With the exponentially increasing prevalence of networked sensors and other devices for collecting data in real-time, automated data analysis methods with theoretically justified performance guarantees are in constant demand. Often a key question with such streaming data is whether they show evidence of anomalous behaviour. This could, e.g., be due to malignant bot activity on a website; early warning of potential equipment failure or detection of methane leakages. These and other motivating examples share a common feature which is not accommodated by classical point anomaly models in statistics: the anomaly may not simply be an 'outlying' observation, but rather a distinctive pattern observed over consecutive observations. The strategic vision for this programme grant is to establish the statistical foundations for Detecting Anomalous Structure in Streaming data settings (DASS). Discussions with a wide-range of industrial partners from different sectors have identified important, generic challenges that cut across distinct DASS applications, and are relevant for analysing streaming data more broadly: Contemporary Constrained Environments: Anomaly detection is often performed under various constraints due, for example, to the restrictions on measurement frequency, the volume of data transferable between sensors and a central processor, or battery usage limits. Additionally, certain scenarios may impose privacy restrictions when handling sensitive data. Consequently, it has become imperative to establish the mathematical underpinning for rigorously examining the trade-offs between, e.g., statistical accuracy, communication efficiency, privacy preservation and computational demands. Handling Data Realities: A substantial portion of research in statistical anomaly detection operates under the assumption of clean data. Nevertheless, real-world data typically exhibit various imperfections, such as missing values, labelling errors in data streams, synchronisation discrepancies, sensor malfunctions and heterogeneous sensor performance. Consequently, there is a pressing need for the development of principled, model-based procedures that can effectively address the features of real data and enhance the resilience of anomaly detection methods. Identifying, Accounting for and Tracking Dependence: Not only are data streams often interdependent, but also anomalous patterns may be dependent across those streams. Taking into account both types of dependence is crucial in enhancing the statistical efficiency of anomaly detection algorithms, and also in controlling the errors arising from handling a large number of data streams in a principled way. Other challenges include tracking the path of an anomaly across multiple data sources with a view to learning causal indicators allowing for precautionary intervention. Our ambitious goal of comprehensively addressing these challenges is only achievable via the programme grant scheme. Our philosophy is to tackle the methodological, theoretical and computational aspects of these statistical problems together. This integrated approach is essential to achieving the substantive fundamental advances in statistics envisaged, and to ensuring that our new methods are sufficiently robust and efficient to be widely adopted by academics, industry and society more generally.

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.