The complex interplay between Coulomb repulsion and Fermi statistics in two dimensional systems is responsible for some of the most dramatic phenomena encountered in solid state physics (High critical temperature superfluidity, Fractional Quantum Hall Effect,..). However, despite decades of efforts, many questions regarding these systems are still unsolved. In FERLODIM, we plan to take advantage of recent progress in ultracold gases, to simulate several fundamental Hamiltonians describing these many-body systems in 1 and 2 dimensions. We will realize two ultra-cold atom machines allowing for a full characterization of the many-body wave function of an ensemble of interacting fermions in periodic potentials, called optical lattices. Our experiments will rely on a high resolution imaging system allowing both for single atom detection and the possibility of tailoring optical potentials of arbitrary shape and geometry. This unique design will allow us to address a variety of physical situations, depending on the geometry of the light induced potentials. One-dimensional problems will be addressed, from spin chains to Luttinger liquids. In pure two dimensional configurations, we will investigate the link between the repulsive Hubbard model, superfluidity and the Mott insulator transition, as well as frustration effects in periodic potentials. Finally we will explore the physics of interacting fermions under rotation in the lowest Landau level, and the connection with fractional Quantum Hall systems.