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This research Project continues the geometric study of algebraic, symplectic and arithmetic varieties (DGICYT PS90-0069, PB93-0790, PB96-0234, BFM2000-0799, BFM2003-06001, BFM2003-02914, and DGI MTM2006-14234, MTM2012-38122-C03/FEDER) as well as the applications to Robotics, Computational Biology and Material Science started in the last project. The increasing interdisciplinarity of modern research and the fact that the boundaries between different areas of mathematics are vanishing, with a constant transfer of problems and techniques between them makes it difficult to progress without a multidisciplinary approach. In the present project we gather together experts in Algebraic, Symplectic and Arithmetic Geometry to stimulate the interaction between them and to allow the study of each object from different points of view.