On the geometry of discontinuous subgroups acting on homogeneous spaces

Ali Baklouti (Université de Sfax)

Abstract: Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup $\Gamma$ for the homogeneous space $M = G/H$ and any deformation of $\Gamma$, the deformed discrete subgroup may utterly destroy its proper discontinuous action on M as H is not compact (except for the case when it is trivial). To emphasize this specific issue, we present in this talk different questions related to the geometry of the parameter and the deformation spaces of any discrete subgroup $\Gamma$ acting properly discontinuously and fixed point freely on $G/H$ for an arbitrary H. 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 $\Gamma\G/H$ in question is assumed to be compact, these spaces have specific properties but can fail to be endowed with a smooth manifold structure.

Estimations de fonctions presque psh sur des variétés de Fano

Adnène Ben Abdesselem (Jussieu et Ecole Polytechnique, Tunisie)

Abstract: On expose une méthode permettant de calculer la constante Tian sur des variétés de Fano. Elle consiste à mettre en évidence une enveloppe inférieure pour les fonctions à sup nul et pluri sous-harmoniques à la métrique près. Cela réduit considérablement les difficultés et permet de conclure, par un calcul facile, sur l'existence de métriques d'Einstein-Kähler si l'invariant de Tian est supérieur à m/(m+1), où m désigne la dimension complexe de la variété.

Titre

Jean-Pierre Demailly (Institut Fourier, Grenoble)

Abstract:

3-connected manifolds of positive 2-curvature

Mohamed Labbi (Université de Bahrein)

Abstract: We show that positive p-curvature is preserved under surgery operations of codimension at least p+3. Consequently, we prove that a compact 3-connected string n-manifold M of dimension at least 9 that is string-cobordant to a manifold of positive 2-curvature has a metric with positive 2-curvature. In particular, if n=11 or 13 then M always has a metric with positive 2-curvature. From another side, we show that a compact non-string 3-connected manifold of dimension at least 8 and with vanishing $\alpha$-genus has a metric with positive 2-curvature. Finally, we discuss some interactions between a natural genus on the string-bordism ring which characterises positive 2-curvature and the Witten genus. This is a joint work with Boris Botvinnick.

The two-body problem at Asymptopia

John M. Lee (University of Washington, Seattle)

Abstract: Solutions to Einstein~Rs 4-dimensional vacuum field equations of general relativity are parametrized, roughly speaking, by solutions to the 3-dimensional constraint equations, which represent "initial data" on a spacelike hypersurface. Using the conformal method introduced by James York and others, the problem of finding solutions to the constraint equations can be reduced to solving a formally determined system of elliptic PDEs on a 3-manifold. An asymptotically hyperbolic initial data sets is a solution to the constraint equations on the interior of a smooth manifold with boundary, whose intrinsic and extrinsic curvatures approach those of the standard spacelike hyperboloid in Minkowski space near infinity. The key feature of such initial data sets is that they admit conformal compactifications obtained by adding a boundary at infinity and conformally rescaling the metric and second fundamental form so that they have smooth (or at least C^2) extensions to the boundary. The asymptotic region near conformal infinity ("Asymptopia") reflects the future asymptotic behavior of the resulting spacetime more closely than an asymptotically flat initial hypersurface does. Gluing techniques have proved to be a fruitful way of constructing new solutions to the constraint equations from old ones. However, when typical gluing techniques are applied to two asymptotically hyperbolic initial data sets, the glued data set will have two disconnected asymptotic regions, which is problematic for physical modeling. In this talk, based on joint work with Jim Isenberg and Iva Stavrov-Allen, I will describe a new gluing construction for asymptotically hyperbolic initial data sets, in which the gluing all takes place near the conformal infinity; it can be interpreted physically as starting with two asymptotically hyperbolic initial surfaces in asymptotically flat spacetimes (perhaps representing two isolated gravitational systems), and gluing them together to produce a new system that contains (slightly perturbed) copies of both of the original ones. The main advantage of our construction is that if we start with two space-times whose asymptotic regions are connected, then the new glued spacetime also has a connected asymptotic region.

Sur la Cohomologie de de Rham p-adique pour les variété algébriques

Zoghman Mebkhout (Jussieu)

Abstract: Le but de cet exposé est de présenter la théorie de de Rham p-adique pour les variétés algébriques définies sur un corps de caractéristique p>0. C'est une occasion de montrer comment les méthodes modernes de la géométrie et de la topologie introduites par A. Grothendieck se sont révélées particulièrement fécondes dans un domaine où les opérateurs de dérivation et d'intégration jouent le role central.

