Abstract
We will compare the four notions of ranks of tensors, polynomials and similar geometric structures. We will concentrate on the border rank and smoothable rank, and show examples when they differ. The classification of tensors of border rank three will be presented, with an emphasis of a notion of "exceptional limit" showing how it serves to provide examples of patological behaviour.
The talk is based on a joint works with Joseph Landsberg and with Weronika Buczynska.