Strong correlations in low dimensional systems

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I describe in these notes the physical properties of one dimensional interacting quantum particles. In one dimension the combined effects of interactions and quantum fluctuations lead to a radically new physics quite different from the one existing in the higher dimensional world. Although the general physics and concepts are presented, I focuss in these notes on the properties of interacting bosons, with a special emphasis on cold atomic physics in optical lattices. The method of bosonization used to tackle such problems is presented. It is then used to solve two fundamental problems. The first one is the action of a periodic potential, leading to a superfluid to (Mott)-Insulator transition. The second is the action of a random potential that transforms the superfluid in phase localized by disorder, the Bose glass. Some discussion of other interesting extensions of these studies is given.