The project is organized around four important topics in fluid mechanics: free surfaces and interfaces, boundary layers, vortex dynamics and fluid-structure interactions. The mathematical and the physical-environmental motivations of the project are connected to the events of the 2013 Mathematics of Panet Earth program which will also suggest new directions of research. The four topics are closely interconnected because they often coexist in the same physical situation and because the mathematical tools (such as multiscale analysis, asymptotic expansions, stability theory) that are needed to analyze them are quite similar. Our main directions of research will be: -Free surfaces and interfaces. We are mostly interested in situations which are singular, either because of the lack of smoothness (examples are wave breaking, the description of shorelines and the influence of rough topographies in shallow water models) or because of the presence of small parameters (examples are continuous but sharp stratification in two fluid models, multiscale models that describe the energy spectrum in wave breaking and compressible fluids with free surface at low Mach number). We expect improvements in the modelling and numerical simulations of these phenomena through the derivation of more accurate asymptotic models. We also plan to develop suitable mathematical tools in order to handle these singular situations rigorously. -Boundary layers. We are interested both in the construction of boundary layers expansions and the study of their stability properties. For the first aspect, we shall study the construction of boundary layers in degenerate situations, for example in the presence of rough boundaries or in situations where boundary layers of different sizes need to be connected (this is crucial to understand oceanic circulation). We shall also study the well-posedness of the Prandtl type equations that arise in oceanics models. For the second aspect, we plan to make progress in the understanding of instabilities in boundary layers either in the classical inviscid limit of the incompressible Navier-Stokes equation with Dirichlet boundary condition by addressing the question of the destabilizing effect of viscosity or in slightly regularized situations like some critical Navier conditions or the alpha-models equations -Vortex dynamics. We shall study both perfect and viscous incompressible fluids using mainly the vorticity equation. Our interest lies in singular domains (flow around rough obstacles for example) or in singularly pertubed domains (flow around small obstacles). In the two-dimensional case, the question of understanding the large time behaviour of perfect and viscous fluids will be also adressed. Another important direction of research will be the study of vortex filaments , the most challenging question being the rigorous understanding of the motion of vortex filaments in the vanishing viscosity limit (the expected asymptotic model is the binormal flow). -Fluid-structure interactions. We first plan to get a better understanding of qualitative properties of the fluid-structure interactions on the most simple models (incompressible fluids with rigid bodies). Typical questions that will be addressed are the uniqueness of weak solutions in 2D for viscous fluids and the study of the smoothness of particles trajectories. Some singular limits like vanishing viscosity limit, vanishing particles limit and mean field limit will be also studied. Finally we plan to make progress in the understanding of more complete models that take into account for example deformable solids or compressible fluids.

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 p-adic Langlands correspondence has become nowadays one of the deepest and the most stimulating research programs in number theory. It was initiated in France in the early 2000's by Breuil and aims at understanding the relationships between the p-adic representations of p-adic absolute Galois groups on the one hand and the p-adic representations of p-adic reductive groups on the other hand. Beyond the case of GL2(Qp) which is now well established, the p-adic Langlands correspondence remains quite obscure and mysterious new phenomena enter the scene; for instance, on the $GLn(F)$-side one encounters a vast zoology of representations which seems extremely difficult to organize. The CLap--CLap ANR project aims at accelerating the expansion of the p-adic Langlands program beyond the well-established case of GL2(Qp). Its main originality consists in its very constructive approach mostly based on algorithmics and calculations with computers at all stages of the research process. We shall pursue three different objectives closely related to our general aim: (1) draw a conjectural picture of the (still hypothetical) p-adic Langlands correspondence in the case of GLn,, (2) compute many deformation spaces of Galois representations and make the bridge with deformation spaces of representations of reductive groups, (3) design new algorithms for computations with Hilbert and Siegel modular forms and their associated Galois representations (in which the p-adic Langlands correspondence is supposed to be realized). This project will also be the opportunity to contribute to the development of the mathematical software SageMath and to the expansion of computational methodologies.

Our project is to treat multiscale models which are both infinite-dimensional and stochastic with a theoretic and computational approach. Multiscale analysis and multiscale numerical approximation for infinite-dimensional problems (partial differential equations) is an extensive part of contemporary mathematics, with such wide topics as hydrodynamic limits, homogenization, design of asymptotic-preserving schemes. Multiscale models in a random or stochastic context have been analysed and computed essentially in finite dimension (ordinary/stochastic differential equations), or in very specific domains, mainly the propagation of waves, of partial differential equations. The technical difficulties of our project are due to the stochastic aspect of the problems (this brings singular terms in the equations, which are difficult to understand with a pure PDE's analysis approach) and to their infinite-dimensional character. These two aspects, combined, typically raise compactness and computational issues. Our aim is to create the new tools, analytical, probabilistic and numerical ones, which are required to understand a large class of stochastic multiscale partial differential equations, that includes some kinetic and dispersive equations. Our aim is to derive reduced equations. In the different regimes we are interested in, and, particularly, in the diffusive regime (diffusion-approximation), that leads to limit equations with white noise. We will investigate the derivation of reduced equations in different context and for various models: - collisional kinetic equations with a Vlasov forcing term induced by (resp. a collisional kernel perturbed by) an external, or coupled, Markov process (kinetic equations for plasmas or fluids, resp. modelling of motion by run-and-tumble), - limit Boltzmann to Navier-Stokes under stochastic forces, - collisional or non-collisional kinetic equations with a stochastic drag force term also induced by an external, or coupled, Markov process (models of sprays in turbulent flows, stochastic Cucker-Smale models, stochastic Landau damping), - dispersive models for the propagation of waves (e.g. Zakharov system, Klein-Gordon-Zakharov system, stochastic NLS equations). The numerical approximation of these models raises the following issues, which we will investigate. First, the construction and analysis of asymptotic-preserving schemes for equations in which the small parameter affects equally the deterministic ans stochastic terms. This concerns the design of schemes unconstrained by the small parameters for the original equations. Numerical schemes for the approximation of the reduced equations furnished by the theoretical analysis are another approach that we will develop. This requires the computation of the coefficients of the reduced equations. We will build accurate Heterogeneous Multiscale Methods (HMM) to that purpose. HMM in our context (kinetic, dispersive stochastic equations) have never been developed. Several questions of numerical analysis are related. Regarding these questions, we will analyse the efficiency of the the numerical schemes in the approximation of invariant measures or auto-correlations, their orders of convergence, and develop strategies to reduce the overall cost, like high-order integrators for invariant distributions, variance reduction strategies in Monte-Carlo methods.