# Abstracts of the talks

Z. ADWAN (University of Texas at Brownsville)

On Microlocal Analyticity and Smoothness of Solutions of First Order Nonlinear PDEs

We study the microlocal analyticity and smoothness of solutions u of of the nonlinear pde u_{t}=f(x,t,u,u_{x}) under some assumptions on the repeated brackets of the linearized operator and its conjugate. We also generalize a result obtained recently by N. Lerner, Y. Morimoto, and C.-J. Xu. This is joint work with S. Berhanu.

S. ASSERDA (University Ibn Tofail, Kenitra, Morocco)

The degree of holomorphic approximation on a totally real set

Let E be a totally real set on a Stein open set $\Omega$ on a complete noncompact Kahler manifold $(M,g)$ with nonnegative holomorphic bisectional curvature such that $(\Omega,g)$ has bounded geometry at E. Then every function $f$ in a $C^p$ class with compact support on $\Omega$ and dbar-flat on E up to order $p-1$, $p\geq 2$ (respectively, in a Gevrey class of order s>1, with compact support on $\Omega$ and dbar-flat on E up to infinite order) can be approximated on compact subsets of E by holomorphic functions $f_k$ on $\Omega$ with degree of approximation equal to $k^{-p/2}$ (respectively, exp (-c(s)k^{1/2(s-1)})).

P. CORDARO (University of Sao Paulo)

Gevrey vectors in locally integrable structures

In this talk we shall present recent results, obtained in collaboration with J.E.Castellanos and G.Petronilho, concerning the analytic (resp. Gevrey) regularity of analytic (resp. Gevrey) vectors associated to system of vector fields defined by locally integrable tube structures. We shall also explain how these results can be applied to the study of regularity of solutions to certain systems of semi-linear PDE.

M. DERRIDJ

Microlocal subellipticity for systems of vector fields

We are interested in the existence of microlocal subelliptic estimates for some systems of complex vector fields which are special classes of locally integrable systems. In the case of codimension 1, the situation is quite clear: the necessary condition of F.Treves for microlocal hypoellipticity, he gave in the seventies, is also sufficient in case the coefficients are analytic,by a result of H Maire. H. Maire gave also an example showing that this is not the case for higher codimension systems. So we study also some classes of vector fields, even in higher codimension case, for which we obtain microlocal suellipticity (and also maximal estimates) hence microlocal hypoellipticity, for these systems and the associated second order operators. This is a joint work with Bernard Helffer.

P. EBENFELT (University of California at San Diego)

Transversality of holomorphic mappings between CR manifolds

Transversality is an important notion in geometry. We shall
consider transversality in the context of mappings between generic
submanifolds in complex space. We will present some new results that extend
previous results and answers questions posed by the speaker and Linda
Rothschild some five years ago. This joint work with Son Doung.

A. ELKHADIRI (University of Kenitra)

Homomorphisms of quasianalytic local rings

Let $\mathcal{C}_n$ be a local quasi-analytic subring of the ring
of germs of $C^\infty$ functions on $\mathbb{R}^n$, and let
$\mathcal{C}=\{ \mathcal{C}_n\,,\, n\in\mathbb{N}\}$. We suppose
that $\mathcal{C}$ is closed under composition. Consider a map
$\varphi : (\mathbb{R}^n, 0)\rightarrow (\mathbb{R}^k, 0)$ vanishing
at zero, where $\varphi$ is a k-tuple $
(\varphi_1,\ldots,\varphi_k)$ and $\varphi_1,\ldots,\varphi_k$ are
in $\mathcal{C}_n$. Then $\varphi$ defines uniquely a map $\phi:
\mathcal{C}_k \rightarrow \mathcal{C}_n$ by composition, and $\phi$
induces a morphism $\hat{\phi}: \hat{\mathcal{C}_k }\rightarrow
\hat{\mathcal{C}_n}$ between completions. We let $\phi_* :
\frac{\hat{\mathcal{C}_k} }{\mathcal{C}_k
}\rightarrow\frac{\hat{\mathcal{C}_n }}{\mathcal{C}_n }$ be the
homomorphism of groups induced by $\phi$ and $\hat{\phi}$ in the
obvious manner. In the analytic case, i.e when each $\mathcal{C}_n$
is the ring of germs of real analytic functions, M. Eakin and A.
Harris gave a condition under which $\phi_*$ is injective. In this
talk we prove that the same statement does not hold for a
quasianalytic system unless this system is analytic.

J. GLOBEVNIK (University of Ljubljana)

Small families of complex lines for testing holomorphic extendibility

Let a, b be two points in the open unit ball B in C^2. It has been known for a while that for each k there is a function f in C^k(bB) which extends holomorphically into B along any complex line passing through either a or b yet f does not extend holomorphically through B. In the talk we will present the recent result that there is no such f which is infinitely smooth and give a rather complete description of pairs of points a, b in C^2 such that whenever an infinitely smooth function on the unit sphere bB extends holomorphically into B along each complex line that passes through either a or b and meets B, then f extends holomorphically through B.

