Stochastic growth phenomena naturally emerge in a variety of physical and biological contexts, such as growth of combustion fronts or bacterial colonies, crystal growth on thin films, turbulent liquid crystals, etc. Even though all these phenomena might appear very diverse at a microscopic scale, they often have the same large-scale behaviour and are therefore said to belong to the same Universality Class. This in particular means that an in-depth analysis of those processes describing these large-scale behaviours is bound to give very accurate quantitative and qualitative predictions about the wide variety of extremely complicated real-world systems in the same class. Over the last 40 years, the Mathematics and Physics communities in a joint effort determined what were widely believed to be the only two universal processes presumed to capture the large-scale behaviour of random interfaces in one spatial-dimension, namely the Kardar-Parisi-Zhang and Edrwards-Wilkinson Fixed Points, and studied their Universality Classes. In a recent work, I established the existence of a third, new universality class, entirely missed by researchers, and rigorously constructed the universal process at its core, the Brownian Castle. The introduction of this novel class opens a number of new stimulating pathways and a host of exciting questions that this proposal aims at investigating and answering. The second pillar of this research programme focuses on two-dimensional random surfaces, which are particularly relevant from a physical viewpoint as they correspond to the growth of two-dimensional surfaces in a three-dimensional space. Despite their importance, two-dimensional growth phenomena are by far the most challenging and the least understood. Very little is known concerning their universal large-scale properties and the even harder quest for fluctuations has barely been explored. The present proposal's goal is to develop powerful and robust tools to rigorously address these questions and consequently lay the foundations for a systematic study of these systems and their features. The last theme of this research plan concerns the Anderson Hamiltonian, also known as random Schrödinger operator. The interest in such an operator is motivated by its ramified connections to a variety of different areas in Mathematics and Physics both from a theoretical and a more applied perspective. Indeed, the spectral properties of the Anderson Hamiltonian are related to the solution theory of (random) Schrödinger's equations or properties of the parabolic Anderson model, random motion in random media or branching processes in random environment. The Anderson Hamiltonian has attracted the attention of a wide number of researchers, driven by the ambition of fully understanding its universal features and the celebrated phenomenon Anderson localisation. This proposal will establish new breakthroughs and tackle long-standing conjectures in the field by complementing the existing literature with novel techniques.

Random motions in random media have been intensively studied for over forty years and many interesting features of these models have been discovered. The aim is to understand the motion of a particle in a turbulent media. Most of the work has been focused on the case where the particle evolves in a static random environment, for which slow-downs and trapping phenomena have been proved. More recently, mathematicians and physicists have been interested in the case of dynamic random environments, where the media can fluctuate with time. Random walks on the exclusion process is probably the canonical model for the field. Much less is known on this model, but exciting conjectures and questions have been made. Some of the most challenging questions concern the possibility of super-diffusive regimes, and the existence of effective traps along the trajectory of the walk. In this project, we aim at, on one hand, adapt the techniques recently developped in one dimension to the multi-dimensional model and, on the other hand, understand the presence or absence of atypical behaviors for this model.

Cyber-physical risks pose new challenges to the UK security and defence community. Nation state actors are increasingly equipping themselves with tools capable of causing damage to cyber-physical systems. During times of elevated threat, e.g. the Ukraine war, it has become common for such tools to be used in the theatre of conflict, but the risk that they will be used outside the affected region also increases, as does the risk that they will fall into the hands of non-state actors and used more widely than against conventional cyber-physical systems. At the same time, society as a whole is developing a growing reliance on social-media, apps, connected IoT devices and both AI and Mixed Reality. By itself, this also has major security implications leading to high volumes of new crimes that need addressing but, when used in combination with cyber-physical attacks, a qualitatively different form of attack is experienced, known as a hybrid threat because of its similarity to the hybrid cyber/kinetic warfare that has emerged as a feature of recent conflicts. Our team will create a cohort of leaders equipped with specialist skills specifically adapted to tomorrow's cyber-physical risks. They will learn how to (1) manage risks that develop jointly across the cyber and physical domains, (2) reduce risks in one domain by intervening in the other domain, (3) understand how criminal groups protect their interest by operating across domains, and (4) pre-empt the propagation of unintended consequences from one domain to the next. These future leaders will also undertake an intense programme of activities designed to nurture key complementary skills. The eXchange, Lead!, Get it Done, and Next Step schemes will be shaped with our industry partners to address important gaps highlighted in a recent government report: "40% of businesses reported that cybersecurity job applicants are deficient in their complementary skills". Beyond fostering technical expertise, the CDT will help students and partners develop an acute awareness of the wider issues and dilemmas posed by all such work in democratic societies. To deliver an exceptional learning experience, we bring a powerful group of industry partners and academic experts from the engineering, natural and social sciences within a single-site research training centre in London. Academically excellent and pro-active within and between cohorts, the 50+ CDT students will specialise in a range of scientific techniques from at least two of the following four 'cyber-physical risk' themes. These were selected in consultation with students and partners based on their relevance, attractiveness and employment opportunities: 1. Futures: forecast how future socio-technical trends will shape the geopolitical implications of cyber-physical risk, understand how to co-design effective control and mitigation measures for different legal, technical, and policy contexts in order to support societal resilience. 2. Cyber-physical systems: support regulation of IoT devices, detect 'below the radar' attacks against cyber-physical control systems; contribute to the resilience of critical infrastructures. 3. Online Communication: prevent the misuse of social media for disinformation campaigns, detect incitement to hate crime in games, analyse crime related communication on the dark web, etc. 4. Simulation and Interaction: develop and apply simulation technologies (e.g., digital twins, VR, XR) for the study of human behaviour in a range of cyber-physical risk scenarios. EPSRC areas directly relevant to the following themes: Artificial intelligence technologies, Building a secure and resilient world, Digital security by design, Digital signal processing, Human communication in information and communication technologies, Human-computer interaction, ICT networks and distributed systems, Image and vision computing, Natural language processing, Operational research, Pervasive and ubiquitous computing.

