Final Activity Report Summary - MAGNETIZED STARS (Analytic and Computational Relativistic Stellar Magneto Hydro Dynamics) There is a growing consensus in explaining soft gamma repeaters (SGRs) via the magnetar model. Magnetars are believed to be neutron stars with strong magnetic field, such as B greater than 1 013 Gauss. The phenomenon of SGRs may allow us to determine fundamental properties of magnetars in the near future. There already exist at least two sources in which quasi-periodic oscillations (QPOs) have been observed in their X-ray tail. The frequency of many of these oscillations is similar to what one would expect for torsional modes of the solid crust in magnetars. During an SGR event, torsional oscillations in the crust could be excited and lead to the observed frequencies. These signals could be the first evidence of a direct detection of oscillations of neutron stars. Under these conditions, in order to understand the feature of oscillations in magnetar and to explain the observed evidence satisfactorily, we tried to consider the torsional oscillations in magnetars in our project. As a magnetar model to examine we adopted the ideal magnetohydrodynamics (MHD) approximation, with which the electric field for the co-moving observer was vanished due to the infinite conductivity. Since the magnetic field outside of a star was generally considered to be dipole, as suggested by one of the observed evidence, we also considered the dipole magnetic fields inside the star to connect them to the exterior magnetic fields. On these magnetar models we added the axial perturbations, which were corresponding to torsional oscillations, with relativistic Cowling approximation, meaning that the metric perturbations were neglect. This approximation proved to be very robust for axial perturbations since, essentially, this type of perturbations did not involve the density variations, which led to variation in the radiative part of the metric. Through our project we could get deep understanding about the torsional oscillations in magnetars. One of them was that we could find that the Alfven oscillations in magnetar could be continuum and had two families of QPOs. With these two families and the crust oscillation that we computed it was possible to explain all observed QPO frequencies and predict the parameters of the star, i.e. mass, radius, crust thickness and strength of magnetic field. Moreover, we could restrict the magnetic field geometry on magnetars. Namely, we proved that the magnetic field had to permeate the whole star and could not be confined in the crust. Future events could enhance our approach and be used to constrain the parameters of magnetars.