Deep geothermal energy allows clean, non-intermittent, heat and/or power production regardless of weather conditions at any hour of the day or night. It will contribute to the decarbonization of our economy reaching its maximum mitigation potential by 2050 (ANCRE, 2015). However, exploitation of subsurface natural resources is faced with an uncertain environment. This is sometimes coined as the geological risk. Whatever the deep geothermal technology - conventional heat mining of deep aquifers, enhanced/engineered geothermal systems or power production in magmatic settings – and its maturity level, this feature makes geothermal operations high-risk projects with substantial initial investments (several M€) related to drilling costs. Even if insurance policies have recently been adapted to new targets, a single exploration failure may deter operators from a region with assumed good potential but complex geology for decades (e.g. the Hainaut aquifer in Northern France in the early 80’s). Better knowledge of the subsurface is then a key bottleneck for the deployment of deep geothermal technologies. It has been observed that the most efficient way to mitigate the geological risk is the collaborative integration of multidisciplinary data and interpretations into a geomodel of the subsurface. In a geothermal context, the first goal of such conceptual models is the prediction of the spatial distribution of temperature. Then, in order to reach economic profitability, deep geothermal projects need power levels that require convective exchanges with the reservoir at high flow rate through production and injection wells. Parallel to that, transient convective processes, which are ubiquitous in high temperature magmatic settings, also control the temperature distribution and the natural state of many sedimentary basins and basement type geothermal plays. Aforementioned conceptual models must consequently be dynamic by nature and integrate subsurface mass and energy transfers controlled by multiscale geological structures. Numerical simulation has become a powerful method for scientific inquiry on par with experimental and theoretical approaches, especially when data are as scarce and heterogeneous as subsurface data. Moreover, much progress has been made during the last decades in static geological modeling, dynamic geothermal reservoir modeling and performance computing with several contributions from CHARMS’ partners. Yet, many developments are still largely independent and confined to academic circles. There is no off-the-shelf software that integrates all of them in a consistent framework. The main objective of CHARMS is to take that step further and deliver the foundations components of an open framework so that integrated dynamic conceptual models of geothermal systems in complex geological settings can be produced from the early phases of exploration, to increase the probability of success, and evolve continuously through collaborative contributions into operational reservoir models to guarantee sustainable exploitation. The project is based on the three following pillars: • a consistent framework to link evolutionary complex geological models and the definition of the nonlinear physics of geothermal flows, • the improvement of the parallel ComPASS platform, which already has promising results, with numerical schemes tailored to accurately model multiphase multicomponent geothermal flows on unstructured meshes with discontinuities (fault, fractures…), • baseline validation tests and industrial cases, including complex well geometries, to assess the usefulness of the new tools. CHARMS gathers scientists who know each other and have a strong experience in subsurface modeling activities. BRGM (French Geological Survey) will lead the project leveraging the numerical expertise of the University of Nice and Paris 6, and the Maison de la Simulations as well as the industrial experience of Storengy (ENGIE Group).

Mean Field Games (MFG) is new and challenging mathematical topic which models the dynamics of a large number of interacting agents. It has many applications: economics, finance, social sciences, engineering,.. MFG are at the intersection of mean field theory, optimal control and stochastic analysis, calculus of variations, partial differential equations and scientific computing. Based on the internationally recognized expertise of its teams, the project intends to achieve major breakthroughs in 4 directions: the mean field analysis (i.e., the derivation of the macroscopic models from the microscopic ones); the mathematical analysis of news MFG models; their numerical analysis; the development of new applications. In this period of quick and worldwide expansion of the MFG modeling, it intends to foster the French leadership in the domain and to attract new French researchers coming from related areas.

