The Zilber-Pink conjecture predicts that an algebraic curve in A_2 has only finitely many intersections with the special curves, unless it is contained in a proper special subvariety.Under a large Galois orbits hypothesis, we prove the finiteness of the intersection with the special curves parametrising abelian surfaces isogenous to the product of two elliptic curves, at least one of which has complex multiplication. Furthermore, we show that this large Galois orbits hypothesis holds for curves satisfying a condition on their intersection with the boundary of the Baily--Borel compactification of A_2.More […]

I will explain how tame topology seems the natural setting for variational Hodge theory. As an instance I will sketch a new proof of the algebraicity of the components of the Hodge locus, a celebrated result of Cattani-Deligne-Kaplan (joint work with Bakker and Tsimerman).

Let k be an algebraically closed complete rank 1 non-trivially valued field. Let X be an algebraic curve over k and let X^an be its analytification in the sense of Berkovich. We functorially associate to X^an a definable set X^S in a natural language. As a corollary, we obtain an alternative proof of a result of Hrushovski-Loeser about the iso-definability of curves. Our association being explicit allows us to provide a concrete description of the definable subsets of X^S: they correspond to radial sets. This is a joint work with […]