The Ax-Schanuel theorem is a function field analogue of the Schanuel’s conjecture for exponentials in transcendental number theory. This theorem is extended Hodge theoretically in the past decade. When phrased geometrically, it roughly says that if the intersection of a given algebraic variety and the graph of a period mapping is "unlikely", then this intersection should be related to group orbits of Hodge structures. We will first review the history of the subject, and recap the notions in mixed Hodge theory, including variations of mixed Hodge structures, their period mappings, and weakly special subvarieties. A version of Ax-Schanuel for principal bundles with flat connections is proved by Blázquez-Sanz, Casale, Freitag and Nagloo. We will explain how their theorem and jet spaces can be used to deduce the Ax-Schanuel theorem for derivatives of mixed period mappings. Zoom link: TBA
