Tensor decomposition and secant varieties

Determining secant varieties (i.e. varieties which are the union of secant lines, planes, etc.) for a given projective variety X is a classical problem in Algebraic Geometry. When X is a Segre Variety, its secant varieties parameterize tensors with assigned tensor rank, and their study is related to the study of tensor decompositions, a quite relevant issue in applied math. In a similar way, secant varieties of Veronese varieties are related to symmetric tensors. In this talk a sketchy view of the state of the art on these problems will be given.