Laboratoire International Associe (LIA) based on an agreement between the Aix-Marseille University, the CNRS (Centre National de la Recherche Scientifique), the University of Aix-Marseille and the Ecole Centrale de Marseille, on the french side, and the INdAM (Istituto Nazionale di Alta Matematica), on the italian side. Laboratoire hYpatia de Sciences Mathematiques (LYSM) will support french-italian collaborations, scientific exchanges and visits, in different areas of pure and applied mathematics, including: - Algebraic Geometry - Analysis - Complex Differential Geometry - Ergodic Theory & Dynamical systems - Logic - Mathematical Physics and Statistical Mechanics - Non Commutative Geometry - Number theory - Probability Theory and Interacting Particles Systems