Abstract: Beginning with Fermat's characterization of primes which are the sum of two squares, I would like to show how this naturally leads to Waring's Problem for Integers and then to Waring's Problem for Homogeneous Polynomials.

One half of this latter problem has been solved in recent years and I will explain the nature of the approach to that solution through the study of the higher secant varieties of Veronese varieties and the study of non-reduced subschemes of projective n-space.

Despite the technical sounding terms above, the talk is aimed at a general audience and I will keep the technicalities to a minimum.