Non-archimedean parametrizations, points counting and applications

We will provide a general overview on some recent work on non-archimedean parametrizations and their applications. We will start by presenting our work with Cluckers and Comte on the existence of good Yomdin-Gromov parametrizations in the non-archimedean context and a $p$-adic Pila-Wilkie theorem. We will then explain how this is used in our work with Chambert-Loir to prove bialgebricity results in products of Mumford curves. We will conclude with our recent work with Cluckers and Forey providing uniform bounds for the number of $\mathbb{F}_q [t]$-points of bounded degree on algebraic varieties.

(30 minute general talk followed by expert talk)