Bounded-rank tensors

Abstract:  The notion of rank for matrices (two-dimensional tensors) has a natural generalisation to higher-dimensional tensors. However, in higher dimensions it is much less well behaved than for matrices. For instance, it is NP-hard to compute, and equations for tensors of bounded rank (the analogue of determinants for matrices) are not known in general. I will present some recent and ongoing work (mostly with Jochen Kuttler and my student Rob Eggermont) on those equations and on the complexity when the rank is fixed but the dimension of the tensor is allowed to vary.