The EPSRC Centre for Doctoral Training in Statistics and Machine Learning (StatML) will address the EPSRC research priority of the 'physical and mathematical sciences powerhouse' through an innovative cohort-based training program. StatML harnesses the combined strengths of Imperial and Oxford, two world-leading institutions in statistics and machine learning, in collaboration with a broad spectrum of industry partners, to nurture the next generation of leaders in this field. Our students will be at the forefront of advancing the core methodologies of data science and AI, crucial for unlocking the value inherent in data to benefit industry and society. They will be equipped with advanced research, technical, and practical skills, enabling them to make tangible real-world impacts. Our students will be ethical and responsible innovators, championing reproducible research and open science. Collaborating with students, charities and equality experts, StatML will also pioneer a comprehensive strategy to promote inclusivity, attract individuals from diverse backgrounds and eliminate biases. This will help diversify the UK's future statistics and machine learning workforce, essential for ensuring data science is used for public good. Data science and AI are now part of our everyday lives, transforming all sectors of the economy. To future-proof the UK's prosperity and security, it is essential to develop new methodology, specifically tailored to meet the big societal challenges of the future. The techniques underpinning such methods are founded in statistics and machine learning. Through close collaboration with a broad range of industry partners, our cohort-based training will support the UK in producing a critical mass of world-leading researchers with expertise in developing cutting-edge, impactful statistical and machine learning methodology and theory. It is well documented in government and learned society reports that the UK economy has an urgent need for these people. The significant level of industry support for our proposal also highlights the necessity of filling this gap in the UK data science ecosystem. StatML will learn from and build upon our previous successful experiences in cohort training of doctoral students (our existing StatML CDT funded in 2018, as well as other CDTs at Imperial and Oxford). Our students will continue to produce impactful, internationally leading research in statistics and machine learning (as evidenced by our students' impressive publication record and our world-leading research environment, as rated by the REF 2021 evaluation), while complementing this with a bespoke cohort-based Advanced Training program in Statistics and Machine Learning (StatML-AT). StatML-AT has been developed from our experience and in partnership with industry. It will be responsive to emerging technologies and equip our students with the practical skills required to transform how data is used. It will be delivered by our outstanding academics from both institutions alongside with industry leaders to ensure that students receive training in cutting edge technologies, along with the latest ideas in ethics, responsible innovation, sustainability and entrepreneurship. This will be complemented by industrial and academic placements to allow the students to develop their own international network and produce high-impact research. Together, StatML and its partners will train 90+ students over 5 cohorts. More than half of these will be funded from external sources, including 25+ by industry, representing excellent value for money. Our diverse cohorts will benefit from a unique and responsive training program combining academic excellence, industry engagement, and interdisciplinary culture. This will make StatML a vibrant research environment inspiring the next methodological advancements to transform the use of data and AI across industry and society.

The CDT will train the next generation of leaders in statistics and statistical machine learning, who will be able to develop widely-applicable novel methodology and theory, as well as create application-specific methods, leading to breakthroughs in real-world problems in government, medicine, industry and science. The research will focus on the development of applicable modern statistical theory and methods as well as on the underpinnings of statistical machine learning. The research will be strongly linked to applications. There is an urgent national need for graduates from this CDT. Large volumes of complicated data are now routinely collected in all sectors of society, encompassing electronic health records, massive scientific datasets, governmental data, and data collected through the advent of the digital economy. The underpinning techniques for exploiting these data come from statistics and machine learning. Exploiting such data is crucial for future UK prosperity. However, several reports from government and learned societies have identified a lack of individuals able to exploit this data. In many situations, existing methodology is insufficient. Off-the-shelf approaches may be misleading due to a lack of reproducibility or sampling biases which they do not correct. Furthermore, understanding the underlying mechanisms is often desired: scientifically valid, interpretable and reproducible results are needed to understand scientific phenomena and to justify decisions, particularly those affecting individuals. Bespoke, model-based statistical methods are needed, that may need to be blended with statistical machine learning approaches to deal with large data. Individuals that can fulfill these more sophisticated demands are doctoral level graduates in statistics who are well versed in the foundations of machine learning. Yet the UK only graduates a small number of statistics PhDs per year, and many of these graduates will not have been exposed to machine learning. The Centre will bring together Imperial and Oxford, two top statistics groups, as equal partners, offering an exceptional training environment and the direct involvement of absolute research leaders in their fields. The supervisor pool will include outstanding researchers in statistical methodology and theory as well as in statistical machine learning. We will use innovative and student-led teaching, focussing on PhD-level training. Teaching cuts across years and thus creates strong cohort cohesion not just within a year group but also between year groups. We will link theoretical advances to application areas through partner interactions as well as through a placement of students with users of statistics. The CDT has a large number of high profile partners that helped shape our application priority areas (digital economy, medicine, engineering, public health, science) and that will co-fund and co-supervise PhD students, as well as co-deliver teaching elements.