Tensors (or multidimensional matrices) represent a type of data structure that is ubiquitous in the sciences. Because of their wide use, there is much interest in understanding their fundamental properties, such as tensor decomposition and tensor rank. Tensor decomposition is a sparse representation of a tensor as a linear combination of rank-one tensors, while tensor rank is the number of summands in a minimal decomposition.
So it is natural to ask - how does one check the rank of a given tensor? Or, how does one find the minimal decomposition of a tensor? Already with these questions many beautiful objects from classical Algebraic Geometry arise, such as the Segre variety and its secant varieties. Moreover, because of the inherent symmetry, Representation Theory becomes an important tool.
After discussing some of the applications of tensor decomposition, I will explain the geometric setting together with the algebraic questions we're interested in. Throughout, I will describe recent contributions to the theory (finding equations of secant varieties) and practice (the development of new algorithms) related to tensor decomposition.