Local analytic classification of q-difference equations

We essentially achieve Birkhoff's program for q-difference equations by giving three different descriptions of the moduli space of isoformal analytic classes. This involves an extension of Birkhoff-Guenther normal forms, q-analogues of the so-called Birkhoff-Malgrange-Sibuya theorems and a new theory of summation. The results were announced in « La variété des classes analytiques d'équations aux q-différences dans une classe formelle » and « Développement asymptotique et sommabilité des solutions des équations linéaires aux q-différences » and in various seminars and conferences between 2004 and 2006.