We live in the "Era of Mathematics" (UKRI, 2018), in which mathematics research has deep and widespread impact. Medical imaging is enhanced using the theory of inverse problems. Predicting sewage contamination in waterways after storms requires solving complicated systems of hydrodynamic equations. Machine learning tools are revolutionising data-intensive computing and, handled with proper mathematical care, have vast potential benefits for science and society. These are examples of the ongoing explosion in mathematical innovation driving, and being driven by, the analysis and modelling of data running through every aspect of life. Cutting-edge research now sits at the interface of data science and mathematical modelling. Methods and fields such as compressed sensing, stochastic optimisation, neural networks, Bayesian hierarchical models - to name but a few - have become interwoven and contributed to the delivery of a new domain of research. We refer to this research interface as "statistical applied mathematics". Established in 2014, the Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa, samba.ac.uk) delivers leading research and training in this space. In the development of this bid, we have consulted widely with academic, industrial, and governmental partners, who consistently report a large and widening gap between demand and supply for highly skilled graduates. Our vision is to create a new generation of statistical applied mathematicians ready to lead high-impact, data-driven, mathematically-robust research in academia and industry. We will nurture a vibrant culture of cohort learning, enabling internationally-leading training in modern mathematical data science. A particularly important research focus will be the synthesis of data-driven methods with robust mathematical modelling frameworks. Tomorrow's industrial mathematicians and statisticians must understand when machine learning tools are (and are not) appropriate to use and be able to conduct the underpinning research to improve these tools by integrating scientific domain knowledge. This research challenge is informed by deep partnerships with a range of industry and government bodies. Our long-term partners such as BT, Syngenta, Novartis, the NHS, and the Environment Agency co-create our vision and our training. They are emphatic that we must address the urgent need for mathematical data science talent in this key strategic area for the UK economy. Many of our students will work directly on industry challenges during their PhD either in their core research or with internships. Our unique Integrative Think Tanks are the key mechanism for exploring new research ideas with industry. These are week-long events where SAMBa students, leading academics, and partners work together on industrial and societal problems. SAMBa graduates will be able to develop and apply new ideas and methods to harness the power of data to tackle challenges affecting society, the economy, and the environment. Our students will move into academia, providing sustainability to the UK's capacity in this field, as well as industry and government, providing impact through societal benefits and driving economic growth. Many alumni now hold permanent positions at leading UK universities and senior positions in a range of businesses. The CDT will be embedded within the University of Bath's Department of Mathematical Sciences, where 98% of the research is world leading or internationally excellent (REF2021). The CDT is supported by 58 academics in maths, with similar numbers of co-supervisors from industry and other departments. The centre will be co-delivered with 22 industry and government partners. A vital international perspective is provided by a worldwide network of 11 academic institutions sharing our scientific vision.

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.