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Magnetic vortices are topologically stable magnetization configurations at the nanoscale. When subject to a current polarized in spin, exchange interaction with the itinerant electrons, modify the configuration of the fixed spins. The spin-transfer torque effect, as experimentally observed, is able to change the magnetization topology. We investigate the relation between the dynamics of magnetic vortices driven by a spin-polarized current and the changes in the magnetization topology. These transitions are characterized by the appearance of magnetic singularities.