This thesis manuscript presents experimental results on Bose-Einstein condensates in optical lattices and in the regime of fast rotation. In an optical lattice an array of inde-pendent 2-dimensional Bose-Einstein condensates is produced from an initially 3-dimen-sional cloud. The number of 2D BECs in this array can be controlled by superimposing a magnetic field gradient along the axis of the lattice and then selectively evaporating the sites. When producing a small number of sites and letting them interfere, dislocations in the interference pattern were observed, which must be due to nontrivial phase defects, as for example free vortices. Such vortices and vortex-antivortex pairs play an important role in the superfluid to normal phase transition in a 2-dimensional gas, the Kosterlitz-Thouless transition. With 3-dimensional condensates the limit of fast rotation is studied in a hybrid potential contain-ing a small quartic term. In this trap rotation frequencies larger than the harmonic trapping fre-quency can be reached. These experiments are a first step to a possible observation of strongly correlated states close to the regime of the fractional quantum Hall effect.
