MathSys addresses two of EPSRC's CDT priority areas in Mathematical Sciences: "Mathematics of Highly Connected Real-World Systems" and "New Mathematics in Biology and Medicine". We will train the next generation of skilled applied mathematical researchers to use and develop cutting-edge techniques enabling them to address a range of challenges faced by science, industry and modern society. Our Centre for Doctoral Training will build on the experience and successes of the Complexity Science DTC at Warwick, while refining the scope of problems addressed. It will provide a supportive and stimulating environment for the students in which the common mathematical challenges underpinning problems from a variety of disciplines can be tackled. The need for mathematically skilled researchers, trained in an interdisciplinary environment, has never been greater and is viewed as a major barrier in both industry and government. This is supported by quotes from reports and business leaders: "Systems research needs more potential future leaders, both in academia and industry" (EPSRC workshop on Systems science towards Engineering, Feb 2011); Andrew Haldane (Bank of England, 2012) said "The financial crisis has taught us the importance of modelling and regulating finance as a complex, adaptive system. That will require skills currently rare or missing in the regulatory community - including, importantly, in the area of complexity science"; Paul Matthews (GlaxoSmithKline) stated "Scientists trained in statistical and computational approaches who have a sophisticated understanding of biologically relevant models are in short supply. They will be major contributors in the task of translating insights on human biology and disease into treatments and cures." Our CDT will address this need by training PhD students in the development and innovation of mathematics in the context of real-world systems and will operate in close collaboration with stakeholders from outside academia who will provide motivating problems and real-world experience. Common mathematical themes will include statistical behaviour of complex systems, tipping points, novel methods in control and resilience, hierarchical aggregation methods, model selection and sufficiency, implications of network structure, response to aperiodic forcing and shocks, and methods for handling complex data. Applications will be driven by local and external partner expertise in Epidemiology, Systems Biology, Crop Science, Healthcare, Operational Research, Systems Engineering, Network Science, Financial Regulation, Data Analysis and Social Behaviour. We believe that the merging of real-world applications with development of novel mathematics will have great synergy; applications will motivate and drive mathematical advances while novel mathematics will allow students to solve challenging real-world problems. The doctoral training programme will follow a 1+3 year MSc+PhD model that has proved successful in the Complexity Science DTC. The first year will consist of six months of taught training, followed by 3-month group research projects on problems set by external partners and a 3-month individual research project, leading to an MSc qualification. This preparation will enable the students to make rapid progress tackling their 3-year PhD research project, under the guidance of one mathematical and one application-oriented supervisor, alongside general skills training and group research projects. We have over 50 suitable supervisors with relevant mathematical expertise, all enthusiastic to contribute; they will be supported by a similar number of application-oriented supervisors from across campus and from external partners. The CDT seeks the equivalent of 7 full studentships per year from EPSRC and has commitment from non-RCUK sources for the equivalent of 3 full studentships per year.

In view of the rapid increase in demand for mobile data services, next generation wireless access networks will have to provide greatly increased capacity density, up to 10 Gbps per square kilometre. This will require a much larger density of very small, cheap and energy-efficient base stations, and will place increasing demand on the bandwidth and energy efficiency of the network, and especially the backhaul network. Recent work on network MIMO, or coordinated multipoint (CoMP) has shown that by ensuring base stations cooperate to serve users, especially those close to cell edge, rather than interferring with one another, inter-user interference can be effectively eliminated, greatly increasing the efficiency of the network, in terms of both spectrum and energy. However this tends to greatly increase the backhaul load. This work proposes a form of wireless network coding, called network coded modulation, as an alternative to conventional CoMP. This also enables base station cooperation, but instead of sending multiple separate information flows to each base station, flows are combined using network coding, which in principle allows cooperation with no increase in backhaul load compared to non-cooperative transmission, while gaining very similar advantages to CoMP in terms of bandwidth and energy efficiency. The objective of the proposed work is to establish the practical feasibility of this approach, and evaluate its benefits, as applied to next generation wireless access networks. To this end it will develop practical signalling schemes, network coordination and management protocols, and, with the help of industrial collaborators, will ensure compatibility with developing wireless standards.

