The GALF project represents an inter-European and transatlantic fundamental research project with the broad objective to study several of the most relevant problems in current Number Theory using p-adic methods. The team consists of internationally renowned and leading experts from Paris, Lille and Bordeaux in France, from Montreal in Canada and from Luxembourg, joining their complementary expertise in the project. A central problem in Number Theory is the relationship between special values of L-functions and fundamental arithmetic invariants such as regulators or Tate-Shafarevich groups. The study of p-adic properties of special cycles on algebraic varieties plays a key role in this theory.The GALF project will investigate various facets of this, including applications of special cycles on Shimura varieties to p-adic analogs of the Birch and Swinnerton- Dyer conjecture, and p-adic aspects of the Plectic Conjecture. Connexions between Plectic Conjecture, Iwasawa theory and Fargues-Fontaine theory shall be examined. Other important GALF research themes to be examined are the role of p-adic and plectic cohomology methods in extending the theory of complex multiplication to other settings like that of real quadratic fields. The Langlands programme is a vast international research effort establishing deep links between various mathematical areas and appearing in various different forms. It relates automorphic forms and representations with Galois representations and hence number theory. The GALF research will target p-adic and mod p aspects of those via Galois deformation techniques. Geometrically defined Hilbert modular forms mod p of parallel and partial weight one play a very special role. In the GALF project, their attached Hecke algebras shall be related to universal deformation rings with a particular focus on the ramification properties at p. A part of the GALF project is to attach an L-function to an overconvergent eigenform mod p of finite slope and to examine the local behavior at p of the attached Galois representation in view of a formulation of a T-adic Main Conjecture. The theory of deformations of Galois representations has a geometric counter part in so-called eigenvarieties. The GALF efforts in this context especially target the local structure of the eigenvariety at classical weight one points. Moreover, properties of companion forms attached to specific weight 1 modular forms, in particular their Fourier coefficients, will be investigated both in the archimedean and the p-adic settings, with applications to explicit Class Field Theory and Kudla's programme.

The goal of this project is to provide a new state-of-the-art Numerical Methods and High Performance Computing (HPC) software for the numerical simulation of Bose-Einstein condensates (BEC). This is a timely objective, since BEC physics is a very dynamic research field, with applications belonging to a future technological era. The project bridges a gap in this field, where modern, HPC numerical codes are nowadays absent. With the purpose to develop robust and reliable numerical simulators, based upon new mathematically sound methods and modern HPC strategies, this project has no worldwide equivalent and will strongly impact studies of BEC physics conducted in both mathematics and physics communities. The project combines mathematical modelling, numerical analysis and simulation in a coherent workflow that brings together 20 (permanent) mathematicians and computer scientists from 4 partners. This includes a solid task-force made of 5 research engineers, who will use their important experience in HPC to support coding effort. The project will also take benefit from the strong interaction with external collaborators, who are expert physicists in BEC systems. This participation ensures the mandatory critical mass required to take up the following challenges: (i) develop new high-order numerical methods with firm mathematical background; (ii) develop an integrated and resilient open-source HPC software that will materialize advances in numerical methods and algorithms for BEC simulation; (iii) apply these codes to numerically reproduce realistic physical configurations that are not possible to simulate with presently existing software. With regard to these objectives, the project fits the call of Numerical Models ANR program, action Basic Research in Modelling and Simulation of Complex Systems, with the purpose to "understand and predict" complex physical phenomena. To cover all these challenging topics, the project is divided in 6 tasks with a total number of 26 subtasks. Each task involves at least two partners, and often three to four partners, to encourage exchanges and communication. Most of the subtasks are novelties in the BEC research field and some of the subtasks are highly challenging (e.g. modeling of stochastic effects, high order methods in HPC codes, simulation of real experiments, etc). The team will be strengthened with 2 PhD students and 5 postdoctoral students. Theoretical results will be disseminated through first rank publications, while developed HPC codes will be made available from a dedicated open-access Web-site. This will allow researchers to adapt the codes for their own purposes and thus maintain competition in this very dynamic field. The project will also be a unique opportunity to set a new vivid community of research in the emerging field of the numerical simulation of BEC systems. In order to ensure the training of new researchers and the resilient character of the developed software, two workshops and a summer school will be organized in the framework of the project.

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 project RANDOM proposes to set a mathematically controlled disorder in a system of dual electromagnetic and acoustic resonators on a surface (i) to discover new photonic and phononic physical behaviors in relation with a random distribution controlled by mathematical rules (order/disorder transition, hyperuniformity, percolation paths, Anderson type localization), (ii) to co-localize photons and phonons on randomly distributed sites and to enhance their optomechanical interaction, (iii) to develop a time resolved pump-probe acousto-optic near-field microscope for phoXonic experiments and (iv) to propose an OM component operating at the telecom wavelength consisting in an acoustic, optical and OM stamping of the phoXonic surface. RANDOM will allow producing a complex coding of a random patterns with numerous degrees of freedom usable in information coding. The typical platform under investigation in the RANDOM project is composed of disordered resonators, made of gold nano-pillars to confine both the EM and the elastic waves. We propose to investigate in this dual phononic/photonic (phoXonic) platform the effect of disorder, and more especially of different mathematical probability laws governing the particle geometrical distributions, on transport phenomena, nanoscale localization, and enhanced OM interactions. Numerically, time and frequency domain techniques will be used to perform calculations of the dispersion curves, transmission, reflection, and absorption spectra, field distributions, density of states and eigenvalues of the metasurface. The involvement of mathematical skills in probability and statistics, as Poisson or Gibbs processes, appears in the distribution of the resonators. The microscopic distribution of the acoustic and EM fields in the structure will be also investigated and analyzed experimentally, based on transient grating and picosecond ultrasonic (elastic side), high-resolved optical microscopy and FTIR spectroscopy (EM side). Finally, all results will be used to deliver an OM component, named OM-QR code, consisting in an OM image of the random distribution. The principle lays on the optical reading (the response) of a mechanical wave (the message) through the OM coupling. Finally, we will demonstrate the properties of a new OM component, called OM-QR code, based on the optical reading (the response) of a mechanical wave (the message) via OM coupling. With such functionality, we will be able to define an uncoded image of the random distribution. As such, the project RANDOM goes beyond the current state of the art regarding (i) fundamental theoretical and experimental OM studies of phononic, photonic, and phoXonic crystals, (ii) instrumental development with realization of a time resolved pump-probe acousto-optics microscope, and (iii) applied science with the demonstration of a new OM component for information coding. To succeed, the project brings together a multidisciplinary consortium of four complementary partners in the fields of acoustics, optics and mathematics. With the working wavelength of this new experimental platform at 1.55 µm, corresponding to GHz phonons and photons around 180 THz, RANDOM offers new research and applications perspectives in the field of information and communication.