The energy-critical defocusing NLS on T^3

We prove global well-posedness in H^1(T^3) for the energy-critical defocusing NLS.Comment: 24 page

The Euler--Poisson system in 2D: global stability of the constant equilibrium solution

We consider the (repulsive) Euler-Poisson system for the electrons in two dimensions and prove that small smooth perturbations of a constant background exist for all time and remain smooth (never develop shocks). This extends to 2D the work of Guo.Comment: 39 page

On the local extension of Killing vector-fields in Ricci flat manifolds

We revisit the problem of extension of Killing vector-fields in smooth Ricci flat manifolds, and its relevance to the black hole rigidity problem

Global solutions for the gravity water waves system in 2d

We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu \cite{WuAG}. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case \cite{GMS2,Wu3DWW}. In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2d.Comment: final version to be publishe