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 projet de recherche collaboratif ``Beyond KAM theory'' is a project in Mathematics. Its goal is the study of dynamical systems both in finite and infinite dimensions in view of applications to partial differential equations and spectral theory. More specifically, we will be interested in systems displaying quasi-periodic behaviors which means displaying quasi-periodic patterns in time or space. A fundamental tool in this approach is the so-called KAM theory (for Kolmogorov, Arnold, Moser) that allows to prove, for certain perturbations of integrable hamiltonian systems, the existence of invariant tori on which the dynamics of these systems is quasi-periodic. KAM theory is a powerful tool: its range of application goes from the study of one-dimensional dynamical systems (circle diffeomorphisms) to that of infinite dimensional hamiltonian systems such as hamiltonian partial differential equations. The domain of application of KAM theory is nevertheless hampered by three classical restrictions: KAM method generally applies to perturbations of simple model systems ; (b) small divisors phenomena impose quantitative non-resonance conditions; (c) the existence of resonances and their geometry often make necessary the introduction of parameters and non-degeneracy assumptions on the way these parameters control the system. One main goal of our project will be, whenever possible and for different types of systems, to go beyond these restrictions. The systems we will consider are finite dimensional hamiltonian systems, diffeomorphisms of the circle, of the disk and of the torus, quasiperiodic cocycles jointly with quasiperiodic Schrödinger operators and hamiltonian partial differential equations. The funding of this project by the Agence Nationale de Recherche, that will last 4 years, will allow the collaboration of mathematicians, with complementary skills (dynamical systems, small divisors, hamiltonian theory, partial differential equations, normal forms, cocycles) and that use in their research KAM theory as a fundamental tool. The partners of the project are: (a) Partner at Nantes, Lab. J. Leray, Univ. Nantes: Benoît Grébert (representative), Eric Paturel, Georgi Popov, Laurent Thomann; (b) Partner at Nice, Lab. J.A. Dieudonné, Univ. Nice Sophia Antipolis: Philippe Bolle, Claire Chavaudret, Laurent Stolovitch (representative); (c) Partner at Paris, Lab. de Prob. et Mod. Aléat., Univ. Pierre et Marie Curie: Artur Avila, Abed Bounemoura, Hakan Eliasson, Bassam Fayad, Jacques Féjoz, Sergei Kuksin, Raphaël Krikorian (coordinator), Laurent Niederman and Jean-Christophe Yoccoz. The funding will allow the organization of one international conference gathering international leading experts, four annual meetings where all the participants of the project will present their current works, and one summer or winter school. This funding will also permit deeper collaborations between the member of the project and, through various invitations, with other worldwide experts; it will allow the members of the project to participate to conferences in the field, and this way, to diffuse and deepen their ideas. The financial support of missions for the members of the project or for their PhD students (in particular to attend the summer or winter schools) is an important point. Finally, this financial support will make possible the hiring of two post-doctoral researchers each financed for one year. The amount of the requested funding is 305 keuros.

The ambition of the ChaMaNe project is to create a research group to achieve significant advances in the field of mathematics for neuroscience. The comprehension of the brain is still far from being achieved and its modeling is extremely complex. In particular, many scales coexist: from proteins and synapses to macroscopic brain areas and functions, from the few milliseconds of an action potential to the hours or years of synaptic plasticity and learning processes. Hence, the variety of questions, with different ranges of difficulty, emerging from this topic is enormous, and only a very small number of them has been addressed. The key questions of the ChaMaNe project will be to understand, on the one hand, the intrinsic dynamics of a neuron and their consequences, and on the other hand the qualitative dynamic that emerges from large neural networks with respect to the intrinsic behavior of individual neurons, interactions between neurons, memory effects, spatial structure ... Answering those very broad questions requires a combination of expertises based on partial differential equation's (PDE) analysis, probability and statistics that will form the consortium. Our strategy will mainly go through a theoretical approach, but we will also rely on data analysis as well as on the development of numerical methods for specific contexts. This approach, based on theory and integrating also applications, will be established in a complementary and fruitful way, allowing us to promote more efficiently interactions with neuroscientists. In recent years, an important literature has been developed around the mathematical models of neuroscience derived from PDEs or stochastic models. The methods developed recently are very effective in some particular cases, but are far from covering the diversity of issues underlying many models whose complexity is the result of essential biological properties. Our ambition is to better understand the various complex patterns resulting from neural models, by strengthening the existing literature and creating new tools and methods. The observed patterns that result from the communication between the neurons is very rich: they can notably reveal phenomena of synchronization of neuron discharges which are more or less rhythmic within a population, or phenomena of propagation of waves of neuron discharges in the brain. One of the major questions that will be tackled in the ChaMaNe project is to understand what may be the mechanisms underlying all these interesting observations and how to give them the most rigorous mathematical framework possible. Would this result from the intrinsic dynamics of each neuron? From memory effects or learning? From the spatial complexity of interconnections between neurons? What is the influence of noise ? Understanding fully these issues as a whole is a titanic task, but with an original approach, combining various theoretical expertise's and an applied component related to data analysis, we have the ambition to bring a number of significant answers to these questions. The models involved are governed by deterministic and stochastic dynamics at different scales, hence, the unification of different theories coming from Partial Differential Equations (PDE), Probability theory and Statistics is essential to elaborate fruitful and promising directions. The strategy of the ChaMaNe project to make significant progress in this area will be through three complementary axes : 1. Single stochastic/deterministic neuron models 2. Scale of large number of neurons : homogeneous models 3. Scale of large number of neurons :spatial interaction or complex connectivity matrix It is important to stress that these three axes are not at all independent and feed one another. A large component of the project is theoretical, but on each part, comparison with data will be made in a complementary way or to complement the theoretical work.

