Multiple zeta values (MZVs) are a fascinating generalization of the Euler-Riemann zeta function to several arguments. In a formal context, they arise as periods of mixed Tate motives over Z and their motivic and formal avatars illuminate their algebraic structure. In physics, they arise in the context of scattering amplitudes in both string theory as well as Feynman amplitudes, and their motivic properties as well as higher genus generalizations help understand the tropical limits of string theory. There exist algebriac and rational relations between MZVs which are interesting from a mathematical, computational and physical perspective. Linear rational relations have been tabulated in the seminal computational work by Broadhurst, Blümlein and Vermaseren, and utilising these linear relations helps speed up calculations significantly in physical applications.
In this talk, we go beyond linear relations. When expressing MZVs as iterated sums and iterated integrals gives one gets algebraic relations known as shuffle and stuffle relations. These relations arise from algebraic properties of Hopf algebras. To such a Hopf algebra one can associate a “geometric object” each and their “intersection” describes a the “variety of MZVs”, which is a toric variety. Studying this toric variety helps us make predictions and conjectures pertaining to the non-linear algebra of MZVs, as well as computationally enumerate number of relations between MZVs that are non-linear, as well as compute them explicitly.