Cooperative game theory is a branch of game theory that offers a conceptually simple and intuitive mathematical framework to model collaborative settings involving multiple decision makers (players). Solutions of cooperative games offer different ways to share the profit or cost among the players in a way that ensures the fairness and stability of the collaboration, while considering the possibility that any subgroup of players has the option to form their own coalition. The focus of this project is on the most generic class of cooperative games - the integer maximisation games. These games arise in settings where the players in each coalition need to solve an integer maximisation problem to achieve the best interests of their coalition. This proposed research addresses a fundamental question of how to distribute payoff under a new paradigm with the presence of uncertainty and in the context of reasonably large games. Often, formulating a real-life application as a cooperative game, where relevant, is not a difficult task. The part that discourages the use of cooperative game theory is the difficulty in undertaking numerical computation of the solutions due to their combinatorial structures. This is particularly true in integer maximisation games where the set of inputs of the problem, i.e., the value that each coalition can create, involves solving an exponentially large number of integer linear programs. The first part of the proposed research provides efficient algorithms for payoff allocation in reasonably large integer maximisation games. In addition, an open-source software package for computing these solutions and showcase real-world applications is made available. This promises to extend the impact to wide groups of practitioners and academics who want to apply cooperative game theory to profit-/cost-sharing applications. The proposed project also aims to study cooperative games with uncertain payoffs. While uncertainty is a natural part of most decision-making problems, the issue has been largely ignored in the literature of cooperative game theory and there is currently no rigorous framework for handling these. We propose a new framework where fundamental concepts such as stability and fairness are redefined in the face of uncertainty.

Cooperative game theory is a branch of game theory that offers a conceptually simple and intuitive mathematical framework to model collaborative settings involving multiple decision makers (players). Solutions of cooperative games offer different ways to share the profit or cost among the players in a way that ensures the fairness and stability of the collaboration, while considering the possibility that any subgroup of players has the option to form their own coalition. The focus of this project is on the most generic class of cooperative games - the integer maximisation games. These games arise in settings where the players in each coalition need to solve an integer maximisation problem to achieve the best interests of their coalition. This proposed research addresses a fundamental question of how to distribute payoff under a new paradigm with the presence of uncertainty and in the context of reasonably large games. Often, formulating a real-life application as a cooperative game, where relevant, is not a difficult task. The part that discourages the use of cooperative game theory is the difficulty in undertaking numerical computation of the solutions due to their combinatorial structures. This is particularly true in integer maximisation games where the set of inputs of the problem, i.e., the value that each coalition can create, involves solving an exponentially large number of integer linear programs. The first part of the proposed research provides efficient algorithms for payoff allocation in reasonably large integer maximisation games. In addition, an open-source software package for computing these solutions and showcase real-world applications is made available. This promises to extend the impact to wide groups of practitioners and academics who want to apply cooperative game theory to profit-/cost-sharing applications. The proposed project also aims to study cooperative games with uncertain payoffs. While uncertainty is a natural part of most decision-making problems, the issue has been largely ignored in the literature of cooperative game theory and there is currently no rigorous framework for handling these. We propose a new framework where fundamental concepts such as stability and fairness are redefined in the face of uncertainty.

Microscopy is vitally important across a wide range of scientific and technological fields. However, despite the multitude of techniques available, there are many materials that are inaccessible to conventional tools: conventional light microscopy is limited to around micron length-scales; electron microscopy often leads to sample damage or charging; and scanning probe methods (such as atomic force microscopy) are limited to small areas on predominantly flat surfaces. Such problems are particularly acute in the case of delicate materials: for example, organic electronic thin films that are damaged by high-energy electrons, or fine polymer structures, where charging obscures the image. The main aim of this proposal is to develop a revolutionary new technique - Scanning Helium Microscopy (SHeM) - that generates images using a low-energy beam of neutral atoms and so obviates the above problems. The new technique has great potential, but it is essential to improve its spatial resolution and to make it possible for non-specialists to perform helium microscopy easily. The applicants are ideally positioned to lead these developments, by exploiting the technology they developed. The research programme is designed to firmly establish helium microscopy as a cutting-edge research tool. The main themes are: 1. To develop a new high resolution microscope that will achieve nanoscale resolution and an imaging rate comparable with scanning probe techniques. The new microscope will make possible a wide range of new experiments. It will be suitable for use by non-specialists and made available to users through a facility-like access model. 2. To establish and promote the nascent field of helium-microscopy by performing a broad range of collaborative experiments, spanning multiple applications. These will establish applicability of the technique, and help to develop the imaging modalities required to optimise image contrast arises from a variety of atom-surface scattering mechanisms. 3. To develop advanced image collection and reconstruction methods, including making use of the compressibility of natural images, to minimise acquisition time and maximise the information content that can be obtained during any given experimental period. By applying such cutting-edge algorithms to a low-signal scanned probe microscopy for the first time, we anticipate the impact of this theme extending far beyond the present project. The programme is inherently collaborative: the new microscope will be developed and constructed at the Cavendish Laboratory (Physics, Cambridge), supported by nano-fabrication of key components in the Materials Physics group, Glasgow. Researchers in Applied Maths (Cambridge) will develop accelerated imaging methods, while a further series of international collaborators have agreed to provide samples, time and expertise, to explore helium imaging in a diverse range of fields. Microscopy with helium will have impact across a wide range of scientific and technological fields, wherever it is difficult to image delicate samples. Applications that are already foreseen include semiconductor devices, composite materials, organic films and the high aspect-ratio structures used in MEMS devices; but the scope for this new microscopy has yet to be fully explored. Success in the project will lead to the commercialisation of a new imaging technology, the impact of which the UK is uniquely positioned to exploit.

