Lecturer: Jacques-Arthur Weil (Université de Limoges, France). Title: Applications of symbolic methods for solving systems of linear ordinary differential equations. Abstract: This course concerns linear differential systems of the form and their applications. In a first part, we introduce tools to describe the algebraic properties of solutions of such systems. This will allow us to establish constructive methods and algorithms. The main theoretical object is the differential Galois group. We present recent methods and algorithms to describe it: how to compute its invariants (as rational solutions of linear differential systems), how to factor such systems (i.e transform them to triangular form when possible), the eigenring method to compute decompositions (i.e put a system in block diagonal form when possible). This is closely linked to the lectures of M. Barkatou. We will then proceed to more advanced methods, centered around the notion of reduced form of a linear differential system : we will discuss the advantage of these forms and how to compute them. We will also mention the analytical methods, with links to the lectures of J.-P. Ramis and C. Mitschi. In a second part, we will show some applications of these methods to the integrability of dynamical systems, specially hamiltonian systems : variational methods, integrability criteria, formal first integrals along solutions (works of Morales and Ramis, Simó, Aparicio-Monforte and myself, etc). If time permits, we will also illustrate these techniques with applications to quantum mechanics (work of Acosta-Humánez, Morales and myself, etc). Of course, all parts will be full of examples and exercises illustrating all these situations.