Public opinion on complex scientific topics can have dramatic effects on industrial sectors (e.g. GM crops, fracking, global warming). In order to realise the industrial and societal benefits of Autonomous Systems, they must be trustworthy by design and default, judged both through objective processes of systematic assurance and certification, and via the more subjective lens of users, industry, and the public. To address this and deliver it across the Trustworthy Autonomous Systems (TAS) programme, the UK Research Hub for TAS (TAS-UK) assembles a team that is world renowned for research in understanding the socially embedded nature of technologies. TASK-UK will establish a collaborative platform for the UK to deliver world-leading best practices for the design, regulation and operation of 'socially beneficial' autonomous systems which are both trustworthy in principle, and trusted in practice by individuals, society and government. TAS-UK will work to bring together those within a broader landscape of TAS research, including the TAS nodes, to deliver the fundamental scientific principles that underpin TAS; it will provide a focal point for market and society-led research into TAS; and provide a visible and open door to engage a broad range of end-users, international collaborators and investors. TAS-UK will do this by delivering three key programmes to deliver the overall TAS programme, including the Research Programme, the Advocacy & Engagement Programme, and the Skills Programme. The core of the Research Programme is to amplify and shape TAS research and innovation in the UK, building on existing programmes and linking with the seven TAS nodes to deliver a coherent programme to ensure coverage of the fundamental research issues. The Advocacy & Engagement Programme will create a set of mechanisms for engagement and co-creation with the public, public sector actors, government, the third sector, and industry to help define best practices, assurance processes, and formulate policy. It will engage in cross-sector industry and partner connection and brokering across nodes. The Skills Programme will create a structured pipeline for future leaders in TAS research and innovation with new training programmes and openly available resources for broader upskilling and reskilling in TAS industry.

Moore's Law states that the number of active components on an microchip doubles every 18 months. Variants of this Law can be applied to many measures of computer performance, such as memory and hard disk capacity, and to reductions in the cost of computations. Remarkably, Moore's Law has applied for over 50 years during which time computer speeds have increased by a factor of more than 1 billion! This remarkable rise of computational power has affected all of our lives in profound ways, through the widespread usage of computers, the internet and portable electronic devices, such as smartphones and tablets. Unfortunately, Moore's Law is not a fundamental law of nature, and sustaining this extraordinary rate of progress requires continuous hard work and investment in new technologies most of which relate to advances in our understanding and ability to control the properties of materials. Computer software plays an important role in enhancing computational performance and in many cases it has been found that for every factor of 10 increase in computational performance achieved by faster hardware, improved software has further increased computational performance by a factor of 100. Furthermore, improved software is also essential for extending the range of physical properties and processes which can be studied computationally. Our EPSRC Centre for Doctoral Training in Computational Methods for Materials Science aims to provide training in numerical methods and modern software development techniques so that the students in the CDT are capable of developing innovative new software which can be used, for instance, to help design new materials and understand the complex processes that occur in materials. The UK, and in particular Cambridge, has been a pioneer in both software and hardware since the earliest programmable computers, and through this strategic investment we aim to ensure that this lead is sustained well into the future.

Most information about the Earth's sub-surface (e.g., rock stratal geometries, temperatures, pressures, composition, fluid content) comes from either seismic or electromagnetic waves. These propagate through the subsurface and either defract, refract or reflect (echo) back to the surface. There they are recorded and interpreted for sub-surface properties. Traditionally such waves emanate from active energy sources (earthquakes in seismology, or actively-induced seismic or electromagnetic sources in industrial subsurface exploration settings). However, in the past five years a revolutionary new set of methods has developed under the general name of 'Wavefield Interferometry', which have changed the nature of seismology fundamentally. In its most popular form, interferometry allows the energy from passive sources like ocean waves, wind, and anthropogenic activity (previously considered to be background noise) to be used to image the Earth. Interferometry allows this 'noise' field to be converted into signals that look like seismograms from active sources, even though no such sources occurred. The resulting seismograms from such virtual (imagined) sources are used to image the real Earth structure. In only five years this has become a standard technique in surface wave tomography of the Earth's crust and upper mantle, and similar techniques are under development for the exploration industry. Indeed, in the seismological community this has been so successful that signals from earthquakes (the previous data source) are now often ignored - only the background energy field (previously considered to be noise) is used for subsurface imaging. A limiting problem exists with such methods, which has only been fully illuminated over the past two years. Theoretically, interferometry works when the noise field comes equally from all directions. This is never the case on Earth for either passive noise fields, or even when active 'bespoke' fields are used in the industrial setting, principally because the dominant form of energy propagation from sources on or near the Earth's surface is through so-called surface waves, waves that hug the Earth's outermost surface as they travel. Surface waves thus dominate the virtual seismograms to an extent that swamps all body wave information. For industrial exploration it is strictly necessary to use body waves. Since interferometry would open new doors in subsurface exploration, it is highly desirable to be able to alter the interferometric methods to be able to work within biased energy fields. Our research group has recently developed a method, called 'directional balancing', that can be integrated within wavefield interferometric methods to correct biases due to the energy field directionality (provisional patents filed; manuscript submitted for publication). This method promises to reduce approximately-horizontally propagating surface wave energy, while enhancing the more vertically-propagating body wave arrivals to a realistic level. The method requires that energy is recorded on an array of receivers (rather than only by a pair of receivers as in standard interferometry). In industrial seismics, arrays of receivers are always available since they form intrinsic components of the seismic acquisition and processing system. Hence, in principle directional balancing is directly applicable to industrial seismic data, using both passive and active sources of energy. This project will develop the directional balancing method to the point of industrial application, and apply it to real, industrial-scale, seismic data sets provided by the industrial partner. By enhancing the body wave arrivals relative to surface waves, these methods promise to make wavefield interferometry techniques applicable to industrial scale seismics, thereby opening new fields of research, development and creating new and exciting possibilities for subsurface exploration.

