Incredible technological advances in data collection and storage have created a world in which we are constantly generating data. From supermarket loyalty cards and social media posts to healthcare records and credit card transactions, a digital footprint exists for every aspect of our lives. The ability of data science to analyse and act upon these complex and varied data sources has the potential to improve and revolutionise our lives in a myriad of ways, for example, through the development of driverless cars and personalised medicine. The great challenge of data science lies in the trade-off between the speed and accuracy with which large volumes of data can be analysed and acted upon within complex data environments. Extracting deeper knowledge from data requires increasingly sophisticated mathematical models. However, applying such models introduces significant computational constraints, forcing data scientists to rely upon simpler models or approximate inference tools. In collaboration with strategic partners, this project will bring together industry experts to investigate new approaches to data science driven by fundamental challenges in modelling and analysing large-scale spatial and security data. The data and issues within this domain are highly-significant to modern society as they cover, for example, issues pertaining to fraud detection and computer hacking, as well as understanding and predicting human behaviour within a Smart City environment. Novel mathematical advances in computational statistics and machine learning will be developed to produce scalable techniques for applying sophisticated mathematical models to large-scale heterogeneous and structured data sources. A key component of this project is reproducibility through the creation of open-source software. These tools will allow data scientists to implement research outcomes to extract key features from complex data and make decisions with high accuracy under uncertainty.

This is an application for partial funding for a 5-day research workshop in probability entitled Random Dynamics and Other Recent Developments, to be held at the University of Sheffield in April 2018. The meeting will benefit UK mathematics by stimulating interactions between UK and international experts on the latest developments and future research directions within probability. Particular emphases is placed on the participation of female academics and early career researchers. The UK is recognized globally as a leading center for research in probability, but there is a constant need to keep abreast of international developments. Our workshop focuses on the inter-related themes of random geometry, random reinforced processes and stochastic dynamical systems. All three of these areas have seen intense recent activity within the international probability community, both for their theoretical significance and for their importance in applications such as data science, social networks and machine learning. The workshop is centered around three mini-courses, given by internationally distinguished researchers, corresponding to the three themes mentioned above. The mini-courses will be complemented by invited individual talks from leading researchers, alongside opportunities for PhD students and early career researchers to give contributed talks or present at a poster session. Additionally, the proposed schedule of the workshop contains time set aside on each day for break-out sessions and discussion groups, to facilitate exploration of new research directions and to support the formation of new collaborations.

We aim to grow the world's leading centre for training in quantum engineering for the emerging quantum technology (QT) industry. We have designed this CDT in collaboration with a large number of academic and industry experts, and included as partners those who will add substantially to the training and cohort experience. Through this process a consistent picture of what industry wants in future quantum engineers emerged: people who can tackle the hardest intellectual challenges, recognising the end goal of their research, with an ability to move from fundamental physics towards the challenges of engineering and miniaturising practical systems, who understands the capabilities of other people (and why they are useful). Industry wants people with good decision-making, communication and management skills, with the ability to work across discipline boundaries (to a deadline and a budget!) and build interdisciplinary teams, with the ability to translate a problem from one domain to another. Relevant work experience, knowledge of entrepreneurship, industrial R&D operations and business practices are essential. By forming a hub of unrivalled international excellence in quantum information and photonics, surrounded by world-class expertise in all areas of underpinning science and technology and the scientific and technological application areas of QT, and a breadth of academic and industry partners, we will deliver a new type of training: quantum engineering. Bristol has exceptional international activity in the areas that surround the hub: from microelectronics and high performance computing to system engineering and quantum chemistry. The programme will be delivered in an innovative way-focussing particularly on cohort learning-and assessed by a variety of different means, some already in existence in Bristol. We believe that we are attempting something new and exciting that has the potential to attract and train the best students to ensure that the resulting capacity is world-class, thus providing real benefits to the UK economy.

Geometry and number theory are core disciplines within pure mathematics, with many repercussions across science and society. They are subjects that have attracted some of the best minds in mathematics since the time of the Ancient Greeks and continue to exert a natural fascination on professional and amateur mathematicians alike. Throughout the history of mathematics, both topics have very often inspired major mathematical developments which have had enormous impact beyond their original applications. The fascination of number theory is exemplified by the story of Fermat's last theorem, the statement of which was written down in 1637 and which is simple enough to be understood by anyone familiar with high school mathematics. It took more than 350 years of hard work and highly significant developments across mathematics before Wiles's celebrated proof was finally published in 1995. Wiles's proof involves a mixture of ideas from number theory and geometry, and the interplay between these topics is one of the most active areas of research in pure mathematics today. The Centre is needed to educate the next generation of academic researchers to maintain the excellence and competitiveness of the UK's universities and also to deliver highly trained mathematicians ready to take their place in financial and other high-tech industries. As shown by our letters of support from the Bank of England, the Satellite Applications Catapult, Heilbronn, Royal Bank of Scotland, and Schlumberger, a wide range of employers have the vision to invest in highly trained pure mathematicians. Our partners all speak highly of the analytical and problem-solving abilities of pure mathematicians trained to PhD level. The students trained in this Centre will be even more highly skilled: the structure of the training programme will encourage independence and leadership and will embed professional development and key skills such as programming, communication skills and public engagement alongside cutting-edge research in topics chosen from geometry and number theory.

Geometry and number theory are core disciplines within pure mathematics, with many repercussions across science and society. They are subjects that have attracted some of the best minds in mathematics since the time of the Ancient Greeks and continue to exert a natural fascination on professional and amateur mathematicians alike. Throughout the history of mathematics, both topics have often inspired major mathematical developments which have had enormous impact beyond their original applications. The fascination of number theory is exemplified by the story of Fermat's last theorem, the statement of which was written down in 1637 and which is simple enough to be understood by anyone familiar with high school mathematics. It took more than 350 years of hard work and significant developments across mathematics before Wiles's celebrated proof was finally published in 1995. Wiles's proof, for which he was awarded the prestigious Abel Prize in 2016, involves a mixture of ideas from number theory and geometry, and the interplay between these topics is one of the most active areas of research in pure mathematics today. For example, the work of Ngo on the Langland's program (for which he was awarded the Fields Medal in 2010, the highest honour in mathematics) and Scholze on arithmetic algebraic geometry (for which he was offered a New Horizons in Mathematics Breakthrough Prize in 2016, and is expected to be awarded the Field Medals this year), show the significant impact of geometric ideas on number theory. In the other direction, number theory has been used to prove conjectures in geometry, including a path proposed by Kontsevich (Fields Medal 1998, Breakthrough Prize 2015) and Soibelman to help solve one of the major open problems in geometry, the SYZ conjecture, which lies at the interface of geometry and theoretical physics. These and other connections between geometry and number theory continue to lead to some of the most exciting research developments in mathematics. This CDT will be run by a partnership of researchers at Imperial College London, King's College London, and University College London, which together form the largest and one of the strongest UK centres for geometry and number theory. By training mathematicians to PhD level in geometry and number theory, and by ensuring that more general skills (for example, computing, communication, teamwork, leadership) are embedded as a demanding and enjoyable part of our programme, this CDT will deliver the next generation of highly trained researchers able to contribute not only to the UK's future educational needs but also to those of the financial and other high-tech industries. Our graduates will contribute directly to national security (GCHQ is, for example, a user of high-end pure mathematics) but also more indirectly as employees in industries which value the creative and novel approach that mathematicians typically bring to problem solving.