Modern computational practice in particle physics has produced an enormous database connecting quantum field theory and particle physics to problems in number theory and algebraic geometry. The appearance of polylogarithms, multiple zeta values and other periods in these computations is testimony to that.

On the mathematical side, these numbers and special functions are of interest in their own right. Amongst their most fundamental properties are the relations they satisfy resulting from the fact that they origin from iterated integrals.

There is a lot of common algebraic structure to be studied in the mathematical theory of these numbers and in physics amplitudes. We will start by reviewing the Hopf algebra structure of renormalization and aim to end with a description of the Hodge-theoretic aspects of Feynman amplitudes.

The purpose of this workshop is to present latest developments in this field in pedagogical lectures that should be accessible to PhD students in a format that provides ample time for discussion.