Our proposal builds on the successful start made by Cambridge Centre for Analysis (CCA), a current EPSRC Centre for Doctoral Training. We propose to develop further our activity in two important and rapidly evolving areas of analysis, namely mathematics of information and statistics of complex systems. Beginning with Newton, for whom the development of calculus and the mathematical understanding of bodies in motion were closely intertwined, the mathematics used to describe real phenomena consistently involves notions of continuity, rate of change, average value, and basic challenges such as the relationship between discrete and continuum objects. This is the domain of analysis, encompassing modelling by partial differential equations and by random processes, and the mathematical theory which guides effective computation for such models. The centrality of mathematical analysis in the relationship between mathematics and its applications has been acknowledged by successive International Reviews of Mathematics, as has the need to increase the capacity of UK PhD training in analysis. Mathematical Analysis and its Applications is an EPSRC Priority Area. Beyond the established and important uses of analysis in modelling physical phenomena, digital technology has created new areas where mathematical analysis, in guiding the extraction of knowledge from massive discrete systems, plays an essential role. These include the fields of high-dimensional statistics and the mathematics of information, including compressed sensing. In each of these, one is looking for a reliable means to interpret massive high-dimensional data. Already several CCA students are working in these areas. Big Data is one of the Eight Great Technologies championed by the Minister for Universities and Science. Statistics and Data to Knowledge are EPSRC Priority Areas. We propose a first year training programme based on our current successful model, now expanded by two further core courses, one in Statistics of Complex Systems and one in Mathematics of Information. These new courses will be paired with postgraduate level courses from the existing Cambridge Masters' (MASt), which students can use to consolidate their understanding. The core courses themselves are based on supervised student team assignments leading to student presentations. The other main components of the first year are research mini-projects (often the route to a PhD project) and an industry workshop. Years two to four are devoted mainly to the PhD thesis. First year training establishes a collaborative ethos in the cohort and, by mixing students with different prior skills, encourages cross-fertilization of ideas across the different threads of analysis. This is sustained in later years through a programme of seminars, workshops and training in transferable skills. The students appreciate that their collective understanding of a given problem using different skills will often exceed each individual's understanding. This makes cohort-based training especially valuable in analysis. We already expose all our students to the role of mathematics and the opportunities for mathematicians in industry and society, and we encourage first-hand engagement with applications through mini-projects, industrial seminars and study weeks, and, for some, PhD projects with industrial partners. The development of core skills and eventually the ability to generate new ideas is the hardest and crucial part of training as a research mathematician. This is necessarily our overriding task, in which we seek synergy and inspiration from user engagement. In the new CDT, our network of industrial connections will be further enhanced, along with our collaborations with Cambridge engineering colleagues, and our links with the Smith Institute for Industrial Mathematics.

Natural and industrial porous media is key to a wide variety of traditional and emerging engineering applications, including but not limited to oil and gas extraction from geological reservoirs, carbon capture and storage, geothermal reservoir engineering, soil sciences, groundwater remediation and protection, biological engineering, food processing, fuel cells, nano-technology, construction engineering, wood processing and printing. We are proposing the formation of a UK wide research network focussed on porous media flow which will sit at the interface between engineering, applied mathematics, applied probability and scientific computing. The overall aim of the network is to transfer techniques, models and scientific insights between engineering and mathematics, as well as promote mobility between academia and industry. Initially, the key areas of scientific research will include: large scale computational modelling, fundamental pore-scale physics, inverse problems and history matching, reservoir simulations, soil science and shallow ecosystems, theoretical biology and physiology, groundwater remediation, subsurface storage of greenhouse gases and nuclear waste, uncertainty quantification, homogenisation and multi-scale methods, visualization, numerical analysis and random field modelling.