This is a mobility project for Dr David Sauzin, a most experienced French CNRS researcher in mathematics, for two years in Pisa, at the SNS (CRM research center E. de Giorgi), to collaborate with prof. Stefano Marmi. The goal of the project is to develope innovative applications of Ecalle's theory of resurgent functions and Borel's theory of monogenic functions in various problems of asymptotic analysis: especially in holomorphic dynamical systems (small divisor problems) and number theory (modular forms and elliptic functions). Few mathematicians have a command of these two powerful theories, which were up to now mostly French specialities and of which Dr. Sauzin has become a recognized expert. This project will be the occasion of a synergy between French and Italian competencies. Dr Sauzin will participate to the activities of the CRM's research group "Holomorphic dynamical systems and arithmetic", developing also scientific relations with prof. Marco Abate. He will also benefit from the unique scientific environment provided by Pisa's SNS, in mathematics as well as physics, e.g. appropriating tools from variational methods and geometric measure theory and developing his interest in perturbative Quantum Field Theory. In Pisa, Dr Sauzin will himself teach a course on resurgent functions and related topics at the SNS, tutor master students and PhD students, and organize an international conference in October 2009 and another one at the end of his stay. This project will contribute to enhance European scientific excellence and enrich the scientific life of one of the best poles in Italy, with interesting prospects in terms of exchanges of students between Pisa and Paris after Dr. Sauzin's re-establishment in France, where this added value to his career will help him to reach a higher position in CNRS.