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.

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)})).

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.

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.

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.

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.