Unlikely Intersections and applications to Diophantine Geometry

This thesis concerns problems of "Unlikely Intersections", i.e. about varieties who are not expected to intersect unless there is a special geometric relation between them. In literature, there are many problems from very different settings that can be viewed in this perspective, starting from the celebrated Mordell conjecture (also known as Faltings' theorem) and the common formulation of these problems in this language gives common strategies to deal with them. The most general conjecture in this setting is the so called "Zilber-Pink Conjecture", raised in somewhat different form independently by Bombieri, Masser and Zannier and by Zilber in the case of tori and by Pink in the more general context of mixed Shimura varieties. This conjecture includes several famous statements, as Mordell-Lang and Manin-Mumford conjecture for semiabelian varieties and as André-Oort conjecture for Shimura varieties. With the aim of studying some particular cases of the conjecture, in this thesis we are going to apply the so called "Pila-Zannier strategy" to two particular cases: the first one in the setting of multiplicative groups, and the second new one in a special family of split semiabelian varieties over a curve. To tackle this problems, we use different ingredients coming from o-minimality, theory of heights of algebraic numbers and deeper results of Diophantine geometry.