Abstracts

In this joint work with Jean Yves Chemin and Isabelle Gallagher, we prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem: if a sequence  ,  of initial data, bounded in some scaling invariant space, converges weakly to an initial data u₀ which generates a global regular solution, does  generate a global regular solution? Because of the invariances of the Navier-Stokes equations, a positive answer in general to this question would imply global regularity for any data so we introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

We study, extending the techniques of Paul H.Rabinowitz, the issue of existence of periodic orbits for Reeb vector-fields in dimension 3 and the issue of existence of invariants of deformation that can be attached them. These problems have of course been already considered in the past by a large number of other mathematicians (not exhaustively: Viterbo, Hofer, Bourgeois, Oancea, Hutchings, Taubes) using other techniques.

Let G be a Lie group and G a lattice in G.  Let (G, G) denote the space of all homomorphic embeddings of G as a lattice into G. By a result of Weil, (G,G) is an open subset in the space of all homomorphisms from G to G, and Wang  showed subsequently that its components  are smooth manifolds. There is also a natural left-action of Aut(G) on (G,G). The lattice G is rigid in G if and only if Aut(G) acts transitively on (G,G). More generally, the orbit space Aut(G)\ (G,G) provides a quantitative measure for the degree of non-rigidity of G in G.  In this talk, we discuss the above setup for simply connected, solvable Lie groups G. We are particularly interested in describing principal situations where G is deformation rigid, that is, situations, where the space Aut(G)\ (G,G) is finite our countable.

We present in this talk different questions related to the geometry of the parameter and the deformation spaces of a discrete subgroup  of a lie group G, acting properly discontinuously and fixed point freely on a homogeneous space G/, where H stands for an arbitrary closed subgroup of G. The topological features of deformations, such as rigidity and stability are also discussed, namely the analogue of the Selberg-Weil-Kobayashi rigidity Theorem in the non-Riemannian setting. Whenever the Clifford-Klein form \G/H in question is assumed to be compact, these spaces have specific properties but can fail to be endowed with a smooth manifold structure.

Dans cet exposé nous nous intéressons aux espaces temps dit globalement hyperboliques Cauchy compacts. Ce sont des espaces temps qui admettent une fonction, dite fonction temps, propre surjective qui croit strictement le long des courbes causales inextensibles. Les niveaux de telles fonctions sont des hypersurfaces de type espaces appelés hypersurfaces de Cauchy. La donnée d'une fonction temps définit naturellement une famille à 1-paramètres d'espaces métriques. Notre but est d'étudier le comportement asymptotiques de ces familles d'espaces métriques. Il y a deux cas de figure à considérer : le premier étant le comportement asymptotique dans le passé; le deuxième est celui du comportement asymptotique dans le future. Plus de conditions géométriques sur l'espace-temps (courbure constante) et les fonctions temps à considérer (convexité) seront nécessaires.

Given a nonnegative  function f of d variables, is it possible to write it as a finite sum of squares f=  of functions having a given degree of regularity? The answer depends on the dimension d and on the required degree of regularity. For any d, a result of Phong and Fefferman asserts that one can choose g_j  C^{1,1} (functions whose gradient is Lipschitz continuous). This result is the best possible for d> 3. On the other hand, for d=1 and for any m, one can choose g_j   ^m. Precise results, based on a joint work with Ferruccio Colombini and Ludovico Pernazza, are given for d=2 and d=3

CR Geometry and Mappings in Several Complex Variables

In this talk, we will discuss the problem of prescribing the Q-curvature in odd dimensions. We will first present a classification of solutions in the case of the Euclidean space, we then investigate the case of prescribing singularities. At the end, if time permits we will show how this problem relates to the study of the fractional Moser-Trudinger inequality.

We discuss the solvability and normalization, in the real analytic and smooth categories, of a class of vector fields in a neighborhood of an invariant torus. The vector fields are supposed to satisfy Siegel type conditions.

In 1968, Artin proved his famous approximation theorem: given any system of real-analytic equations, if there exists a formal solution to such a system at a given point, then there exists a real-analytic solution that is as close as we want in the Krull topology to the formal solution. One question that naturally thereafter rises is whether the conclusion of Artin’s approximation theorem is still preserved if the system of equations is coupled with a specific PDE. In 1978, Milman investigated such a question when th PDE consists of the standard CR operator in ²ⁿ= ⁿ: he showed that any formal solution of a system of real-analytic equations and of the standard CR equations in ⁿ can be approximated (in the Krull Topology) by a sequence of convergent solutions of the system of analytic and CR equations. In this talk, we will discuss recent results generalizing Milman’s theorem when the standard CR operator in ⁿ is replaced by the tangential CR operator associated to a real-analytic CR manifold.

Conformal metrics of prescribed Q-curvature on 4-manifolds

In this talk we address the question of existence, on a four dimensional Riemannian manifold, of conformal metrics of prescribed Q-curvature. The Q-curvature is a generalization on four manifolds of the two dimensional Gauss curvature. This scalar quantity turns out to be helpful in the understanding of the topology and geometry of four dimensional manifolds. This problem amounts to solving a fourth order nonlinear PDE involving the Paneitz Operator. This PDE enjoys a variational formulation; however the corresponding Euler-Lagrange functional does not satisfy the Palais-Smale condition. In this talk we will report on recent existence results obtained through a Morse theoretical approach to this non-compact variational problem combined with a refined analysis of the singularities of the corresponding gradient flow.

La régularité des applications de transport optimales en  géométrie riemannienne est liée à la positivité d'un tenseur introduit par Ma, Trudinger et Wang en 2005. Nous expliquerons comment surgit ce tenseur et en quoi sa positivité contraint la géométrie de la variété.

Highlights of Salah Baouendi's contributions to mathematics and to the mathematical community.

Rigidity of CR maps between Shilov boundaries of bounded symmetric domains

Rigidity of CR maps between real hypersurfaces has attracted considerable attention, yet for the higher codimension CR manifolds only little is known. In this joint work with S.Y.Kim we consider maps between Shilov boundaries of bounded symmetric domains of higher rank. The result states that any such CR embedding is the standard linear embedding up to CR automorphisms. Our basic assumption extends precisely the well-known optimal bound for the rank one case. There are no other restrictions on the ranks, in particular, the difficult case when the target rank is larger than the source rank is also allowed.