ERC FUNDED PROJECTS

"This project aims to develop the field of Harmonic Analysis, and more precisely to study problems at the interface between Fourier Analysis and PDEs (and also some Geometry). We are interested in two aspects of the Fourier Analysis: (1) The Euclidean Fourier Analysis, where a deep analysis can be performed using specificities as the notion of "frequencies" (involving the Fourier transform) or the geometry of the Euclidean balls. By taking advantage of them, this proposal aims to pursue the study and bring novelties in three fashionable topics: the study of bilinear/multilinear Fourier multipliers, the development of the "space-time resonances" method in a systematic way and for some specific PDEs, and the study of nonlinear transport equations in BMO-type spaces (as Euler and Navier-Stokes equations). (2) A Functional Fourier Analysis, which can be performed in a more general situation using the notion of "oscillation" adapted to a heat semigroup (or semigroup of operators). This second Challenge is (at the same time) independent of the first one and also very close. It is very close, due to the same point of view of Fourier Analysis involving a space decomposition and simultaneously some frequency decomposition. However they are quite independent because the main goal is to extend/develop an analysis in the more general framework given by a semigroup of operators (so without using the previous Euclidean specificities). By this way, we aim to transfer some results known in the Euclidean situation to some Riemannian manifolds, Fractals sets, bounded open set setting, ... Still having in mind some applications to the study of PDEs, such questions make also a connexion with the geometry of the ambient spaces (by its Riesz transform, Poincaré inequality, ...). I propose here to attack different problems as dispersive estimates, ""L^p""-version of De Giorgi inequalities and the study of paraproducts, all of them with a heat semigroup point of view."

940 540 €

Start date: 2015-06-01, End date: 2020-05-31

"Invariance under gauge transformation groups provides the natural structure explaining the laws of physics. In life sciences, new mathematical tools are needed to estimate approximate invariance and establish general but approximate laws. Rephrasing Poincaré: a geometry cannot be more true than another, it may just be more convenient, and statisticians must find the most convenient one for their data. At the crossing of geometry and statistics, G-Statistics aims at establishing the mathematical foundations of geometric statistics and exemplifying their impact on selected applications in the life sciences. So far, mainly Riemannian manifolds and negatively curved metric spaces have been studied. Other geometric structures like quotient spaces, stratified spaces or affine connection spaces naturally arise in applications. G-Statistics will explore ways to unify statistical estimation theories, explaining how the statistical estimations diverges from the Euclidean case in the presence of curvature, singularities, stratification. Beyond classical manifolds, particular emphasis will be put on flags of subspaces in manifolds as they appear to be natural mathematical object to encode hierarchically embedded approximation spaces. In order to establish geometric statistics as an effective discipline, G-Statistics will propose new mathematical structures and theorems to characterize their properties. It will also implement novel generic algorithms and illustrate the impact of some of their efficient specializations on selected applications in life sciences. Surveying the manifolds of anatomical shapes and forecasting their evolution from databases of medical images is a key problem in computational anatomy requiring dimension reduction in non-linear spaces and Lie groups. By inventing radically new principled estimations methods, we aim at illustrating the power of the methodology and strengthening the ""unreasonable effectiveness of mathematics"" for life sciences."

2 183 584 €

Start date: 2018-09-01, End date: 2023-08-31

This research project focuses on the structure, classification and rigidity of three closely related objects: group actions on measure spaces, orbit equivalence relations and von Neumann algebras. Over the last 15 years, the study of interactions between these three topics has led to a process of mutual enrichment, providing both striking theorems and outstanding conjectures. Some fundamental questions such as Connes' rigidity conjecture, the structure of von Neumann algebras associated with higher rank lattices, or the fine classification of factors of type III still remain untouched. The general aim of the project is to tackle these problems and other related questions by developing a further analysis and understanding of the interplay between von Neumann algebra theory on the one hand, as well as ergodic and group theory on the other hand. To do so, I will use and combine several tools and develop new ones arising from Popa's Deformation/Rigidity theory, Lie group theory (lattices, boundaries), topological and geometric group theory and representation group theory (amenability, property (T)). More specifically, the main directions of my research project are: 1) The structure of the von Neumann algebras arising from Voiculescu's Free Probability theory: Shlyakhtenko's free Araki-Woods factors, amalgamated free product von Neumann algebras and the free group factors. 2) The structure and the classification of the von Neumann algebras and the measured equivalence relations arising from lattices in higher rank semisimple connected Lie groups. 3) The measure equivalence rigidity of the Baumslag-Solitar groups and several other classes of discrete groups acting on trees.

876 750 €

Start date: 2015-04-01, End date: 2020-03-31

"The aim of this project of 5 years is to create a research group on geometric control methods in PDEs with the arrival of the PI at the CNRS Laboratoire CMAP (Centre de Mathematiques Appliquees) of the Ecole Polytechnique in Paris (in January 09). With the ERC-Starting Grant, the PI plans to hire 4 post-doc fellows, 2 PhD students and also to organize advanced research schools and workshops. One of the main purpose of this project is to facilitate the collaboration with my research group which is quite spread across France and Italy. The PI plans to develop a research group studying certain PDEs for which geometric control techniques open new horizons. More precisely the PI plans to exploit the relation between the sub-Riemannian distance and the properties of the kernel of the corresponding hypoelliptic heat equation and to study controllability properties of the Schroedinger equation. In the last years the PI has developed a net of high level international collaborations and, together with his collaborators and PhD students, has obtained many important results via a mixed combination of geometric methods in control (Hamiltonian methods, Lie group techniques, conjugate point theory, singularity theory etc.) and noncommutative Fourier analysis. This has allowed to solve open problems in the field, e.g., the definition of an intrinsic hypoelliptic Laplacian, the explicit construction of the hypoelliptic heat kernel for the most important 3D Lie groups, and the proof of the controllability of the bilinear Schroedinger equation with discrete spectrum, under some ""generic"" assumptions. Many more related questions are still open and the scope of this project is to tackle them. All subjects studied in this project have real applications: the problem of controllability of the Schroedinger equation has direct applications in Nuclear Magnetic Resonance; the problem of nonisotropic diffusion has applications in models of human vision."

785 000 €

Start date: 2010-05-01, End date: 2016-04-30