H. GUEMRI (University of Tunis)

Existence results for critical semilinear equations on Heisenberg group domains

Following the work of G. Citti and F. Uguzzoni who studied Yamabe type problems on Heisenberg group domains, we shall study a certain critical semilinear equation on the one dimensional Heisenberg group.

J. HOUNIE (University of Sao Carlos)

Extension of CR functions on rough tubes

In this talk we will describe some generalizations of Bochner's extension theorem. In the classical setting, the function that one wishes to extend is defined on a tube $U+iR^m$ over an open set $U$ of $R^m$ and the case of tubes over embedded manifolds of class $C^2$ is also known. Here we consider tubes $X+iR^m$ where $X$ is not necessarily a manifold. This is joint work with S. Berhanu.

X. HUANG (Rutgers university)

A codimensional two real submanifold with a symmetric elliptic CR singular point

We study the pseudonormal form for a real submanifold with a CR
singularity. We also generalize a result of Moser using the rapid
convergent power series method.

B. LAMEL (University of Vienna)

Automorphism groups of minimal CR manifolds

We show that the local automorphism group of a minimal CR manifold M is a finite dimensional Lie group if and only if M is holomorphically nondegenerate. (Joint work with R. Juhlin)

L. LANZANI (University of Arkansas at Fayeteville)

Div-Curl type inequalities for Hodge systems: Old and New

I will review a 2005 result (joint with E.M. Stein) concerning L^1-to-L^p inequalities for the canonical solution of the classical Hodge system:
dZ=f; d^*Z=g with the data subject to the compatibility conditions: df=0; d^*g=0. (Here d denotes the exterior derivative operator in Euclidean space).
I will then discuss recent work (joint with A. Raich) that aims at generalizing these inequalities to Hodge-type systems for higher order differential operators.

C. LAURENT (University of Grenoble I)

Stability of embeddability under perturbation of the CR structure for compact CR manifolds

Given a compact CR manifold, we are looking for sufficient geometrical conditions such that the embeddability property remains stable under small perturbations of the CR structure.

J. LEITERER (Humboldt university)

Estimates for the Cartan lemma on holomorphic matrices

We start with the well-known Cartan lemma
on factorization of holomorphic matrices
defined on the intersection of two rectangles.
We prove a uniform estimate for the factors
depending on the size of the rectangles (which is
quite simple, but sharp, in some sense).
Then we discuss such estimates for more general
cocycles of holomorphic matrices.

L. LEMPERT (Purdue University)

Direct images under not necessarily proper maps

Consider a holomorphic submersion $p:Y\to S$ of complex manifolds,
with fibers $Y_s$, and a holomorphic vector bundle $E\to Y$.
Often there is a natural way to organize the spaces $H^0(Y_s, E)$,
$s\in S$, of holomorphic sections into a complex vector bundle
over $S$. For example, according to Grauert, this is so if $p$
is proper and the spaces of sections all have the same dimension.
In a joint work with R. Sz\"oke, we study non proper (or improper?)
submersions. Assuming $E$ has a Hermitian metric, and there is
a volume form at hand with respect to which to integrate, in a
non proper situation it is better to use the spaces $H_s$ of
holomorphic $L^2$ sections of $E|Y_s$. The question is
whether one can organize these Hilbert spaces into a Hilbert
bundle over $S$. I will discuss what type of structure the
collection $H_s$ has, and give examples when this structure
is equivalent to a Hilbert bundle.

N. LERNER (University of PARIS VI)

Hypoellipticity for a class of kinetic equations

We prove some hypoellipticity results with sharp exponents for a class of kinetic equations related to the linearization of the Boltzmann equation without angular cutoff. The method of proof is using some elements of the Wick calculus of pseudodifferential operators.

R. MEZIANI (University of Kenitra)

Automorphisms of holomorphic foliations

Let F_\Omega be a germ at the origin of C^n of holomorphic singular foliation defined by \Omega = 0 (with \Omega integrable). We are interested in the group Aut(F_\Omega) of germs at the origin of C^n of holomorphic diffeomorphisms \Phi fixing the foliation F_\Omega : \Phi * \Omega ^ \Omega =0. Some of them fix the leaves of our foliation (we denote them by Fix(F_\Omega)). In the case where F_\Omega has an holomorphic first integral we caracterize geometric properties of the foliation by studying the quotient
Aut(F_\Omega ) / Fix(F_\Omega).

G. MENDOZA (Temple University)

Complex b-manifolds and their boundaries

A complex b-manifold is a manifold M with boundary together with an involutive sub-bundle $L$ of the complexification of its b-tangent bundle such that L and its conjugate are in direct sum. The boundary of such a manifold inherits a structure which in the compact case resembles that of a circle bundle of a Hermitian holomorphic line bundle over a compact complex manifold. In my talk I will give a brief introduction to complex $b$-manifolds, describe how the structure on the boundary is obtained, and present classification theorems for such structures generalizing the classification of complex line bundles by their Chern class and of holomorphic line bundles by the Picard group. These classification theorems permit the construction of new complex $b$-manifolds out of a given one.

