The question at the heart of this proposal is one of the most ancient problems in science and engineering: How densely can a volume be filled with objects of a particular shape? Such particle packings are of utmost importance for all industries involved in granular processing and appear in a broad range of current scientific and engineering fields such as self-assembly of nano-particles, liquid crystals, glassy materials, and bio-materials. In fact, understanding the macroscopic behaviour of matter from the properties of its individual constituents is one of the central problems in materials science. Packings of hard objects are one of the simplest matter states, but, nevertheless, pose considerable theoretical challenges. Finding the densest packing is an outstanding mathematical problem that originated with Kepler's famous conjecture on regular cannonball piles. Much less is known about non-spherical shapes, despite the fact that all shapes in nature deviate from the ideal sphere. Recently, it has been conjectured that the sphere - the shape with the highest symmetry - is in fact the worst packing object among all convex shapes in both disordered and regular arrangements. This implies that packing densities can be optimized by searching in the space of object shapes. A deeper understanding of this optimization problem would lead to immediate benefits in many industrial sectors, especially pharmaceutical and chemical industries, which rely on storage and transport of large amounts of granular material. Particle packings are in general athermal and thus represent non-equilibrium states of matter. As a consequence, the well established framework of statistical mechanics, which is able to successfully predict the phases and macroscopic properties of many-particle systems at equilibrium, does not apply. Moreover, assemblies of non-spherical particles are characterized by strong orientational correlations, in addition to positional ones, which has thus far prohibited any systematic theoretical investigation. Searches for the optimal packing of non-spherical shapes have focused instead on empirical studies on a case-by-case basis using experiments or computer simulations. These studies suffer from the generic shortcoming that the final packing state is strongly protocol dependent leading to a large variance of obtained packing densities even for the same shape. Novel theoretical tools are therefore needed. The overall aim of this proposal is to provide a general framework to predict the density of packings from the shape of the particles and to understand the organization principles of these packings. To do this, we will generalize our recently developed approach based on a coarse-grained volume function. This will allow us to address the problem of optimizing packing fractions in industry relevant scenarios and to explore novel states of matter due to anisotropic building blocks. The proposal thus has far-reaching consequences both on the practical problem on how to efficiently store granular material as well as on our fundamental understanding of matter away from equilibrium.

Semiconductor structures containing different materials are grown as thin film multilayers by techniques such as molecular beam epitaxy (MBE). MBE produces layers with excellent control of thickness but is limited to total thicknesses of just a few microns. In addition, growth takes place on a substrate, which is a highly crystalline template of a material such as gallium arsenide. After growth, the thin film layer remains bonded to the substrate. However, if one of the layers deposited is a so-called sacrificial layer soluble in a solvent (such as a weak acid) then all the layers deposited on top of it can be removed from the substrate. This process is called epitaxial lift off (ELO) and is advantageous in applications where the substrate is either not required or even hinders the operation of the device. Often using ELO means that the substrate can be recycled, which can reduce operating costs. An additional use for ELO is that the layers can be assembled into complex structures with many different types of materials. ELO layers can be transferred to intermediate flexible plastic substrates and patterned before assembly, so very complex structures can be produced. II-VI semiconductors are materials with a number of very useful properties, for example bandgaps ranging from 0 to 5eV. Other II-VI semiconductors have useful magnetic properties, for example some (e.g. CrS) are ferromagnets and others (e.g. MnS) are antiferromagnets. At Heriot-Watt University (HWU), we developed ELO for II-VI compounds using MgS sacrificial layers. The original method could only be used on small sample sizes (3mm square) but demonstrated many useful applications. Within the last few months we have developed a number of breakthroughs in II-VI ELO which show it has much more potential. In particular, we can remove pieces several square cm in size using a flexible plastic carrier. An additional very useful property is that when two ELO layers touch they will combine together, or stack, with the adhesion between layers so strong that they cannot be separated without breaking them. This proposal aims to develop this technology in 3 ways. First, we will show that ELO is easily extended to whole semiconductor wafers, and ELO layers can be transferred on flexible plastic carriers and patterned into small components. The components can be transferred again (stamped) to a final destination. All of this will be done with high (~100%) yield. Second, we will demonstrate the advantages of II-VI ELO by assembling 5 different demonstrator devices requested by our colleagues at HWU. We will supply these for evaluation as part of their own on-going research programmes. The devices include two types of sensors (temperature, and electric or magnetic fields), an optical diode, which only allows light propagation in one direction, a frequency doubler and a photonic bandgap structure. These structures are very difficult to produce by normal thin film growth techniques, but are easily produced by stacking ELO layers. The final strand of the programme develops the potential of ELO in different ways. The ability to move electrons or holes between ELO and adjacent layers would increase the number of applications: for example allowing us in future to develop photovoltaics or detectors. We will measure the electrical transport properties across ELO junctions between ZnSe and different materials and if possible modify them with different surface treatments. One surface treatment developed at HWU protects the II-VI layer surface after growth against contamination. At HWU it has worked for several months. We aim to show that it can be used to transport HWU ELO layers to City College, New York and show that it is possible to combine materials which are not available in the same MBE system and make ELO available to other groups.

