Asymptotic properties of moment polytopes of tensors

forthcoming

In AsympTensorPolytope I will study the behavior of moment polytopes of families of tensors, as well as their ability to prove bounds on (asymptotic) tensor rank and subrank. I aim to determine properties of the moment polytopes of the unit tensors of varying ranks, as well as matrix multiplication tensors, in particular whether they are distinct from the generic polytopes of their respective format. To achieve this, I will use a combination of the various different descriptions of moment polytopes, which come in representation-theoretic, symplectic-geometric, intersection-theoretic, or more combinatorial forms. In particular, I will study the behavior of moment polytopes under taking direct sums and Kronecker products of tensors, as well as recently obtained computational results. The project has the potential of proving new bounds on the complexity of various tensors, as well as furthering our understanding of Strassen's asymptotic spectrum of tensors. The techniques here can also be extended to understand other settings, such as symmetric or antisymmetric tensors (bosonic or fermionic systems), algebras and quiver representations. As a result, AsympTensorPolytope will have an impact in other contexts such as the complexity of matrix multiplication, quantum information theory and combinatorics.