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.

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.

We are living in an unprecedented age where vast quantities of our personal data are continually recorded and analysed, for example, our travel patterns, shopping habits and fitness routines. Our daily lives are now tied into this evolving loop of data collection, leading to data-based automated decisions, that can make recommendations and optimise our routines. There is tremendous economic and societal value in understanding this deluge of unstructured disparate data streams. A key challenge in Artificial Intelligence (AI) research is to extract meaningful value from these data sources to make decisions that can be trusted and understood to improve society. The PASCAL research programme is focused on developing an end-to-end framework, from data to decisions, that naturally accounts for data uncertainty and provides transparent and interpretable decision-making tools. The algorithms developed throughout this research project will be generally-applicable in a wide range of application domains and appropriate for modern computer hardware infrastructure. All of the research and associated algorithms will be widely available through high-quality open-source software that will ensure the widest possible uptake of this research within the international AI research community. PASCAL will focus on two primary applications areas: cybersecurity and transportation, which will stimulate and motivate this research and ensure wide-spread impact within these sectors. To drive through the impact and uptake of this research within these sectors, we will work closely with committed strategic partners, GCHQ, the Heilbronn Institute of Mathematical Research, Transport Research Laboratory, the University of Washington and the Alan Turing Institute. Cybersecurity - The proliferation of computers and mobile technology over the last few decades has led to an exponential increase in recorded data. Much of this data is personally, economically and nationally sensitive and protecting it is a key priority for any government or large organisation. Threats to data security exist on a global scale and identifying potential threats requires cybersecurity experts to evaluate and extract critical intelligence from complex and evolving data sources. In order to model and understand the intricate patterns between these data sources requires complex mathematical models. The PASCAL programme will develop new algorithms that maintain the richness of these mathematical models and use them to provide interpretable and transparent decision recommendations. Autonomous vehicles (AV) - The transition to AVs will be the most significant global change in transportation for the past century. The economic benefit and successful implementation of this technology within the UK requires a thorough understanding of the risks posed by driverless vehicles and what new procedures are required to ensure human safety. Through PASCAL, we will develop a framework to artificially-generate realistic traffic scenarios to test AVs under a wide range of road conditions and create criteria to safely accredit AV vehicles in the UK.

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.