# Martin Orr: Quantitative reduction theory and unlikely intersections

## RTG Seminar: Martin Orr (Warwick), July 9, 2020

**Quantitative reduction theory and unlikely intersections**

Abstract:

The Zilber-Pink conjecture predicts that a family of abelian surfaces over a one-dimensional base, with generic endomorphism ring $\mathbb Z$, contains at most finitely many fibres with quaternionic multiplication. I will discuss a partial proof of this conjecture (joint with Christopher Daw). The talk will focus on one of the new ingredients in this proof: a quantitative version of Borel and Harish-Chandra’s construction of fundamental sets for arithmetic group actions.