Since the end of the 19th century, the analogy between number fields and function fields has played a crucial role in arithmetic geometry. The interpretation of this analogy in the geometric framework has led to the definition of arithmetic varieties over the ring of integers of a number field. To make this analogy more satisfactory, it is important to consider the Archimedean and ultrametric places of a number field in the same way. The work of Arakelov in the '70s has initialized the comprehension of the role the archimedean embeddings of a number field should play in order to compactify the arithmetic variety, giving rise to the theory of Arakelov geometry. These ideas have inspired many new results, including the proof by Faltings of the Mordell conjecture. The slope theory of Bost belongs to Arakelov geometry. This theory had a profound influence on Diophantine geometry, showing how to prove explicit results in an intrinsic and elegant way. It has shed lights on some new arithmetic invariants which are comparable to the successive minima of Minkowski but more relevant in a geometric point of view. Based on these results, the recent works of the French school have established a new geometry of numbers, which we may call absolute, on any algebraic extension of the rational field Q. Fruitful interactions of this theory with other domains, such as number theory (Siegel's lemmas in transcendence theory) or algebraic geometry (algebraicity of formal varieties) have led to many applications in the study of Diophantine problems. The birational arithmetic geometry, and in particular the study of the arithmetic volume function of Hermitian line bundles on projective arithmetic varieties, has also benefited from these advances. Several members of the project have already significantly contributed to these developments. Starting from the existing results, the aim of our project is to develop this absolute geometry of numbers, opening new directions of research (for example, making links with the theory of error correcting codes) and to explore further applications in Arakelov geometry (birational invariants, counting rational points) and Diophantine geometry (transcendence criteria, theory of linear forms in logarithms, Lehmer's problem). To carry over this program, we plan to organize several workshops (at least one workshop per year), a summer school and an international conference at the end of the project. A particular attention will be paid to the participation of PhD students and young researchers as well as the diffusion of the results obtained in the project by communications in seminars and conferences and also through an internet website to increase the visibility of our activities.

Self-healing Ceramic-Matrix Composites (SH-CMCs) have extremely long lifetimes even under severe thermal, mechanical and chemical solicitations. They are made of ceramic fibres embedded in a brittle ceramic matrix subject to multi-cracking, yielding a “damageable-elastic” mechanical behaviour. The crack network resulting from local damage opens a path to fibre degradation by corrosion and ultimately to failure of the composite, e.g. under static fatigue in high-temperature oxidative conditions. But these materials have the particularity of protecting themselves against corrosion by the formation of a sealing oxide that fills the matrix cracks, delaying considerably the fibres degradation. Applications encompass civil aeronautic propulsion engine hot parts and they represent a considerable market; however this is only possible if the lifetime duration of the materials is fully certified. Numerical modelling is an essential tool for such an aim, and very few mathematical models exist for these materials; fulfilling the needs requires a strong academic-level effort before considering industrial valorisation. Therefore, the ambition of this innovative project is to provide reliable, experimentally validated numerical models able to reproduce the behaviour of SH-CMCs. The starting point is an existing image-based coupled model of progressive oxidative degradation under tensile stress of a mini-composite (i.e. a unidirectional bundle of fibres embedded in multi-layered matrix). Important improvements will be brought to this model in order to better describe several physic-chemical phenomena leading to a non-linear behaviour: this will require an important effort in mathematical analysis and numerical model building. A systematic benchmarking will allow creating a large database suited for the statistical analysis of the impact of material and environmental parameter variations on lifetime. Experimental verifications of this model with respect to tests carried out on model materials using in-situ X-ray tomography – in a specially adapted high-temperature environmental & mechanical testing cell – and other characterizations are proposed. The extension of the modelling procedure to Discrete Crack Networks for the large-scale description of the material life will be the next action; it will require important developments on mesh manipulations and on mathematical model analysis. Finally, experimental validation will be carried out by comparing the results of the newly created software to tests run on 3D composite material samples provided by the industrial partner of the project. The project originality lies in a multidisciplinary character, mixing competences in physico-chemistry, mechanics, numerical and mathematical modelling, software engineering and high-performance computing. It aims creating a true computational platform describing the multi-scale, multidimensional and multi-physics character of the phenomena that determine the material lifetime. Important outcomes in the domain of civil aircraft jet propulsion are expected, that could relate to other materials than those considered in this study.

Lithium-bromide systems can play a strategical role in the recovery of unused low exergy level energy source, in particular in the current context where the priority relationship between initial investment and running constraints (cost and electricity consumption reduction/deletion) have shifted. Despite this strategical interest and, among the low Coefficient of Performance, the systems are bulky and the investment cost is high compared to conventional system. To improve the performance of these system, the project proposes to study a new concept of generation of liquid films in the wall allowing to reduce considerably their thickness and to improve the performances of boiler by taking advantage of the boiling phenomenon which can occur there. The application targeted is a lithium-bromide absorption heat-pump of type II driven by a low exergetic level source (˜ 40°C). By the design of modular experimental channels, the development of a new laser-induced fluorescence methodology to access the volumetric mean temperature and LiBr concentration, complex phenomena occurring inside the experimental set-up will be analysed, characterised and modelized. In particular, the impact of the confinement of the fluid inside the channel, the working pressure and the wall superheat on the exchanger performances will be analysed and quantifyed. Novel asymptotic models to capture the drying dynamic of water and water-bromide solution, as well as the interplay between heat and mass transfer will be developed. Tools and performances maps developed in the context of this project will then be exploited to design a compact low-pressure water-based heat- exchangers and a rational design of a LiBr boiler will be proposed.