Empirical Mode decomposition introduced by Huang is a recent signal analysis tool, with a lack of well defined foundations. The main of this talk if to try to build a mathematical framework for this tool.

(joint with C. Fermanian Kammerer and I. Gallagher) --- A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Our approach puts into light microlocal directions and completes, with the Littlewood-Paley theory developed in previous works in collaboration with P. Gérard, C.-J. XU and I. Gallagher a microlocal analysis of the Heisenberg group.

We will discuss a generalization of the Rudin- Carleson theorem to a class of real analytic vector fields in the plane. This is a joint work with Jorge Hounie.

Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing $C^{\infty}$ regularity of the sub-Riemannian minimal surface along its Legendrian foliation. This is a joint work with Giovanna Citti and Maria Manfredini.

In this talk, we shall consider the regularity problem for a calss of two-dimensional degenerate Monge-Ampere equations. We know that, in this case, the solution is smooth if the principal curvature of the solution is positive. Here we prove that, under the same conditions, the solution would be Gevery regularity.

This is a joint work with N. Hanges (NY). In this talk we shall present a new approach for the study of analytic hypoellipticity for sum of squares operators based on hyperfunction theory. We shall also present examples which illustrate this technique.

In my talk I will recall some results on the estimates for real vector fields and hypoellipticiy for second order differential equations, particularly the famous paper of L. Hormander and some of the numerous results related to (and far from being complete). Then I consider the complex case particularly few of the results obtained in the last years.

In this talk, we will discuss various gap properties for proper holomorphic maps between generalized balls in complex projective spaces. Many examples and new conjectures will be presented.

in this talk, we will show how geometric methods of microlocal origin may help to study problems concerning kinetic equations, such as the the return to the equilibrium. A typical example for which these methods apply is the Fokker-Planck equation. We will show some results in this direction, and in particular explain the concept of hypocoercivity.

We will discuss a notion of generalized CR function for tubes over measurable sets and discuss when it it is possible to extend CR functions to the convex hull preserving the CR character. This is joint work with G. Hoepfner and L.A. Carvalho dos Santos

We talk about an analogue of Alexander's Theorem for holomorphic mappings of the unit ball in a complex Hilbert space: Every holomorphic mapping which takes a piece of the boundary of the unit ball into the boundary of the unit ball and whose differential at some point of this boundary is onto is the restriction of an automorphism of the ball (it is actually enough to assume that the mapping is only Gateaux-holomorphic). We will also show in some examples why the differential condition is necessary (it is not in the finite-dimensional case).

In the first par of this talk I will review some classical constructions of Cauchy-type singular integrals (for domains in higher dimensional complex space C^n) with kernels that are holomorphic (analytic) in the parameter: these are the higher dimensional analogues of the Cauchy integral for a planar domain (the well-known single and double layer potential operators have kernels that fail to be holomorphic in the parameter, so from the point of view of complex analysis they are not suitable higher dimensional generalizations of the one-dimensional Cauchy kernel.) In the second part of this talk I will present recent joint work with E.M. Stein concerning L^p-boundary regularity of these (holomorphic) Cauchy-type integrals for a class of domains with restricted boundary regularity: this is the higher-dimensional version of Calderon's celebrated result for Lipschitz curves. Time permitting, I will also discuss some applications to Hardy spaces of holomorphic functions.

In this talk, the author will present some recent results on the strictly pseudoconvex pseudo-Hermitian CR manifolds. They include the eigenvalue estimates for sub-Laplacian and CR-Obata type theorem.

Let $L$ be a complex vector field in the plane such that $L$ is elliptic everywhere except along a simple closed closed curve $\Sigma$. We assume that $L$ is of infinite type along $\Sigma$ and that $L\wedge\overline{L}$ vanishes to first order on $\Sigma$. For such vector fields, the equation $Lu=F(u,x,y)$ will be discussed. Series and integral representations of the solutions will be established in a tubular neighborhood of $\Sigma$. An application to local deformation of surfaces in $\mathbb{R}^3$ with positive curvature near a planar point will be given.

There have been extensive studies on the regularity of solutions to the Boltzmann equation coming from the grazing collisions without angular cutoff assumption, for both spatially homogeneous and inhomogeneous cases. As a further study on the problem in the spatially homogeneous situation, in this talk, we consider the Gevrey regularity of ${\mathcal C}^\infty$ solutions with the Maxwellian decay, by using analytic techniques developed in a series of recent collaborations by Alexandre, Morimoto, Xu, Ukai, Yang. The content of this talk is based on the joint work with Ukai.

In this talk, we discuss the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces.

We show that various notions of local homogeneity for CR-manifolds are equivalent. In particular, if germs at any two points of a CR-manifold are CR-equivalent, there exists a transitive local Lie group action by CR-automorphisms near every point.