Building correct communicating systems (i.e. concurrent and distributed systems) is a hard task. Such systems often present non-deterministic behaviours that are hard to reproduce. This means that, whenever there is a bug in one of such systems, fixing it is a time consuming and costly task. Furthermore, distributed systems are nowadays widespread in our society, including key sectors such as banking or E-healthcare. Thus, bugs in communicating systems can be very damaging, threatening large economic costs, and the safety and security of the industries that rely on them. To guarantee the correctness of communicating systems, many tools and theories have been developed. Session types are among the most influential theories for verifying the absence of concurrency bugs in communicating systems. Session types can guarantee that process implementations only follow the specified structured sequence of actions (send/receive). These specifications effectively represent communication protocols, and session type theories provide techniques to verify that they are absent of concurrency bugs. Among the most influential extensions of session types is the theory of Multiparty Session Types (MPST), which enables the modelling of protocols among an arbitrary number of participants, and has been integrated with many mainstream languages. However, most of the tools based on session types are not following the exact theories that are presented in the scientific publications, since they also need to take into account engineering issues. Furthermore, most of the session type theories are proven correct using complex pen-and-paper soundness proofs. Such proofs can contain errors, and the literature shows examples of these cases. This is a big threat to the validity of large bodies of work. Proof assistants are tools that can guarantee the correctness of these soundness arguments. Encoding languages, tools, and theories in a proof assistant is called mechanisation, and it has led to large influential developments in computer science, such as CompCert, a certified C compiler that has proven to present significantly less bugs than other comparable compilers, and that is currently being used in the context of critical systems. But proof mechanisation is significantly hard, and not much work exists on mechanising MPST. One of the main hurdles in mechanising MPST is the notion of process equivalence, or bisimilarity. Informally, two processes are bisimilar if they match each others actions, according to their Labelled-state Transition System (LTS) semantics. One main techniques for proving bisimilarity is known as the bisimulation proof method, which relies on finding a relation between two processes that guarantees that they will match each other's moves according to their LTS. These proofs are widespread in concurrency theory, and are key to successful mechanisations of MPST. We need a way to simplify the mechanisation of LTS semantics, and bisimilarity proofs. We will study common bisimulation proof techniques and algorithms, and find and implement a suitable candidate for mechanisation. Our main goal is to automate as much as possible of the mechanisation of the LTS semantics. One of the key challenges is to find the suitable definitions: pen-and-paper proofs can often overlook details that are important in a mechanisation. For example, termination is often informally justified in pen-and-paper proofs, but the specific details are key to a successful mechanisation. We will mechanise a generic framework for LTS semantics and bisimilarity in the Coq proof assistant, and use it in two case studies from the project partners. These case studies will serve both as a way to evaluate success, and as main driving elements of this project. Finally, we will study the extraction of certified implementations within our framework, thus contributing to increase the safety and reliability of distributed systems.

The Centre for Doctoral Training MAC-MIGS will provide advanced training in the formulation, analysis, and implementation of state-of-the-art mathematical and computational models. The vision for the training offered is that effective modern modelling must integrate data with laws framed in explicit, rigorous mathematical terms. The CDT will offer 76 PhD students an intensive 4-year training and research programme that equips them with the skills needed to tackle the challenges of data-intensive modelling. The new generation of successful modelling experts will be able to develop and analyse mathematical models, translate them into efficient computer codes that make best use of available data, interpret the results, and communicate throughout the process with users in industry, commerce and government. Mathematical and computational models are at the heart of 21st-century technology: they underpin science, medicine and, increasingly, social sciences, and impact many sectors of the economy including high-value manufacturing, healthcare, energy, physical infrastructure and national planning. When combined with the enormous computing power and volume of data now available, these models provide unmatched predictive tools which capture systematically the experimental and observational