The rôle of infinite discrete subgroups of Lie groups originated in Fuchsian equations and in crystallographic groups, and gradually grew as its arithmetical, ergodical, dynamical, and geometrical aspects developed along the years. The objective of the 4 years project "Dynamics and geometric structures" is to uncover new and remarkable relations between the dynamical and geometrical facets of those groups. This includes a systematic investigation of the generalization of Fuchsian groups, the study of the advanced structure of their moduli space coming from the thermodynamic formalism as well as the intertwinings of geometrical and analytical properties of space-times with their infinity. The project has been organized along five different but interrelated directions we now present. Anosov representations: it is has long been an unresolved question to find the class of discrete groups that should be the higher rank generalization to Fuchsian or convex-cocompact subgroups. It is now widely agreed that the class of Anosov representations invented by Labourie answers this question. Anosov representations have now been characterized in a number of ways. The team wants to address the question of finding the higher rank equivalent of geometrically finite subgroups and also to investigate properties of dynamical systems associated with discrete subgroups. Homogeneous geometry: since the early developments of hyperbolic geometry, the connections between geometry and discrete groups are many. Recent illustrations are the way how Anosov subgroups give rise to geometric structure and the understanding of Lorentzian manifolds of constant curvature. One aim of the project is to explore furthermore those connections. More specifically, the foreseen tasks are to parametrize some moduli spaces of representations using geometric coordinates, to understand compactifications of Riemannian locally symmetric spaces but also to explore Lorentzian manifolds "with particles" which is physically more relevant. Length spectrum: any discrete subgroup gives rise to a length spectrum that in general determines completely the discrete subgroup. In the spirit of Thurston's asymmetric metric on Teichmüller spaces, the team is going to examine the length spectrum comparison in the setting of Anosov representations, and particularly for the Hitchin components, a generalization of the Teichmüller component due to Hitchin. The entropy is the exponential growth rate of the length spectrum, the teams is going to bring to light rigidity results of the entropy as well as its local and global behaviors in that context of "higher Teichmüller spaces". Pressure metric: the above mentioned length spectrum is often realized as the closed orbits lengths of a flow and hence topological and dynamical invariants of this flow can be accessed. Among those is its pressure and thus there is an associated pressure metric on the deformation space of representations. The local behavior of that metric as well as the other numerical quantities (lengths, intersections numbers, etc.) are among the scheduled subjects. Renormalized volume: the renormalized volume is a way to define a "volume" in a context where the volume is infinite. Is has been well studied in the context of convex cocompact hyperbolic 3-manifolds and has strong links with the geometry of the Teichmüller space; in the context of quasifuchsian groups it is related to the Liouville action. The related questions the team wants to inquiry in depth are the fine geometry of hyperbolic 3-manifolds, the possibility to define the Liouville action for the Hitchin component and also to determine a renormalization procedure for the Hitchin component themselves. The project is organized around 5 partners: Lille, Luxembourg, Nice, Paris, and Strasbourg.

This project focuses on the very active field of Lipschitz geometry of singularities. Its essence is the following natural problem. It has been known since the work of Whitney that a real or complex algebraic variety is topologically locally conical. On the other hand it is in general not metrically conical: there are parts of its link with non-trivial topology which shrink faster than linearly when approaching the special point. A natural problem is then to build classifications of the germs up to local bi-Lipschitz homeomorphism, and what we call Lipschitz geometry of a singular space germ is its equivalence class in this category. There are different approaches for this problem depending on the metric one considers on the germ. A real analytic space germ (V,p) has actually two natural metrics induced from any embedding in RN with a standard euclidean metric: the outer metric is defined by the restriction of the euclidean distance, while the inner metric is defined by the infimum of lengths of paths in V. Lipschitz geometry of singular sets is an intensively developing subject which started in 1969 with the work of Pham and Teissier on the Lipschitz classification of germs of plane complex algebraic curves. The project presented here is motivated by several important results obtained in this area during the last decade by Birbrair, Fernandes, Gabrielov, Gaffney, Grandjean,Houston, Lê, Neumann, Parusinski, Paunescu, Pichon, Ruas, Sampaio and others. We think in particular about the surprising discovery