Title: Dimensions of secant varieties

Abstract:  A variety  is the set of solutions of a polynomial system of equations. Considering the union of all linear subspaces spanned by k points on a variety V, one obtains the k-th secant variety of V. Determining the dimension of a secant variety is an interesting and challenging problem. We illustrate this in two instances. The first one concerns the Waring rank. Any homogeneous polynomial f of degree d can be written as a sum of d-th powers of linear forms. The minimum number of summands in such a decomposition is the Waring rank of f. It admits a geometric interpretation using secant varieties. In the second instance we use linear algebra to solve the problem in some cases. The general problem (of decomposing tensors as sums of pure tensors) is open.