| The classical Torelli map is the modular map from the moduli space of smooth projective curves of genus g to the moduli space of principally polarized Abelian varieties of dimension g, mapping a curve into its Jacobian variety together with its theta divisor. The Torelli theorem asserts that this map is injective and its image has been described in several different ways (the so-called Schottky problem). I will present a joint work in colaboration with Silvia Brannetti and Filippo Viviani in which we establish tropical analogues of these classical results. In particular we construct the moduli space of tropical curves of genus g and the moduli space of Abelian varieties of dimension g and we define the tropical Torelli map. We then study the fibers of this map (tropical Torelli) as a Corollary of a Theorem of Caporaso-Viviani ans we characterize its image (tropical Schottky). If time permits I will show how to apply these results to give a solution to a question raised by Namikawa on the compactified classical Torelli map. |