# Unlikely intersections and the Andre-Oort conjecture

# Unlikely intersections and the Andre-Oort conjecture

**In-Person and Online Talk**

**Zoom link: https://princeton.zoom.us/j/99136657600**

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The Andre-Oort conjecture concerns special points of a Shimura variety S - points that are in a certain sense "maximally symmetric". It states that if a variety V in S contains a zariski-dense set of such points, then V must itself be a Shimura variety. It is an example of the field now known as "unlikely intersections theory" which seeks to explain "arithmetic coincidences" using geometry. In fact, there is a very natural sense in which the Andre-Oort conjecture can be seen as an analogue of Faltings theorem concerning rational points on curves.

The proof of this conjecture involves a wide range of disparate mathematical ideas - functional transcendence, mondromy, point counting in transcendental sets, upper bounds for arithmetic complexity (heights of special points), and p-adic hodge theory. We will survey these concepts and how they relate to each other in the proof, aiming to give an overview of the relevant ideas. We will also discuss the current status of the field, now spearheaded by the (still extremely open!) Zilber-Pink conjecture, and what is required to make further progress.