A Diophantine equation, named after the ancient Hellenistic mathematician Diophantus of Alexandria, is a polynomial equation in which all the coefficients are integers (whole numbers) or rational numbers (fractions). The most fundamental question, given a Diophantine equation, is whether it has a solution, that is a collection of integers or rational numbers which satisfy this equation. To decide whether a given Diophantine equation has a solution can be extremely hard, in spite of extensive mathematical machinery that was developed over centuries to attack these questions. A famous example is Fermat's Last Theorem. Despite the relative simplicity of its statement that for any integer n greater than two, the sum of two positive nth powers can not be an nth power, a proof has eluded the efforts of mathematicians for more than 350 years. It has spawned numerous new developments and was finally completed by Andrew Wiles at the end of the 20th century. Equations define not just number theoretic, but also geometric objects. A particularly successful approach, developed in the 20th century, tries to investigate solutions to Diophantine equations via the corresponding geometric objects. The modern study of Diophantine equations using these geometric techniques is called arithmetic (or Diophantine) geometry. Another branch of number theory, in which UK mathematicians play a world leading role, is called additive combinatorics. One of the aims of this discipline is to understand subsets of the integers by decomposing them into structured and random looking parts, with the main challenge arising from the fact that this is usually not a clean dichotomy, but rather a full spectrum. Extremely fruitful connections between these two fields were initiated very recently by applying certain results and techniques from additive combinatorics to questions in arithmetic geometry, thus expanding our knowledge of Diophantine equations significantly. The central aim of this project is to enhance the impact of these techniques by making them available in a much wider context that is natural in arithmetic geometry.

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Driven by structural and functional abnormalities in the muscle of the heart, Heart Failure (HF) is a complex syndrome affecting around 900,000 people in the UK. HF results in a fundamental reduction in the ability of heart muscle to effectively pump and deliver blood to the body. While this deficiency in pump function is easily observed using medical imaging, dissecting the underlying cause of this reduced performance in terms of its implications for muscle structure and function remain open challenges. Further, predicting how disease will progress or respond to therapy remains an unmet need that would substantially improve patient care. Using state-of-the art biomechanical modelling, the "Adaptive, Multi-scale, Data-Infused Biomechanical Models for Cardiac Diagnostic and Prognostic Assessment" project will address these challenges by providing a modelling framework for assessment of the heart. Uniting measurements from microscopy, rheology, and medical imaging, this project aims to create biomechanical models that provide detailed information on the structure and function of the heart aiding diagnosis. Further, the platform will provide infrastructure for predictive modelling, simulating the response and adaptation of the heart over time. Biomechanical models will be systematically validated using animal heart models, providing rich data for understanding the biomechanics in vivo and ex vivo. The framework will, further, be directly translated through a novel study in patients with hypertrophic cardiomyopathy, providing a test bed for validation of predictive models of disease progression and response to therapy through virtual surgery.

