UBC Discrete Math Seminar: Thomas Karam

Tensors are higher-dimensional generalisations of matrices, and likewise the main notion of complexity on matrices - their rank - may be extended to tensors. Unlike in the matrix case however, there is no single canonical notion of rank for tensors, and the most suitable notion often depends on the application that one has in mind. The most frequently used notion so far has been the tensor rank (hence its name), but several other notions and their applications have blossomed in recent years, such as the slice rank, partition rank, analytic rank, subrank, and geometric rank. Unlike their counterparts for the rank of matrices, many of the basic properties of the ranks of tensors are still not well understood. After reviewing the definitions of several of these rank notions, I will present a number of results of a type that arises in many cases when one attempts to generalise a basic property of the rank of matrices to these ranks of tensors: the naive extension of the original property fails, but it admits a rectification which is simultaneously not too complicated to state and in a spirit that is very close to that of the original property from the matrix case.