Classification of meromorphic differential equations

Abstract. K = C(z})  is the field of the convergent Laurent series in z (or, equivalently, the germs of meromorphic functions at z = 8). The classification of differential modules (or matrix differential equations) over K is one of the major achievements of the theory of asymptotic analysis and multisummation. This lecture tries to explain the ideas in a didactic way.  k-summation is introduced in an elementary way and the more involved multisummation is used a black box. Differential Galois of meromorphic differential equations and moduli problems are explained in this context.