The continuous demand for device miniaturization poses technological and economic barriers that cannot be answered by current fabrication techniques. This proposal is aimed at the development of a simple technique for the fabrication of crossbar electrode arrays for non-volatile memory devices based on a modulated block copolymer/nanoparticle (BCP/NP) assembly approach, where the ability to control the interfacial interactions between the NPs and the BCP domains under an electric field is crucial for obtaining the desired structure. Through a tight collaboration between experimental chemists, theoreticians, and an electrical engineer we intend to unravel the fundamental behavior ofBCP/NP assembly under the influence of a directing electric field, and then to utilize the structures formed for the creation of an ultrahigh-density, multi-component memory device.

Fluid movement driven by a density difference is very common. When a freezer is opened, or a window on a winter's day (a ventilation flow), you may have noticed that the dense, cold air rushes across your feet. This effect can be felt even if you are on the other side of the room, the cold air warming a little as it mixes with the warmer air above, but remaining sufficiently cool and distinct as it flows like a flood across the floor. These are part of a very broad family of fluid flows present across our homes, industries, and the wider environment, known as gravity-currents. Ventilation flows are important to understand for the spread of pathogens and disease, and cold-fronts are essentially the same but on the scale of 100-1000km. In industry, accidental spills of hazardous gas must be planned for, and suitable defences put in place. A very dangerous subset of gravity-currents are particle-driven currents, the suspended particle load providing the driving density and facilitating immense destructive power. For example, powder-snow avalanches are a hazard in mountainous regions, easily burying people and buildings. Pyroclastic density currents, searing hot clouds of ash released by volcanos and flowing out across the ground, famously buried Pompeii, leaving a city of people entombed in volcanic rock. Massive submarine turbidity-currents, >1000km long and moving at up to 10m/s, carry nutrients and carbon into the deep ocean, and have destroyed numerous cables and pipes carrying internet data or energy. Smaller (though still substantial) turbidity-currents will pose an increasing hazard to the UK as we develop deep-marine wind power, which must be connected back to shore by cables. The feasibility of these and other developments rely on our ability to predict and mitigate natural hazards. The front the current pushes aside the ambient fluid, and it is the dynamics here which determine the rate of advance of the current. In addition, this region is a principal source of mixing, and for some currents it is also a region in which there is intense erosion of the bed. As the current mixes with the fluid around it, it becomes more dilute, and the current becomes bigger while simultaneously having a reduced driving density. Conversely, as it erodes the bed the driving density increases. Thus, the front exerts a very strong control on the advance of the current, and the mixing and erosional processes are a critical part of this. However, to date these processes have not been included in the mathematical models that are designed to predict these currents, which has limited their applicability to flows over short distances so that the mixing does not substantially affect on the overall density. Additionally, the front of the current is the most dangerous part: the same processes that enable the rapid erosion of the bed can facilitate immense destructive power. In this fundamental scientific study, I will develop novel mathematical models that capture the dynamics of the front of a gravity-current, including the mixing and erosional processes. First, experimental work using newly developed techniques will yield data of unprecedented quality for a cool, temperature driven current, measuring the details of the vortices and mixing in both the head of the current and throughout. Additional experiments will focus on capturing the details of the erosional processes in sediment-driven currents. Informed by these measurements, I will capture the vital aspects of the dynamics of the head within a new mathematical model, for the first time including the mixing and erosional processes. Finally, the model of the head will be combined with a model for the rest of the current, which I developed previously, to give a complete model that can predict the motion of the current. This urgently required project represents a substantial leap-forward in our understanding and predictive power for this important and dangerous class of flows.