In 1956 A.D. Alexandrov conjectured that if an ovaloid in R^3 satisfies an elliptic equation between its principal curvatures and ist unit normal, then any other compact surface of genus zero immersed in R^3 that satisfies the same equation is a translation of this ovaloid. Our aim in this talk is to give a proof of this conjecture. As a particular case, we also obtain an affirmative solution another classical problem: the characterization of round spheres as the unique elliptic Weingarten spheres immersed in R^3, generalizing in this way previous results by Hopf, Alexandrov, Chern or Hartman and Wintner among others. This is a joint work with Jose A. Galvez.

The Mixmaster Universe has been proposed by Misner (1969) as a model for a chaotic big bang cosmological singularity. This cosmological model describes Bianchi IX spatially homogeneous, anisotropic vacuum space-times. In 1970, Belinskii, Khalatnikov, and Lifshitz (BKL) conjectured that particle horizons form towards the big bang. In other words, backwards light-cones remain spatially bounded, and spatially separate regions causally decouple.   We prove this is indeed the case, for Lebesgue almost every solution. More specifically, the answer to this question depends on the convergence speed towards the Mixmaster attractor. Ringström (2001) showed that this convergence occurs at all, and that curvature blow-up occurs towards the singularity (``cosmic censorship''). We introduce a novel expanding measure in order to prove that the convergence is fast enough to guarantee the formation of particle horizons for Lebesgue almost every solution.