evidence available. Because they are based on sound deductive principles, they are also the only effective tool in many problems where data is either sparse or, as is often the case, acquired in conditions that differ from the relevant real-world scenarios. Developing and exploiting these models requires a broad range of skills - from abstract mathematics to computing and data science - combined with expertise in application areas. MAC-MIGS will equip its students with these skills through a broad programme that cuts across disciplinary boundaries to include mathematical analysis - pure, applied, numerical and stochastic - data-science and statistics techniques and the domain-specific advanced knowledge necessary for cutting-edge applications. MAC-MIGS students will join the broader Maxwell Institute Graduate School in its brand-new base located in central Edinburgh. They will benefit from (i) dedicated academic training in subjects that include mathematical analysis, computational mathematics, multi-scale modelling, model reduction, Bayesian inference, uncertainty quantification, inverse problems and data assimilation, and machine learning; (ii) extensive experience of collaborative and interdisciplinary work through projects, modelling camps, industrial sandpits and internships; (iii) outstanding early-career training, with a strong focus on entrepreneurship; and (iv) a dynamic and forward-looking community of mathematicians and scientists, sharing strong values of collaboration, respect, and social and scientific responsibility. The students will integrate a vibrant research environment, closely interacting with some 80 MAC-MIGS academics comprised of mathematicians from the universities of Edinburgh and Heriot-Watt as well as computer scientists, engineers, physicists and chemists providing their own disciplinary expertise. Students will benefit from MAC-MIGS's diverse network of more than 30 industrial and agency partners spanning a broad spectrum of application areas: energy, engineering design, finance, computer technology, healthcare and the environment. These partners will provide internships, development programmes and research projects, and help maximise the impact of our students' work. Our network of academic partners representing ten leading institutions in the US and Europe, will further provide opportunities for collaborations and research visits.

The aim of this fellowship is to answer a key research question for power systems engineering: "As the UK and other countries move towards transport electrification, how can potentially millions of electric vehicles be successfully integrated into power system operations?" Electric vehicles have become increasingly cost competitive, due to the cost of lithium-ion battery packs falling by approximately 77% over the last 6 years. The UK has over 100,000 electric vehicles, but it is estimated 26 million will be needed to meet 2050 emissions targets. The UK government is strongly supporting this, announcing a ban on the sale of diesel and petrol cars and vans after 2040, and the Faraday Challenge, £246 million towards electric vehicle battery development. If electric vehicle charging is left uncoordinated, the large-scale adoption of electric vehicles is expected to cause significant power system challenges. Peak demand is expected to increase on the order of 20GW (approximately a 40% increase), necessitating new power plants and large-scale transmission infrastructure upgrades. A significant impact is also expected at the local distribution network level. The My Electric Avenue project identified that without smart charging, transport electrification will necessitate new investment to reinforce 32% of UK low voltage distribution network feeders (312,000 feeders). This motivates the need for smart charging - coordinated scheduling of the charging times and powers of electric vehicles. However, existing strategies do not facilitate or incentivise this coordination, particularly at the local distribution network level. Top-down regimes that directly curtail charging impose an external cost on electric vehicle owners and manufacturers, and will slow adoption. Mechanisms that instead incentivise coordination are a promising approach, but require careful engineering design, since they influence power system operation in real time. Through this fellowship, a networked market platform will be designed which can incentivise aggregate and localised coordination between millions of electric vehicles, while managing local power network voltage and thermal constraints in real time. This will be achieved by combining recent advances in multi-agent control, power engineering and networked matching market theory, to design new algorithms suitable for large-scale implementation. The project is supported by two industry partners, EDF Energy, the second largest electricity supplier in the UK with over 5 million customers, and Upside Energy, a UK virtual demand side response aggregator. The proposed market platform has the potential to provide significant value by alleviating the need for generation and transmission infrastructure investments, increasing network efficiency and increasing energy security.

