Toroidal b-divisors and Monge-Ampère measures

A b-divisor on a projective algebraic variety X is an element in the projective limit of divisors, indexed over all proper modifications of X, and satisfying some functoriality properties. Amongst all b-divisors, the so called "toroidal" ones can be studied using convex-geometrical and tropical techniques. The aim of the talk is to give an overview of the theory of toroidal b-divisors and to give interesting complex-geometric and arithmetic applications.