Tensors play a wide role in numerous application areas as Signal Processing for Telecommunications, Arithmetic Complexity or Data Analysis. In some applications tensors may be completely symmetric, or symmetric only in some modes, or may not be symmetric. In most of these applications, the decomposition of a tensor into a sum of rank-1 terms is relevant, since tensors of interest have a reduced rank. Most of them are structured i.e. they are either symmetric or enjoy some index-invariance. Lastly, they are often real, which raises open problems concerning the existence and calculation of the decompositions. These issues build the basic bricks of the research program we propose. The classes of tensors described above have a geometric translation in terms of classical algebraic varieties: Segre, Veronese, Segre-Veronese varieties and Grassmannians and their secant varieties. A complete description of equations for those secant varieties and their dimensions is still not known (only dimensions of secant varieties to Veronsean are classified), although they have been studied by algebraic and differential geometers and algebraists for a long period up to now.

The aim of this research project is:

• to attack both the description of the ideal of those secant varieties and their dimensions, starting from low dimensions and low degrees,
• to propose algorithms able to compute the rank of structured tensors. Workshops in Palo Alto (CA-USA, 2008) and in Nice (FR, 2009) showed that Italy and France are among the most active in Europe in the field of tensor decompositions.

Both the coordinator of this project and the hosting organization have already obtained results in this field regarding equations and algorithms. Hence this program is crucial for the development of those research areas in the European Community, along with the numerous international collaborations already existing. The impact of this project will be visible in both academic and industrial worlds.