One of the most amazing results of twentieth century mathematics was the discovery by Alonzo Church and Alan Turing that there are problems in mathematics which cannot be solved, in the sense that there is no algorithm to solve them. This means that no matter how powerful a computer you use to help you, there are some mathematical problems that you will still not be able to solve. If the problem that cannot be solved is in the form of a "yes" or "no" question, then we call it an undecidable problem. On the other hand, if it can be solved, then we say that it is a decidable problem. There are many decision problems that arise naturally in algebra. Important examples include the word problem, which asks us to decide whether two different algebraic expressions are equal to each other, and the membership problem, which asks us to decide whether one element in an algebraic structure can be expressed in terms of another collection of elements. Being able to solve problems like these is important when studying infinite algebraic structures. The main topic of this project is to investigate a range of decision problems like these for three classes of algebraic objects called groups, monoids and inverse semigroups. These three classes arise naturally in the study of symmetry and partial symmetry in mathematics. An important tool for defining infinite groups, monoids and inverse semigroups, is given by the theory of presentations in generators and relations. The idea is that the elements of the group or monoid are represented by strings of letters, called words. We are also given a set of defining relations, which are rules telling us that certain pairs of words are equal to each other. Two words are then equal if one can be transformed into the other by applying the relations. For example, if we use the letters x and y, and we have a single defining relation xy=yx, then the words xyx and yxx are equal since xyx = (xy)x = (yx)x = yxx. On the other hand, the words xy and yy are not equal. The problem of determining whether or not two words are equal to each other is the word problem mentioned above. When we define a monoid or group using a presentation, by increasing the number of relations we can increase the complexity of the monoid or group that we define. If there are no relations these are called free monoids and groups, and because of their simple structure several natural decision problems, like the word problem, can be seen to be decidable in these cases. In contrast, it is known that there are monoids, groups, and inverse semigroups which are defined by finitely many generators and relations, but have undecidable word problem. The situation is the similar for the many other decision problems arising in algebra. It is natural to ask whether groups or monoids which are close to being free, in some sense, will have good algorithmic properties. An important positive result of this kind for groups is Magnus's theorem which shows that groups defined by a single defining relation all have decidable word problem. On the other hand, it was recently discovered that there are inverse monoids defined by a single defining relation that have undecidable word problem. However, it remains an important longstanding open problem whether the word problem is decidable for one-relator monoids. There are many fascinating open problems like this one which ask fundamental questions about where the boundary between decidability and undecidability lies for finitely presented groups, monoids and inverse semigroups. In this project we will explore a range of interrelated problems of this kind. This will be done by developing geometric and topological methods, which use the "shape" of these algebraic objects, or the way they interact with spaces, to shed light on their algorithmic properties.

