Lectures on the Ax–Schanuel conjecture

Functional transcendence results have in the last decade found a number of important applications to the algebraic and arithmetic geometry of varieties X admitting flat or hyperbolic uniformizations: Pila and Zannier’s new proof of the Manin–Mumford conjecture, the proof of the Andre–Oort conjecture for Ag, and the generic Shafarevich conjecture for hypersurfaces of Lawrence–Venkatesh, to name a few. The key insight (originally stemming from work of Pila and Zannier) is the use of o-minimality to pass between the geometry of X and that of its uniformizing space.