Résumé
I will start to motivate few recent results on Poincaré series from a naive personal viewpoint. Then I will report on joint work with Yannick Bonthonneau, Matthieu Léautaud, Gabriel Rivière where we consider Poincaré series on the torus which count orthogeodesics between convex bodies. When the convex are analytic, the Poincaré series has analytic extension through the imaginary axis as a multivalued holomorphic function with an infinite number of branching points. We determine explicitely the monodromy around these singular points.
