Tensors of Minimal Border Rank

A class of tensors, called "concise (m,m,m)-tensors  of minimal border rank", play an important role in proving upper bounds for the complexity of matrix multiplication. For that reason Problem 15.2 of "Algebraic Complexity Theory" by Bürgisser, Clausen and Shokrollahi is to classify tensors of minimal border rank. It is now now understood that this question is equivalent to notoriously difficult questions in algebraic geometry and commutative algebra. However the full question is not what is needed for complexity theory. I will explain recent progress on the question in several directions. A surprising role is played by degree seven zero dimensional local Gorenstein schemes.