Matrix rank is well-known to be multiplicative under the Kronecker product, additive under the direct sum, normalised on identity matrices and non-increasing under multiplying from the left and from the right by any matrices. In fact, matrix rank is the only real matrix parameter with these four properties. In 1986 Strassen proposed to study the extension to tensors: find all maps from k-tensors to the reals that are multiplicative under the tensor Kronecker product, additive under the direct sum, normalized on diagonal tensors, and non-increasing under acting with linear maps on the k tensor factors. Strassen called the collection of these maps the "asymptotic spectrum of k-tensors". Strassen proved that understanding the asymptotic spectrum implies understanding the asymptotic relations among tensors, including the asymptotic rank. In particular, knowing the asymptotic spectrum means knowing the arithmetic complexity of matrix multiplication, a central problem in algebraic complexity theory.
I will give an overview of known elements in the asymptotic spectrum of tensors, including our recent construction which is based on information theory and moment polytopes. This recent construction is joint work with Matthias Christandl and Peter Vrana.
Then I will introduce the analogous object in graph theory: the asymptotic spectrum of graphs. I will explain the relation to Shannon capacity and give an overview of known elements in the asymptotic spectrum of graphs.