Alessandro Oneto: Waring's Problem: Secant varieties and defectiveness

In 1770, Waring stated that any natural number can be written as sum of at most 9 cubes of positive integers, or at most 19 fourth-powers, and so on for any given degree. From an Algebraic point of view, we can consider a sort of Waring's Problem for Polynomials about how to write an homogeneous polynomial of given degree as sum of powers of linear forms. Similarly, a Waring's Problem for Tensors about how to write a given tensor as sum of decomposable tensors.

The main purpose of this seminar is to relate these largely unknown problems to some very classical geometric objects: Secant Varieties of Veronese varieties and Segre varieties.