N. Mir (University of Rouen)

Mini-course: Holomorphic mappings of real-algebraic CR manifolds

We will discuss several algebraic properties of holomorphic mappings between real-algebraic CR submanifolds in complex space.

S. NIVOCHE (University of Nice)

Polynomial convexity

We discuss a version of the Hilbert Lemniscate Theorem in C^n : any polynomially convex compact subset K of C^n can be approximated externally
by special polynomial polyhedra P defined by proper polynomial mappings from C^n to C^n with almost all their zeros in P. We precise this version when the compact set is balanced. We give several applications of these results about the pluricomplex Green function and functions in L^+.

F. SAHRAOUI (University of Sidi Bel Abbes)

The dynamics of holomorphic germs tangent to the identity near a smooth curve of fixed points

Let $ f \in $End (\C^2 ,O)$ be tangent to the identity
and with a order $\nu(f)\geq 2$. We try to study the dynamics of
$f$ near the set of his fixed points. Using some results of
Abate, we prove that if the set of fixed
points of $f$ is smooth at the origin, $f$ is tangential to this
set, and the origin is not singular, then there are no parabolic
curves for $f$ at the origin. After that and using some
techniques and results of Hakim, we prove that if the set of fixed points of
$f$ is smooth at the origin and this last one is a singular
point of $f$, with the pure order of $f$ $\nu_0(f) = 1$,
then there exist $\nu(f)-1$ parabolic curves for $f$ at the
origin. Finally and using always the same results of Hakim, we prove that if $O$ is
dicritical, then there exist infinitely many parabolic curves.

S. SAHUTOGLU (University of Toledo)

Irregularity of the Bergman projection on worm domains in $C^n$

We construct higher-dimensional versions of the Diederich-Fornaess worm
domains and show that the Bergman projection operators for these domains are not
bounded on high-order $L^p$-Sobolev spaces for $1\leq p<\infty.$ This is joint work
with David Barrett.

M.-C. SHAW (University of Notre-Dame)

The Cauchy-Riemann equation on curved manifolds

In this talk we study the closed range property and boundary regularity of the Cauchy-Riemann equations on domains in complex manifolds with emphasis on the role of the curvature condition. We will discuss the case for pseudoconcave domains as well as the recent results on product domains (joint work with Debraj Chakrabarti).

L. STOLOVITCH (University of Nice)

Normal forms of analytic perturbations of quasihomogeneous vector fields : Rigidity and analytic invariants sets.

We study germs of holomorphic vector fields which are "higher order"
perturbations of a quasihomogeneous vector field in a neighborhood of the origin
of $\Bbb C^n$, fixed point of the vector fields. We define a "diophantine
condition'' on the quasihomogeneous initial part $S$ which ensures that if such
a perturbation of $S$ is formally conjugate to $S$ then it is also
holomorphically conjugate to it. We study the normal form problem relatively to
$S$ as well as the existence of germs of analytic invariant sets at the origin.

B. STENSONES (University of Michigan at Ann Arbor)

Regions of attraction

We shall look at some open problems and answers about regions of attraction in C^2 and C^3.

F. TREVES (Rutgers university)

Remarks on the solvability of complex vector fields with critical points

The lecture will discuss what is known (to the lecturer) about the local solvability (or lack thereof) of various classes of smooth, complex vector fields that vanish on some nonempty subset of the base manifold. The basic definitions, local solvability, ellipticity, hypoellipticity will be recalled. A special class of vector fields, based on the microlocal definition of 'principal type', wil be shown to have advantages: stability and the role of the linear part of the vector field at its critical set. The latter points to the importance of vector fields in Euclidean space with coefficients that are complex linear functions. In 3 dimensions or higher nothing is known even in such a simple situation - except when the linear functions are real. In 2D the known facts about the linear coefficients class, as well as the class of smooth vector fields that are elliptic off their critical set, will be explained.

D. ZAITSEV (Trinity College)

Dynamics of one-resonant biholomorphisms

This is a joint work with F. Bracci.
We construct a simple
formal normal form for holomorphic
diffeomorphisms in $C^n$ whose
differentials have one-dimensional family of resonances in
the first $m$ eigenvalues, $m\leq n$ (but more resonances are
allowed for other eigenvalues).
We further provide invariants and give conditions
for the existence of basins of attraction.

C. ZUILY (University of Paris Sud)

Cauchy problem, smoothing and Strichartz estimates for the water waves equations

In this joint work with T.Alazard and N.Burq we are interested with the equations describing the water waves. This is a system of hyperbolic type which has been investigated by many authors. In our work we first improve the former results on the Cauchy problem in term of regularity indices for the initial conditions and second we show in the case of 2d waterwaves that these equations enjoy dispersive properties as the Kato smoothing effect and the Strichartz estimates.