# A dynamical variant of the Pink-Zilber conjecture

This talk will present results from the paper of Ghioca and Nguyen "A dynamical variant of the Pink–Zilber conjecture". Pink-Zilber conjecture relates to the theory of unlikely intersections in Diophantine Geometry and is a generalisation of Manin-Mumford conjecture. The latter states there can't be too many special points in the variety unless this variety is special as well. For example, for abelian varieties special = torsion, so Manin-Mumford will state that if the set of torsion points is dense in a subvariety X of an abelian variety, then X is a torsion subvariety (translation of an abelian subvariety by a torsion point). Instead of searching torsion points inside X, we can look at the intersections of X with torsion subvarieties - Pink-Zilber states that if X intersects the set of torsion subvarieties of some chosen dimensions in a dense subset, then X is contained in a torsion subvariety.

The analogue of Zilber-Pink conjecture described in the article considers a map of affine spaces A^n \to A^n of a particular, quite simple form and states that an irreducible variety X is contained in a periodic subvariety of A^n if X intersects the union of periodic subvarieties of some chosen dimension in a dense subset. The proof is by induction and involves some basic algebraic geometry/number theory as properties of finite morphisms and Weil heights.

The talk will begin with a presentation of some different variants of Manin-Mumford and Pink-Zilber conjecture and then outline the sketch of the proof of the main theorem from the article of Ghioca and Nguyen.

*This is an informal seminar on arithmetic algebraic geometry and related fields.*

*The purpose is to present work in progress or to learn about some existing piece of mathematics. Typically the talks are parts of a longer series.*