Rank v.s. border rank

The rank of a matrix admits numerous definitions, many of which fail to coincide when generalized to tensors (although all give rise to the same set of rank one tensors). The rank of a tensor T is defined to be the minimum number r of rank one elements $a_i$ such that T=$a_1+...+a_r$. The set of tensors of rank at most r is not closed, and by definition the set of tensors of border rank at most r is the Zariski closure of this set. In this talk I will survey what is known about ranks and border ranks of tensors and symmetric tensors. While border rank is more natural from the perspective of algebraic geometry (and has been studied extensively), I'll show that the geometry assciated to rank also can be beautiful. Recent results in the talk are joint with Z. Teitler.