Abstract:

Abstract: The notion of rank for matrices (or $2$-tensors) is well understood but the generalizations to higher order tensors is much less well understood and is an active area of research with applications in Biology, Statistics, Computing and Signal Processing.

I will explain the nature of the problem, the notions of "tensor rank" and "border rank" and how problems in this area have been studied through the lens of Higher Secant Varieties of Segre Varieties and the cohomology of non-reduced varieties whose support is a union of linear spaces. I will mention some recent progress on the problems that have been done by me and my collaborators M. V. Catalisano and A. Gimigliano and also some conjectures and open problems. If time permits I will also explain some recent work of Landsberg, Weyman and ourselves on the defining ideals of these higher order secant varieties.