There are around 304 million lakes globally. These provide essential resources for human survival and are an important component of global biogeochemical cycles. Lakes are also fragile systems that are sensitive to multiple pressures including nutrient enrichment, climate change and hydrological modification, making them important 'sentinels' of environmental perturbation. However, traditional monitoring has only produced data from a tiny fraction of the global population of lakes and disentangling the causes of change requires consistently-produced data from a large number of lakes, along with measurements of possible causes of change. Satellite observations (remote sensing) and the establishment of a global lake observatory would produce a step-change in our ability to detect and attribute the causes of changes in lakes world-wide. This is now possible for three reasons: (1) the improved wavebands, spatial resolution and frequency of data collection from satellite sensors is now sufficient to monitor inland waters; (2) formulae to correct for atmospheric properties and to convert the detected reflected light to useful lake properties have been developed; and (3) computing power has increased to the point that allows near real time and archived information from satellites to be processed. GloboLakes will analyse 20 years of data from more than 1000 large lakes across the globe to determine 'what controls the differential sensitivity of lakes to environmental perturbation'. This is an ambitious project that is only possible by bringing together a consortium of scientists with complementary skills. These include expertise in remote sensing of freshwaters and processing large volumes of satellite images, collation and analysis of large-scale environmental data, environmental statistics and the assessment of data uncertainty, freshwater ecology and mechanisms of environmental change and the ability to produce lake models to forecast future lake conditions. The eight objectives of GloboLakes are to: (i) develop remote sensing algorithms to estimate lake biogeochemical and physical parameters; (ii) make these algorithms operational and process satellite data; (iii) compile integrated spatio-temporal information on climatic and catchment data for >1000 lakes; (iv) integrate data and assess uncertainty in data sources; (v) detect spatial and temporal patterns in lake water quality; (vi) attribute the causes of lake response to environmental conditions; (vii) forecast lake sensitivity to environmental change; (viii) apply data to lake management and the monitoring of freshwater resources. The project will focus on the retrieval of surface water temperature as this has a fundamental effect on lake ecology, the concentration of coloured dissolved organic matter and suspended solids that derive largely from the catchment, the abundance of phytoplankton measured as the concentration of the pigment, chlorophyll a, and the abundance of cyanobacteria (blue-green algae) that can potentially be toxic. Knowledge of the conditions of lakes and their sensitivity to change is also extremely valuable for the management of lakes and reservoirs and GloboLakes will provide information and products specifically for environmental managers. A satellite due to be launched during the course of the project, called Sentinel 2, will provide even greater spatial resolution allowing data to be collected and exploited from even smaller lakes. This will be investigated by GloboLakes and incorporated into the framework of a global lake observatory.

Deltas are economic and environmental hotspots, food baskets for many nations, and home to a large portion of the world population. They sustain rich, biodiverse ecosystems and related services. Most deltas are also international and regional transportation hubs that support intense economic activity. Yet, deltas are deteriorating at an alarming rate due to climate impacts (e.g., sea level rise and flooding), human-induced catchment changes (e.g., water and sediment flow reduction), and local exploitation (e.g., sand, groundwater, and hydrocarbon extraction). The international science community recognizes the need to develop a solid knowledge base for protecting these vulnerable coastal systems, and this BF initiative leads the way by coordinating and enhancing innovative international work towards the development of a science-based framework for delta sustainability. The project will develop a versatile modeling framework that may be applied from local to national levels to evaluate the unique functioning, critical stressors, and vulnerability of the world’s deltas. The framework will ingest social, economic, physical and ecosystem data into an open-access repository and will allow planners to model and deliver optimized, viable solutions for their region. In areas for which detailed data are sparse, an infrastructure for critical data gathering will be developed and modeling and prediction tools will be customized. The framework will initially be applied to three case-studies for which local and regional partnerships are already in place, including the Ganges-Brahmaputra-Meghna (GBM), Mekong, and Amazon deltas. The team represents the BF-G8 countries: Brazil, Canada, China, France, Germany, Norway, India, Japan, UK, and USA, and includes partners in the Netherlands, Vietnam, and Bangladesh. It is composed of government and university researchers, and NGO’s, working closely with policymakers. The training of graduate students and post-docs able to work across disciplinary boundaries and countries will also be a unique legacy of the project.