Roughly speaking the principle of unlikely intersections states that if the intersection of two objects is larger than expected, there should be an underlying geometric reason. This somewhat vague principle includes some of the most important diophantine results in the last 35 years.

Arithmetic dynamics is a relatively new area combining number theory and algebraic dynamical systems. Several highly interesting examples of the unlikely intersection principle in arithmetic dynamics start to emerge recently.

In this talk, I will discuss some (possibly all, if time permits) of the following examples together with their dynamical analogues:

-The Manin-Mumford Conjecture (Raynaud's Theorem)

-The Bombieri-Masser-Zannier Bounded Height Conjecture (Habegger's Theorem).

-The Mordell-Lang Conjecture (proved by Faltings, McQuillan, and Vojta)

-André's result which is part of the more general André-Oort Conjecture