For any smooth complex projective variety $X$ and any smooth very ample hypersurface $Y\subset X$, we develop the technique of genus zero relative Gromov-Witten invariants of $Y$ in $X$ in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps which relates these relative invariants to the Gromov-Witten invariants of $X$ and $Y$. Given the Gromov-Witten invariants of $X$, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of $Y$ whose homology and cohomology classes are induced by $X$.

We formulate a conjecture giving a link between the various rings parametrizing the $2$-dimensional potentially semistable $p$-adic representations of ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ with Hodge-Tate weights $(0,k-1)(k\in \mathbf {Z},1<k<p)$ having the same reduction modulo $p$ and the representations of ${\rm GL}\sb 2(\mathbf {Z}\sb p)$ that are used, via compact induction, to build the smooth irreducible representations of ${\rm GL}\sb 2(\mathbf {Q}\sb p)$. We prove this conjecture for semistable representations and $k$ even. In doing this, we obtain precise results on the restriction to ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ of the representations of ${\rm Gal}(\overline {\mathbf {Q}}/\mathbf {Q})$ over $\overline {\mathbf {F}}\sb p$ that are associated to modular forms on $\Gamma\sb 0(pN)(p\nmid N)$ of weight smaller than $p$. In an appendix, G. Henniart determines which smooth irreducible representations $\lambda$ of ${\rm GL}\sb 2(\mathfrak {O}\sb F)$ are typical for ${\rm GL}\sb 2(F)$ (where $F$ is a locally compact non-Archimedean field and $\mathfrak {O}\sb F$ its ring of integers), in the sense that there is a component $s(\lambda)$ in the Bernstein decomposition (for the category of smooth representations of ${\rm GL}\sb 2(F)$) such that $\lambda$ appears only in smooth representations of ${\rm GL}\sb 2(F)$ belonging to $s(\lambda)$. We need Henniart's results, in the case $F=\mathbf {Q}\sb p$, to state the aforementioned conjecture.

We prove a vanishing theorem for a convex cocompact hyperbolic manifold which relates its L²-cohomology and the Hausdorff dimension of its limit set. The borderline case is shown to characterize the manifold completely.

We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifold, based on M. Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.

The main results of this paper are concerned with the "bad" behavior of the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces. This leads to some kind of ill-posedness of the corresponding Cauchy problem: the flow map cannot be of class $C\sp 2$ in any Sobolev space.

In this paper, we prove that there exist no blow-up solutions of the critical generalized Korteweg‒de Vries (gKdV) equation with minimal L²-mass, assuming an L²-decay on the right on the initial data.