In this talk we will focus on geometric and algebraic properties of tensors relating to their rank and border rank. In particular, we will:
1) present geometric properties of Coppersmith-Winograd tensor (that characterize it),
2) provide tensors with a large ratio of rank to border rank (previously conjectured to be bounded by two),
3) introduce a new method to bound border rank from below (slightly improving the known lower bounds for matrix multiplication),
4) introduce local variants of secant varieties (relating the study of border rank to geometric properties of the Hilbert scheme),
5) show that optimal border rank algorithms for matrix multiplication can be assumed to have a very special form,
6) present new necessary conditions for small border rank and compute border ranks of flag tensors (answering questions of Iliev, Manivel and Leitner).
All results are based on a joint work with Joseph Landsberg.