The achievements of modern research and their rapid progress from theory to application are increasingly underpinned by computation. Computational approaches are often hailed as a new third pillar of science - in addition to empirical and theoretical work. While its breadth makes computation almost as ubiquitous as mathematics as a key tool in science and engineering, it is a much younger discipline and stands to benefit enormously from building increased capacity and increased efforts towards integration, standardization, and professionalism. The development of new ideas and techniques in computing is extremely rapid, the progress enabled by these breakthroughs is enormous, and their impact on society is substantial: modern technologies ranging from the Airbus 380, MRI scans and smartphone CPUs could not have been developed without computer simulation; progress on major scientific questions from climate change to astronomy are driven by the results from computational models; major investment decisions are underwritten by computational modelling. Furthermore, simulation modelling is emerging as a key tool within domains experiencing a data revolution such as biomedicine and finance. This progress has been enabled through the rapid increase of computational power, and was based in the past on an increased rate at which computing instructions in the processor can be carried out. However, this clock rate cannot be increased much further and in recent computational architectures (such as GPU, Intel Phi) additional computational power is now provided through having (of the order of) hundreds of computational cores in the same unit. This opens up potential for new order of magnitude performance improvements but requires additional specialist training in parallel programming and computational methods to be able to tap into and exploit this opportunity. Computational advances are enabled by new hardware, and innovations in algorithms, numerical methods and simulation techniques, and application of best practice in scientific computational modelling. The most effective progress and highest impact can be obtained by combining, linking and simultaneously exploiting step changes in hardware, software, methods and skills. However, good computational science training is scarce, especially at post-graduate level. The Centre for Doctoral Training in Next Generation Computational Modelling will develop 55+ graduate students to address this skills gap. Trained as future leaders in Computational Modelling, they will form the core of a community of computational modellers crossing disciplinary boundaries, constantly working to transfer the latest computational advances to related fields. By tackling cutting-edge research from fields such as Computational Engineering, Advanced Materials, Autonomous Systems and Health, whilst communicating their advances and working together with a world-leading group of academic and industrial computational modellers, the students will be perfectly equipped to drive advanced computing over the coming decades.

The Internet is expanding towards mobile wireless connectivity rapidly. However, to enable this for increasing numbers of users and connected devices, and increasingly bandwidth-, processing power- and energy-hungry applications, will require a transformation in the way in which current mobile and wireless networks perform. Shorter wireless distances (small cells, picocells, femtocells) and different network types for the connection (WiFi, 3G, 4G, 5G) depending on the availability and suitability for different applications, is a process that is already happening and expected to continue. This will manifest itself with simpler remote radio heads providing coverage in otherwise difficult to penetrate locations (and the main processing functions gathered together in a centralised pool of base station baseband units), and with the appearance of new wireless standards. NIRVANA takes this evolution and proposes a transformative step: the incorporation of fast, hardware-based, network monitoring, and intelligence (using the monitoring/gathered information) close to the pool of base stations. The proximity of the intelligence enables low-overhead control of a range of operational functions, which allow users to be moved from one connection type to another, according to their application and the load on the network, and to match the network's resources precisely to user needs. It allows energy efficiency to be optimised throughout the network and in the mobile device, too. The latter is augmented by locating the computing resources for a "mobile cloud" near the base station pool. Some processing is offloaded to the mobile cloud instead of being done on the mobile, and even some mobile-to- mobile communication may be done within this cloud - saving the mobile device (and the network) energy that would have been used in radio transmissions. Finally, among the new wireless connection types to be investigated, millimetre-wave communications, using the most up-to-date releases of the wireless local area network standard (802.11ad/j), will be fashioned into a device-to-device mesh network, for mobile distributed caching, which will be shown to further enhance the capacity of the network and its energy efficiency.

As recently discussed by the Wall Street Journal, the remarkable success of the internet may be attributed to the tremendous capacity of unseen underground and undersea optical fibre cables and the technologies associated with them. Indeed, the initial surge in web usage in the mid 1990s coincides with the commissioning of the first optically amplified transatlantic cable network, TAT12/13 allowing ready access to information otherwise inaccessible. Tremendous progress has been made since then, with the introduction of wavelength division multiplexing, where multiple colours of light are used to establish independent connections through the same fibre and coherent detection, the optical analogue of an advanced radio receiver able to detect both amplitude and frequency (or phase) modulation simultaneously enabling the information carrying capacity to be doubled and the required signal power to be reduced. To manage the costs, communication networks typically aggregate connections between many users onto a single communications link within the core of the network, avoiding the tremendous costs associated with dedicated links for all users across vast distances. Typically the trade of between cost and reliability has resulted in traffic from several thousand customers being aggregated onto a single fibre resulting in bit rates in the region of 100 Gbit/s per wavelength channel to support broadband connections of around 10 Mbit/s. However, this has resulted in intensities in optical fibres that are a million times greater than sunlight at the surface of the Earth's atmosphere and so the signal is significantly distorted by nonlinearly (a similar effect to overdriving load speakers). This distortion limits the maximum amount of information which may be transmitted across and optical fibre link, and unless combated, the nonlinear response will result in a capacity crunch, limiting access to the internet to today's levels. This project aims to allow the continued increase of the bandwidth of these fibre networks underpinning modern communications, including 17.6 million UK mobile internet connections and 70% penetration of home broadband connections. To increase capacity we will maximise spectral use, by adapting techniques found in mobile phones for use in fibre networks, resolving the significant issues associated with processing data with 1,000,000 times greater bandwidth using a balance of digital and analogue electronic and optical processing. This will reduce cost, size and power consumption associated with producing Tb/s capacities per wavelength. Critically, the project will develop techniques to understand and mitigate the nonlinear signal distortions. Nonlinear distortions occur within a channel, between channels and between each channels and noise originating in the optical amplifiers. By transforming the signal mid way along the link, we will exploit the nonlinear response of the second half of the fibre link to cancel the nonlinear distortion of the first to minimise the impact of nonlinear distortion associated with the channels themselves, and optimise the configuration of the system to minimise the nonlinear interaction with the noise, resulting in orders of magnitude increases in the maximum capacity of the optical fibre system.