Most nonlinear wave or dispersive equations which arise from physics describe propagation in a medium which is neither homogeneous or spatially infinity. Only recently toy nonlinear models have been studied on a curved background, sometimes compact or rough. In line with our recent work on domains with boundaries, we aim at gathering a better understanding of standard existence and uniqueness questions, and then at studying the ultimately relevant behavior of solutions, like blow-up or asymptotics for large time ; such a precise description of solutions in the large should be partly dépendent on the underlying geometry (boundaries, multiple obstacles, interfaces…). Such problems have been thoroughfully investigated for the linear wave equation, prompted by the birth of radar/sonar technologies (or computed tomography), and motivated in part numerous développements in microlocal analysis. Understanding how to extend such tools for a better understanding of nonlinear phenomena (dispersion or focusing), and combining them with recent nonlinear techniques which proved crucial in dealing with (relatively rigid) toy models will allow us to significantly progress in our mathematical understanding of physically relevant models. Furthermore, gaining a more fine-grained understanding of the behaviour of solutions to dispersive models, including realistic ones, is now increasingly within reach; understanding domains of validity, choices of boundary conditions, (non)linear instability and more generally going beyond existence questions, in relation with the derivation of models are among the main goals of the project.

Decades of research on earthquakes have yielded meager prospects for earthquake predictability: we cannot predict the time, location and magnitude of a forthcoming earthquake with sufficient accuracy for immediate societal value. Therefore, the best we can do is to mitigate their impact by anticipating the most “destructive properties” of the largest earthquakes to come: longest extent of rupture zones, largest magnitudes, amplitudes of displacements, accelerations of the ground. This topic has motivated many studies in last decades. Yet, despite these efforts, major discrepancies still remain between available model outputs and natural earthquake behaviors. Here we argue that an important source of discrepancy is related to the incomplete integration of actual geometrical and mechanical properties of earthquake causative faults in existing rupture models. Indeed, our group has been among the pioneers to show that faults are 3D features, systematically embedded in a permanent damage zone where crustal rocks are intensely faulted and hence are compliant. Faults also are systematically segmented laterally in a generic manner, and this segmentation divides their planes and produces strength and stress heterogeneities in a deterministic manner. As faults grow over the long-term and become more “mature”, some of their properties evolve: the damage zone enlarges and its compliance increases, the fault segments become more tightly connected, the fault plane roughness decreases as might also do the fault friction. All these fault properties and their changes in relation to fault maturity markedly modify the earthquake behavior. In particular, earthquakes on mature and immature faults produce different amplitudes of slips and ground motions, whereas earthquake slips and speeds are systematically largest on the most mature sections of the ruptured zones. These intimate connections between fault and earthquake properties mean that a synoptic understanding of earthquake mechanics cannot be successful until it more fully includes actual fault properties. This is not done at present, as most current earthquake models either ignore fault properties or oversimplify them. We thus aim to take benefit of the fault data and knowledge we have gained in the last decades, and of our strong experience in earthquake modeling, to develop a new generation of rupture and ground motion (GM) models, based on a novel paradigm: 3D fault zones with generic macroscopic properties (especially permanent damage and lateral segmentation) whose inhomogeneous and anisotropic characters evolve depending on both overall and along-strike fault maturity. Furthermore, since most major fault properties are deterministic, even generic, they must result from some common, scale-invariant physics. The understanding of that physics should advance generic earthquake and GM models that could be run for the vast majority of faults and earthquakes worldwide. These new models should thus open a novel avenue in earthquake modeling. We first aim to document the compliance of rocks in natural permanent damage zones. These data –key to earthquake modeling– are presently lacking. A second objective is to introduce the observed macroscopic fault properties –compliant permanent damage, segmentation, maturity– into 3D dynamic earthquake models we have developed in prior works. A third objective is to compute Ground Motions (GM) from these new fault-based earthquake models and propagate them into non-linear media, using codes that we have developed. A fourth objective is to conduct a pilot study aiming at examining the gain of prior fault property and rupture scenario knowledge for Earthquake Early Warning (EEW). We expect that integrating actual fault properties in dynamic rupture, GM, and EEW models will decrease the discrepancies between models outputs and natural earthquake behavior, and hence allow a more accurate anticipation of the “destructive properties” of forthcoming events.