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Sergei SHANDARIN, Roya MOHAYAEE Abstract:The currently popular cosmological models assume that the most of the mass in the universe able to cluster is in the form of cold dark matter. The nature of dark matter is one of the biggest challenges in modern cosmology and physics. The smallest elements of the structure in these models are dark matter halos. The vast majority of these halos are merged into galaxies and clusters of galaxies and most of them are destroyed by the tidal forces there. However, the high resolution Nbody simulations indicate that in some models as many as 10^15 of these halos may survive in our galaxy until present time. The density profile of these halos must contain sharply peaked maxima corresponding to the caustics in the infalling dark matter. We derive the maximum density in the caustic regions in the presence of very small but finite thermal velocities. These density peaks may play an important role if the dark matter particles annihilate. 
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Patrick IGLESIAS Abstract: I will present the construction of the "space of rays" of a riemmannian, or pseudo riemmannian, manifolds. Beginning with the standard construction, by homoegenisation of the variatiational associated problem, I'll tell how to obtain the space of unparametrized geodesics, the space of rays, by reducing the space of parametrized geodesics by the group of affine real transformations. I will then define "sheaf of rays" as legendrian submanifolds in the sphere bundle, projecting as lagrangian submanifolds in the space of rays. I'll show, finally, how caustics can be interpreted as well as apparent contour in the sphere bundle than as nontransverse locus of lagrangian manifolds in the space of rays. I'll discuss some related constructions: caustics of sheaf of light or caustics of flux of matter, diffusion of light through symplectomorphism... Uriel FRISCH 
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Thomas BUCHERT Abstract: Caustic formation in an irrotational dust continuum that moves under inertia has provided the key to a classification of generic building blocks of largescale structure in the Universe, following the successful Zel'dovich approximation to capture the early nonlinear stages of gravitational instability. We review this kinematical picture, comment about its dependence on initial conditions, and illustrate caustic metamorphoses in one, two, and three spatial dimensions. The role of (Newtonian) gravity in this picture, as we understand it to date, is discussed and exemplified using higherorder Lagrangian perturbation schemes and numerical simulations. Also, mathematical aspects concerning Lagrangian and Legendrian singularities are put into perspective. Stéphane COLOMBI Abstract: I'll discuss about various methods to solve directly VlasovPoisson equations in phasespace. After a general introduction, where I will list all the possible approaches known up to now, I'll pay particular attention to the cold case, which is quite relevant in cosmology. I'll then talk in details about 1D1D solvers I have been studying with C. Alard and J. Touma, respectively: one relying on the ``cloudy'' method and another one relying on the waterbag method. Finally, I'll discuss about the appearance and the evolution of caustics. 
°°°°° Scientific discussions, excursions and beach

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Roya MOHAYAEE, Sergei SHANDARIN, Raphael GAVAZZI, Bernard FORT Abstract: Caustics are highdensity structures that form frequently in collisionlessmedia. Under selfgravity, cold dark matter flows focus onto caustics which are yet to be resolved in numerical simulations and or observed in the real world. If detected, caustics would provide a strong evidence for dark matter and would rule out alternative models such as those with modified dynamics. Here, we demonstrate how they might be observed in the weak lensing data. We evaluate the shear distortion and show that its radial profile is marked by a characteristic sawtooth pattern due to the caustics in dark matter haloes that form by selfsimilar accretion. We discuss the observational complications, mainly due to the poor knowledge of the virial radii of the haloes and demonstrate that a superposition ofabout 600 clustersize haloes would give a signaltonoise ratio which is sufficiently large for the detection of caustics with groundbased observations. This number is reduced to 200 for spacebased observations. These bounds can be easily achieved by the ongoing wide field optical surveys such as CFHTLS and the future spacebased projects SNAP and DUNE which have to be accompanied by a Xray followup of the selected clusters for a precise determination of their virial radii. Joe SILK 