In the diagnosis and treatment of disease, clinicians base their decisions on understanding of the many factors that contribute to medical conditions, together with the particular circumstances of each patient. This is a "modelling" process, in which the patient's data are matched with an existing conceptual framework to guide selection of a treatment strategy based on experience. Now, after a long gestation, the world of in silico medicine is bringing sophisticated mathematics and computer simulation to this fundamental aspect of healthcare, adding to - and perhaps ultimately replacing - less structured approaches to disease representation. The in silico specialisation is now maturing into a separate engineering discipline, and is establishing sophisticated mathematical frameworks, both to describe the structures and interactions of the human body itself, and to solve the complex equations that represent the evolution of any particular biological process. So far the discipline has established excellent applications, but it has been slower to succeed in the more complex area of soft tissue behaviour, particularly across wide ranges of length scales (subcellular to organ). This EPSRC SoftMech initiative proposes to accelerate the development of multiscale soft-tissue modelling by constructing a generic mathematical multiscale framework. This will be a truly innovative step, as it will provide a common language with which all relevant materials, interactions and evolutions can be portrayed, and it will be designed from a standardised viewpoint to integrate with the totality of the work of the in silico community as a whole. In particular, it will integrate with the EPSRC MultiSim multiscale musculoskeletal simulation framework being developed by SoftMech partner Insigneo, and it will be validated in the two highest-mortality clinical areas of cardiac disease and cancer. The mathematics we will develop will have a vocabulary that is both rich and extensible, meaning that we will equip it for the majority of the known representations required but design it with an open architecture allowing others to contribute additional formulations as the need arises. It will already include novel constructions developed during the SoftMech project itself, and we will provide many detailed examples of usage drawn from our twin validation domains. The project will be seriously collaborative as we establish a strong network of interested parties across the UK. The key elements of the planned scientific advances relate to the feedback loop of the structural adaptations that cells make in response to mechanical and chemical stimuli. A major challenge is the current lack of models that operate across multiple length scales, and it is here that we will focus our developmental activities. Over recent years we have developed mathematical descriptions of the relevant mechanical properties of soft tissues (arteries, myocardium, cancer cells), and we have access to new experimental and statistical techniques (such as atomic force microscopy, MRI, DT-MRI and model selection), meaning that the resulting tools will bring much-need facilities and will be applicable across problems, including wound healing and cancer cell proliferation. The many detailed outputs of the work include, most importantly, the new mathematical framework, which will immediately enable all researchers to participate in fresh modelling activities. Beyond this our new methods of representation will simplify and extend the range of targets that can be modelled and, significantly, we will be devoting major effort to developing complex usage examples across cancer and cardiac domains. The tools will be ready for incorporation in commercial products, and our industrial partners plan extensions to their current systems. The practical results of improved modelling will be a better understanding of how our bodies work, leading to new therapies for cancer and cardiac disease.

Living organisms construct a tremendous variety of structures across a range of sizes, from large bones to microscopic cell components in order to carry out their life processes. Despite this variation in size, the assembly of all of these objects ultimately relies on the generation of molecules that are nanometres in scale (a billionth of a metre, or 1/100,000th of the thickness of a human hair). These biological "building blocks", composed of compounds such as sugars and proteins are produced by enzymes, the molecular machinery of all living organisms. In order to generate these complex larger structures, living organisms have developed a range of methods for moving these enzymes to specific locations where the structures need to be formed. The ability to manipulate and study objects on nanometre scales is called nanotechnology, and is particularly interesting since at this size range, materials display new properties that are radically different from when they exist in their bulk form. By finding ways of harnessing these unusual properties, it is expected that they can be used to create entirely new types of technologies and devices. The basic idea of being able to move enzymes to particular locations as a means of controlling the construction of objects on this scale would therefore be extremely useful if it could be applied by us to assemble highly miniaturised devices, such as electronic components or circuits. Harnessing enzymes for this purpose is particularly appealing since they are able to conduct a wide range of chemical reactions very efficiently and generate few unwanted by-products. Furthermore, they function under mild conditions and do not rely on rare or toxic materials. In contrast, many of the current techniques used in nanotechnology are derived from the electronics industry are not only limited in the types of chemistry they can achieve due to the harshness of the conditions under which they operate, but are also very power consuming. Accordingly, the aim of this research project is to use enzymes that are able to promote the formation and deposition of materials to generate nanometre-scale patterns on a variety of surfaces. To achieve this aim, enzymes will be used together with an instrument called a "scanning probe microscope". This instrument uses miniature electrical motors to move a very sharp tip, the "probe" of the instrument, which is only a few nanometres wide. The instrument is also able to control the movement of this probe with nanometre precision. This ability to move and position the probe with such fine control makes it possible to use it to "write" patterns on surfaces. By attaching these enzymes to the tips of these probes, the chemical reactivity of the enzymes can be directed to deposit their materials as nanoscopic patterns. This new method of writing nanopatterns will be further facilitated by developing modified versions of these enzymes so that they will perform efficiently on a scanning probe. For example, they may be modified to deposit a wider range of compounds, or to be more resistant to damage so they may be used for a longer period of time before needing to be replaced. The materials that are produced will then be tested to determine their electrical properties so that they can then be applied for the construction of miniaturised electronic devices. Furthermore, experiments will be carried out using many scanning probes writing patterns simultaneously, which will demonstrate how this new method of nanofabrication could be used for the mass production of chemically complex miniaturised devices.