The famous Mumford—Tate conjecture asserts that, for every prime number $\ell$, Hodge cycles are $\mathbb{Q}_{\ell}$ linear combinations of Tate cycles, through Artin's caparisons theorems between Betti and étale cohomology. The algebraic Sato—Tate conjecture, introduced by Serre and developed by Banaszak and Kedlaya, is a powerful tool in order to prove new instances of the generalized...
For both conceptual and practical reasons it is useful to have estimates on the number of points of algebraic varieties over Q, usually phrased in terms of asymptotics as the height of points increases. I will present a new such estimate, improving previous results by Bombieri, Pila, Heath-Brown, Browning, Salberger, Walsh and others. Time permitting, I will present an application to bounding...
