The convolution product of two generic conjugacy classes of the unitary group is described by a probability distribution on the space of central measures of the group. In this talk, I will report a recent result which provides an explicit manifestly positive formula for the density of this measure by using a relation with the quantum cohomology of Grassmannians. In the same vein as the hive model of Knutson and Tao, this formula is given in terms of a subtraction-free sum of volumes of explicit polytopes. As a consequence, this expression also provides a positive formula for the volume of moduli spaces of U_n-valued flat connections on the three-holed two dimensional sphere, which was first given by Witten in terms of an infinite sum of characters. This talk is based on a joint work with Quentin François.
