In the theory of the ranks of tensors, one line of research consists in generalising basic properties of the rank of matrices to appropriate statements that often require significant reformulation. This talk will focus on extensions of tensors that maintain the rank of the original tensor - for various rank notions - and on how constrained or flexible these extensions are. We will take as our starting point the answer that this question has for matrices, and discuss several potential generalisations: some that we will show to be true, some that we will show to be false in a major way, and finally some that we will view as conjectures and which have strong connections to several existing questions on the ranks of tensors that we will finish by describing.

Prior to joining the faculty of Shanghai Jiao Tong University in 2025, Thomas Karam was a postdoctoral researcher at the University of Oxford (2022-2025), a doctoral student at the University of Cambridge (2018-2022), and a student up to masters level at the Ecole Normale Supérieure of Paris (2014-2018). His main existing contributions pertain to the basic theory of the ranks of tensors, the behaviour of Boolean polynomials on finite fields, the unification of density theorems in Ramsey theory and the analogues of Brunn-Minkowski type inequalities in the torus.