ERC FUNDED PROJECTS

The numerical simulation of complex compressible flow problem is still a challenge nowaday even for simple models. In our opinion, the most important hard points that need currently to be tackled and solved is how to obtain stable, scalable, very accurate, easy to code and to maintain schemes on complex geometries. The method should easily handle mesh refinement, even near the boundary where the most interesting engineering quantities have to be evaluated. Unsteady uncertainties in the model, for example in the geometry or the boundary conditions should represented efficiently.This proposal goal is to design, develop and evaluate solutions to each of the above problems. Our work program will lead to significant breakthroughs for flow simulations. More specifically, we propose to work on 3 connected problems: 1-A class of very high order numerical schemes able to easily deal with the geometry of boundaries and still can solve steep problems. The geometry is generally defined by CAD tools. The output is used to generate a mesh which is then used by the scheme. Hence, any mesh refinement process is disconnected from the CAD, a situation that prevents the spread of mesh adaptation techniques in industry! 2-A class of very high order numerical schemes which can utilize possibly solution dependant basis functions in order to lower the number of degrees of freedom, for example to compute accurately boundary layers with low resolutions. 3-A general non intrusive technique for handling uncertainties in order to deal with irregular probability density functions (pdf) and also to handle pdf that may evolve in time, for example thanks to an optimisation loop. The curse of dimensionality will be dealt thanks Harten's multiresolution method combined with sparse grid methods. Currently, and up to our knowledge, no scheme has each of these properties. This research program will have an impact on numerical schemes and industrial applications.

1 432 769 €

Start date: 2008-12-01, End date: 2013-11-30

The purpose of this project is to study basic questions in algebraic and Kähler geometry. It is well known that the structure of projective or Kähler manifolds is governed by positivity or negativity properties of the curvature tensor. However, many fundamental problems are still wide open. Since the mid 1980's, I have developed a large number of key concepts and results that have led to important progress in transcendental algebraic geometry. Let me mention the discovery of holomorphic Morse inequalities, systematic applications of L² estimates with singular hermitian metrics, and a much improved understanding of Monge-Ampère equations and of singularities of plurisuharmonic functions. My first goal will be to investigate the Green-Griffiths-Lang conjecture asserting that an entire curve drawn in a variety of general type is algebraically degenerate. The subject is intimately related to important questions concerning Diophantine equations, especially higher dimensional generalizations of Faltings' theorem - the so-called Vojta program. One can rely here on a breakthrough I made in 2010, showing that all such entire curves must satisfy algebraic differential equations. A second closely related area of research of this project is the analysis of the structure of projective or compact Kähler manifolds. It can be seen as a generalization of the classification theory of surfaces by Kodaira, and of the more recent results for dimension 3 (Kawamata, Kollár, Mori, Shokurov, ...) to other dimensions. My plan is to combine powerful recent results obtained on the duality of positive cohomology cones with an analysis of the instability of the tangent bundle, i.e. of the Harder-Narasimhan filtration. On these ground-breaking questions, I intend to go much further and to enhance my national and international collaborations. These subjects already attract many young researchers and postdocs throughout the world, and the grant could be used to create even stronger interactions.

1 809 345 €

Start date: 2015-09-01, End date: 2020-08-31

"Many physical models involve nonlinear dispersive problems, like wave or laser propagation, plasmas, ferromagnetism, etc. So far, the mathematical under- standing of these equations is rather poor. In particular, we know little about the detailed qualitative behavior of their solutions. Our point is that an apparent com- plexity hides universal properties of these models; investigating and uncovering such properties has started only recently. More than the equations themselves, these univer- sal properties are essential for physical modelisation. By considering several standard models such as the nonlinear Schrodinger, nonlinear wave, generalized KdV equations and related geometric problems, the goal of this pro- posal is to describe the generic global behavior of the solutions and the profiles which emerge either for large time or by concentration due to strong nonlinear effects, if pos- sible through a few relevant solutions (sometimes explicit solutions, like solitons). In order to do this, we have to elaborate different mathematical tools depending on the context and the specificity of the problems. Particular emphasis will be placed on - large time asymptotics for global solutions, decomposition of generic solutions into sums of decoupled solitons in non integrable situations, - description of critical phenomenon for blow up in the Hamiltonian situation, stable or generic behavior for blow up on critical dynamics, various relevant regularisations of the problem, - global existence for defocusing supercritical problems and blow up dynamics in the focusing cases. We believe that the PI and his team have the ability to tackle these problems at present. The proposal will open whole fields of investigation in Partial Differential Equations in the future, clarify and simplify our knowledge on the dynamical behavior of solutions of these problems and provide Physicists some new insight on these models."

2 079 798 €

Start date: 2012-04-01, End date: 2017-03-31

My goal is to study several important open mathematical problems in non-equilibrium (NEQ) systems and to build a bridge between these problems and NEQ aspects of soft sciences, in particular biological questions. Traffic on this bridge is going to be two-way, the mathematics carrying a long history as a language of science towards the soft sciences, and the soft sciences fruitfully asking new questions and building new paradigms for mathematical research. Out-of-equilibrium systems pose several fascinating problems: The Fourier law which says that resistance of a wire is proportional to its length is still presenting hard problems for research, and even the existence and the convergence to a NEQ steady state are continuously posing new puzzles, as do questions of smoothness and correlations of such states. These will be addressed with stochastic differential equations, and with particlescatterer systems, both canonical and grand-canonical. The latter are extensions of the well-known Lorentz gas and the study of hyperbolic billiards. Another field where NEQ plays an important role is the study of glassy systems. They were studied with molecular dynamics (MD) but I have used a topological variant, which mimics astonishingly well what happens in MD simulations. The aim is to extend this to 3 dimensions, where new problems appear. Finally, I will apply the NEQ studies to biological systems: How a system copes with the varying environment,adapting in this way to a novel type of NEQ. I will study networks of communication among neurons,which are like random graphs with the additional property of being embedded, and the arrangement of genes on chromosomes in such a way as to optimize the adaptation to the different cell types which must be produced using the same genetic information. I will answer such questions with students and collaborators, who will specialize in the subprojects but will interact with my help across the common bridge.

2 135 385 €

Start date: 2012-04-01, End date: 2017-07-31