To be announced

Mouayn Zouhair (University Béni Mellal)

Abstract:

On the orbital function of groups of isometries in negative curvature

Marc Peigné (Université de Tours)

Abstract: We study the growth of quotients of Kleinian groups G, i.e. discrete, groups of isometries of a Cartan-Hadamard manifold with pinched negative curvature. More precisely, we will consider quotient groups acting on non-simply connected quotients of a Cartan-Hadamard manifold, so the classical arguments of Patterson- Sullivan's theory are not available here ; this forces us to a more elementary approach, yielding as by-product a new elementary proof for instance of the classical result which states that the orbital functions of a kleinian group G is less than , $e^{\delta_GR}$ where ${\delta_GR}$ is the Poincaré exponent of G. We will give also explicit examples illustrating the fact that the orbital function of such a quotient group may grow quicker than $e^{\delta_GR}$.

Joint work with Francoise Dal'Bo and Andrea Sambusetti

A Poincaré uniformization type-theorem on compact four-manifolds

Mohameden Ould Ahmedou (University Giessen)

Abstract: In this talk we report on some progress for the study the problem of existence of conformal metrics with constant $Q$-curvature on closed four-dimensional Riemannian manifolds. This problem amounts to solve a fourth-order nonlinear elliptic equation involving the Paneitz operator. The corresponding equation has a variational structure, however the associated Euler-Lagrange functional is not bounded from below nor from above in many situations. Furthermore, it does not satisfy the Palais-Smale condition in general. Using an algebraic topological argument, combined with a refined analysis of the loss of compactness, we solve the problem in many cases where blow-up does occur. Precisely, we prove that if the kernel of the Paneitz operator consists only of constant functions, then the above problem is solvable in cases left open after the celebrated works of Chang-Yang (Annals of Maths 1995) and Djadli-Malchiodi (Annals of Maths 2008).

Evolution of starshaped hypersurfaces by general curvature functions

Rachid Regbaoui (Université de Brest)

Abstract: pdf version

Mathematical models of competition and coexistence of microorganisms

Tewfik Sari (Université de Mulhousse et INRIA Sophia-Antipolis)

Abstract: In mathematical ecology, the competitive exclusion principle, sometimes referred to as Gause's Law,is a proposition which states that two species competing for the same resources cannot coexist. Competitive exclusion is predicted by a number of mathematical models, such as the Chemostat models of competition. However, competitive exclusion is rarely observed in natural ecosystems, and many biological communities appear to violate Gause's Law. We discuss some recent results in this field and propose some mechanisms that ensure coexistence of the species.

On the realizability of unstable modules as the cohomology of spaces

Lionel Schwartz (Université Paris 13)

Abstract: The talk will present the main results concerning the representability of unstable modules as the cohomology of spaces. We will discuss some conjectures.

French summary: On présente une un historique des résultas obtenus sur la question de la réalisabilté d'un module instable comme cohomologie d'un espace et quelques conjectures.

Umbilics of surfaces in the Minkowski 3-space.

Farid Tari (Durham University)

Abstract: The Carathéodory conjecture states that any smooth closed and convex surface in the Euclidean 3-space has at least two umbilic points. Various attempts were made to prove this conjecture (see [Guilfoyle and Klingenberg, arXiv:0808.0851v1, 2008] for the latest results on the problem using the mean curvature flow on the space of oriented lines in $\mathbb R^3$). We prove that any closed and convex surface in the Minkowski 3-space of class $C^3$ has at least two umbilic points. For ovaloids, we can even specify the nature of the umbilic points. The induced metric on a closed surface in the Minkowski 3-space must degenerate at some point on the surface. We denote by the $LD$ (the locus of degeneracy) the set of such points. On the $LD$, the tangent plane to the surface is lightlike. This means that the ``normal vector'' to the surface is tangent to the surface. This should complicate matters (we do not have, for instance, a shape operator on the $LD$). However, the presence of the $LD$ makes the situation easier than in the Euclidean case. We use some elementary topological arguments and Poincaré-Hopf theorem to prove the result. (The preprint can be downloaded form http://maths.dur.ac.uk/~dma0ft/Publications.html)