Mathematics Colloquium

In 1965, Lang conjectured that if a complex curve in an n-dimensional torus (\mathbb{C}^{\times})^n over the contains infinitely many torsion points, then that curve must be a coset of a one-dimensional subgroup. That conjecture has since been proven many times, and has sparked off a wealth of activity in a field thats now known as "unlikely intersections". The idea is that intersections that are unlikely for dimension reasons (like a curve intersecting the 0-dimensional set of torsion points) should be very rare, unless they are forced to happen for some algebraic reason. 

Langs conjecture has been generalized to abelian varieties as the Manin-Mumford conjecture, and to Shimura Varieties as the Andre-Oort conjecture. Even the famous Mordell conjecture on the finiteness of rational points in hyperbolic curves can be seen to fit into the same vein. 

We will survey these problems and the recent developments on them, and explain how they tie into functional transcendence, and model theory - specifically the theory of o-minimality.