Unlikely Intersections and Connections to Geometry

The field of unlikely Intersections presents a robust paradigm for problems in which several subjects intermingle: arithmetic, o-minimality, and hodge theory. The goal of this talk will be to introduce some of those connections. My aim is to introduce the subject, and accomplish the following:

(1) Introduce by examples one of the main open problems in the field: The Zilber-Pink conjecture

(2) Describe what o-minimality is, and where it enters the picture

(3) Describe how Hodge Theory presents a natural home for the classical questions, especially in its mixed incarnations

(4) Introduce non-abelian hodge theory, which presents a vast and largely unexplored area for these ideas.

The above is obviously insanely ambitious for a 1-hour talk, and I will fail. My goal will be to paint pictures much more than give precise definitions, and have people leave the talk with questions that are easy-to-ask.