by Birbrair and Fernandes that complex singularities of dimension at least two are in general not metrically conical for the inner metric, which started a series of works leading to the complete classification of the inner Lipschitz geometry of germs of normal complex surfaces by Birbrair, Neumann and Pichon (coordinator of the project), and building on it, to major progress in the study of the outer metric. Another important result is the proof by Parusinski (member of our team) and Paunescu of the Whitney fibering conjecture in the analytic setting, based on a relation between Zariski equisingularity and the arc-wise analytic equisingularity which is of similar nature as Lipschitz equisingularity. Our project has two main objectives: (1) Building classifications of Lipschitz geometry in larger settings such as non-isolated and higher dimensional complex singularities, function germs, and in the global, semi-algebraic and o-minimal settings, (2) Developing bridges between Lipschitz geometry and three other aspects of singularity theory: - equisingularity; - embedded topology; - arc spaces and resolution theory. These three topics are classical areas of singularity theory, but their relations with Lipschitz geometry remain almost unexplored. However, some results have been obtained in all three areas. For example, a long standing question asking if Zariski equisingularity can be interpreted from a Lipschitz point of view got recently (2014) a positive answer for complex surface singularities, but remains open in higher dimensions. Another example is the recent result that the outer Lipschitz geometry of a normal surface singularity determines its multiplicity. This result gives an approach to the famous Zariski multiplicity question from a Lipschitz point of view which, again, needs to be explored in higher dimensions; it also emphasizes the importance of studying the relations between Lipschitz geometry and embedded topology of a hypersurface. As a last example, recent works show the key role played by wedges of arcs in the Lipschitz classifications of real and complex singularities, and of the resolution of singularities in the complex setting. In view of these recent results, as well as several other ones, we strongly believe that Lipschitz geometry will give a new point of view on each of them and will help to solve several important open problems which are a priori not of metric nature.

The understanding of the deep mechanisms responsible of normal or anomalous energy diffusion in chains of coupled oscillators is one of the most important questions of non-equilibrium statistical mechanics, in the mathematical and theoretical physics literature. A phenomenological description has been proposed (see e.g. [Derrida-Dhar-Saito]) and some theoretical predictions are available ([Spohn]) but mathematical proofs are highly challenging. During the last few years, to attack the problem in a rigorous way, it has been proposed to replace the purely deterministic chains by hybrid models: a conservative stochastic noise (in energy and in some cases in momentum) is superposed to the Hamiltonian dynamics. It turns out that, as proved by Basile, Bernardin and Olla in the harmonic case, these noisy Hamiltonian systems behave qualitatively similarly to the deterministic systems. The energy diffusion properties satisfied by the hybrid models are due to a subtle interplay between the Hamiltonian part and the stochastic part of the dynamics and are not reproduced by purely stochastic dynamics. Even if the study of hybrid systems is simpler than the study of purely deterministic chains, the former are the source of many important challenging problems that have not yet been solved, in particular in the anharmonic case. We plan to work on these problems for the models introduced in [Basile-Bernardin-Olla] and also for simplified perturbed Hamiltonian systems considered recently by Bernardin and Stoltz. These stochastically perturbed systems will be studied thanks to the techniques introduced by Varadhan and Yau in the middle of the nineties and currently investigated in the context of scaling limits for stochastic interacting particle systems. The development of these techniques is a very active field of research nowadays and the hybrid models give rise to several new interesting difficulties. Shortly, we are interested in: hydrodynamic limits (given by hyperbolic system of conservation laws) in the Euler time scale after the shocks; behavior of the thermal conductivity in the weak coupling (small interaction) limit and when the noise intensity goes to zero; effect of the disorder (e.g. random masses) on the transport properties of the noisy system (interplay between the noise and the Anderson's type localization phenomenon); study of the anomalous energy fluctuations. To achieve these advances we will combine complementary skills and techniques from four mathematicians specialized and well trained in scaling limits of interacting particles systems, two mathematical physicists specialized in non-equilibrium statistical mechanics and one physicist. A PhD student, who will defend her thesis on June 17, 2014, will complete the team. The project being at the interface of physics and mathematics the exchange of ideas from the two communities is for us a necessity.