Identifiability is a subtle property of projective varieties. Both its existence and its failure, when non-expected, have deep geometrical meanings. In the seminar I will survey the state of the art focusing on recent results with Alex Massarenti that allow to reduce the identifiability to non defectivity for almost all subgeneric ranks.  This will also show an interesting link with Bronowski's work at the beginning of 20th century.

The study of the decompositions of the powers of a quadratic form, also called representations, is a very classical problem. Many examples appear several times even in old literature, especially for the real case. Our purpose is to deal with this problem by a modern point of view, with the final aim of determining its rank and its border rank. The main instrument we used is the apolarity theory, by which it is possible to determine suitable decompositions of a given form, just analyzing its apolar ideal, that in the case of the powers of quadrics results to be generated by harmonic polynomials. This approach also allows us to determine the border rank in the case of three variables.

In Commentationes Geometricae Euler asked when a set of points in the plane is the intersection of two curves, that is, using the modern terminology, when a set of points in the plane is a complete intersection. In the same period, Cramer asked similar questions so that this type of questions is presently known as the Cramer-Euler problem. In this paper, we consider a generalization of the Cramer-Euler problem: characterize the possible complete intersections lying on a Veronese surface, and more generally on a Veronese variety. The main result describes all possible reduced complete intersections on Veronese surfaces. We formulate a conjecture for the general case of complete intersection subvarieties of any dimension and we prove it in the case of the quadratic Veronese threefold. Our main tool is an effective characterization of all possible Hilbert functions of reduced subvarieties of Veronese surfaces.