Typical ranks of random order-three tensors

The determination of tensor rank is a crucial task in many fields. We introduce typical ranks highlighting the difference between the real and the complex case, before focusing on the case of order-three random tensors. Through a result of Friedland, we give a geometric interpretation of tensor rank and link typical ranks to linear sections of the Segre variety. Exploiting tools coming from integral geometry we also show some heuristics on typical ranks of a given family of tensors. In the second part of the talk, we will explain how the rank of a real random 3x3x5 tensor depends on the number of real lines on a random cubic surface.

This is joint work with P. Breiding (University of Osnabrück) and S. Eggleston (University of Osnabrück).