In numerous research areas across engineering and the applied sciences there are nonlinear structures or systems for which there are inadequate or multiple competing mathematical models. This is often caused by a poor understanding of the physics at the appropriate scale. Two examples of this are the study of shimmy oscillations in aircraft landing gear, and the onset of chatter during high-speed machining. In these areas experimental investigations are of fundamental importance in order to resolve the details that the models cannot. However, systematically investigating nonlinear dynamics in an experiment is fraught with difficulties due to the potential for sudden changes in the dynamics as system parameters are varied; moreover, the changes may be qualitative as well as quantitative. For example, for particular choice of parameters a system may have a single stable steady-state whereas for another choice it may become bistable. Of particular interest are the boundaries between different types of qualitative behaviour (so-called bifurcations). This proposal seeks to make the determination of these boundaries in a physical experiment a normal and routine task by leveraging ideas from control theory and dynamical systems, and so reveal previously unseen dynamical phenomena. Control-based continuation is a new method, developed in Bristol in 2008, for systematically characterising the qualitative behaviour of a physical nonlinear experiment. Its potential has been recognised by experimenters in other institutions, who are now applying it to their own systems. The key idea is that dynamical features of the system, such as stability changes or the onset of mixed-mode oscillations, can be found and tracked directly in a physical experiment using a combination of feedback control and numerical path following techniques. Thus a `map' or bifurcation diagram showing different regions of qualitative behaviour can be traced out. To enable the widespread uptake of control-based continuation three key objectives must be satisfied. (1) It must be possible to determine the local linearisation of a steady-state, thus providing additional dynamical information and a means to perform on-line controller design/adaptation. (2) The underpinning numerical methods must be made fast and robust. (3) The scalability of control-based continuation to multi-degree-of-freedom systems must be demonstrated. This proposal seeks to address all three points.

The proposed EPSRC CDT in the Science and Applications of Graphene and Related Nanomaterials will respond to the UK need to train specialists with the skills to manipulate new strictly two-dimensional (2D) materials, in particular graphene, and work effectively across the necessary interdisciplinary boundaries. Graphene has been dubbed a miracle material due to the unique combination of superior electronic, mechanical, optical, chemical and biocompatible properties suitable for a large number of realistic applications. The potential of other 2D materials (e.g. boron nitride, transition metal and gallium dichalcogenides) has become clear more recently and already led to developing 'materials on demand'. The proposed CDT will build on the world-leading research in graphene and other 2D nanomaterials at the Universities of Manchester (UoM) and Lancaster (LU). In the last few years this research has undergone huge expansion from fundamental physics into chemistry, materials science, characterization, engineering, and life sciences. The importance of developing graphene-based technology has been recognized by recent large-scale investments from UK and European governments, including the establishment of the National Graphene Institute (NGI) at UoM and the award of 'Graphene Flagship' funding by the European Commission within the framework of the Future and Emerging Technologies (Euro1 billion over the next 10 years), aiming to support UK and European industries.Tailored training of young researchers in these areas has now become urgent as numerous companies and spin-offs specializing in electronics, energy storage, composites, sensors, displays, packaging and separation techniques have joined the race and are investing heavily in development of graphene-based technologies. Given these developments, it is of national importance that we establish a CDT that will train the next generation of scientists and engineers who will able to realise the huge potential of graphene and related 2D materials, driving innovation in the UK, Europe and beyond. The CDT will work with industrial partners to translate the results of academic research into real-world applications in the framework of the NGI and support the highly successful research base at UoM and LU. The new CDT will build directly on the structures and training framework developed for the highly successful North-West Nanoscience DTC (NOWNANO). The central achievement of NOWNANO has been creating a wide ranging interdisciplinary PhD programme, educating a new type of specialist capable of thinking and working across traditional discipline boundaries. The close involvement of the medical/life sciences with the physical sciences was another prominent and successful feature of NOWNANO and one we will continue in the new CDT. In addition to interdisciplinarity, an important feature of the new CDT will be the engagement with a broad network of users in industry and society, nationally and internationally. The students will start their 4-year PhD with a rigorous, bespoke 6-month programme of taught and assessed courses covering a broad range of nanoscience and nanotechnology, extending beyond graphene to other nanomaterials and their applications. This will be followed by challenging, interdisciplinary research projects and a programme of CDT-wide events (annual conferences, regular seminars, training in transferable skills, commercialization training, outreach activities). International experience will be provided by visiting academics and secondments to overseas partners. Training in knowledge transfer will be a prominent feature of the proposed programme, including a bespoke course 'Innovation and Commercialisation of Research' to which our many industrial partners will contribute, and industrial experience in the form of 3 to 6 months secondments that each CDT student will undertake in the course of their PhD.