling skills has intensified since SAMBa's inception in 2014. Industry, government bodies and non-academic organisations at the forefront of technological innovation all want to achieve competitive advantage through the analysis of data of all levels of complexity. This need is as much of an issue outside of academia as it is for research and training capacity within academia and is reflected in our doctoral training approach. The sense of urgency is evidenced in recent government policy (cf. Government Office for Science report "Computational Modelling, Technological Futures, 2018"), through the EPSRC CDT priority areas of Mathematical and Computational Modelling and Statistics for the 21st century as well as through our own experience of growing SAMBa since 2014. We have had extensive collaboration with partners from a wide range of UK industrial sectors (e.g. agri-science, healthcare, advanced materials) and government bodies (e.g. NHS, National Physical Laboratory, Environment Agency and Office for National Statistics) and our portfolio is set to expand. SAMBa's approach to doctoral training, developed in conjunction with our industrial partners, will create future leaders both in academia and industry and consists of: - A broad-based first year developing mathematical expertise across the full range of Statistical Applied Mathematics, tailored to each incoming student. - Deep experience in academic-industrial collaboration through Integrative Think Tanks: bespoke problem-formulation workshops developed by SAMBa. - Research training in a department which produces world-leading research in Statistical Applied Mathematics. - Multiple cohort-enhanced training activities that maximise each student's talents and includes mentoring through cross-cohort integration. - Substantial international opportunities such as academic placements, overseas workshops and participation in jointly delivered ITTs. - The opportunity for co-supervision of research from industrial and non-maths academic supervisors, including student placements in industry. This proposal will initially fund over 60 scholarships, with the aim to further increase this number through additional funding from industrial and international partners. Based on the CDT's current track record from its inception in 2014 (creating 25 scholarships to add to an initial investment of 50), our target is to deliver 90 PhD students over the next five years. With 12 new staff positions committed to SAMBa-core areas since 2015, students in the CDT cohort will benefit from almost 60 Bath Mathematical Sciences academics available for lead supervisory roles, as well as over 50 relevant co-supervisors in other departments.

This proposal aims to develop a framework for the theoretical understanding of singularities in solutions to nonlinear partial differential equations and as such bridges theoretical and applied mathematics. In science and technology, singularities often correspond to the limiting behaviour of a physics, engineering or economics model and hence are of paramount importance in understanding its behaviour. For example, certain materials (such as CuAlNi crystals) will try to accommodate prescribed boundary deformations by developing infinitely fine internal oscillations, so-called microstructure. Such materials have many important applications, for example in shape-memory alloys, which remember their shape even after being deformed, and will return to it once they are heated above a certain temperature. Other highly oscillatory situations encountered in nature are turbulent flows. In reality, the finest scale for such oscillations is bounded by the emergence of atomistic effects below a certain threshold, but often this atomistic-to-continuum length scale is so small that macroscopically we can assume that the frequency is nearly infinite and thus, the usual continuum mechanics models hit their boundary of modelling validity. In particular, infinitely fast oscillations are not expressible as functions and one needs to switch to a more advanced framework. Other examples are models describing damage and delamination. Here, one wants to infer the behaviour of a material that has suffered some structural damage or attrition, which, however, might not be macroscopically visible. Many engineering challenges in modern technologies can be attributed to such effects (for example in the recent widely-publicised case of cracking in the wing ribs of the new Airbus A380). Interest in singularities occurring in PDEs has never been greater. As so many technological applications depend on predictability and insight into singularities, it is imperative to push towards a greater understanding of the underlying mechanisms. The state of knowledge at the moment is unsatisfactory and many effects are only poorly understood. In the research outlined in this proposal we aim to provide a set of tools to tackle some of the most pressing problems in the theory of singularities and will push for a greater understanding of the underlying effects causing the formation of singularities. Technically, we will base the development on a recently developed tool, the so-called "microlocal compactness form" that allows to capture and investigate a variety of singular effects in a unified way. In the course of the project we will specifically consider the following questions: - We will consider singularities in hyperbolic conservation laws and aim to make progress on the important open questions in the field. - We will investigate how the hierarchy of microstructure can be efficiently described and this description harnessed in homogenisation theory and the modelling of damage and delamination processes. We will also explore the ramifications of such new results on some fundamental questions in the Calculus of Variations (e.g. Morrey's conjecture). - We will further the theoretical understanding of compensated compactness as a tool in the analysis of PDEs. Finally, in collaboration with engineers, we will consider the implications for real-world applications and will use the theoretical insights gained in the course of this work to improve the practical understanding of singularities in applications of science, technology, and engineering.