In complexity theory, one of the main open problems is bounding the asymptotic complexity of matrix multiplication, known as omega. The key input for deriving the existing bounds on omega is a minimal border rank tensor. I will report on a recent characterization of these tensors which shows that they are of algebraic origin and hence amenable to algebra-geometric methods such as deformation theory. The talk will be accessible: no prior knowledge of tensors, complexity theory or algebraic geometry is necessary.