I will recall the Zilber-Pink conjecture for Shimura varieties and give my perspective on current progress towards a proof.

I will give an overview of new ideas showing up in arithmetic intersection theory based on some exciting talks that appeared at the very recent conference "Global invariants of arithmetic varieties". I will also outline connections to globally valued fields and some classical problems.

We will survey work of Birman-Craggs, Johnson, and Sato on the abelianization of the level 2 congruence group of the mapping class group of a surface, and of the corresponding Torelli group. We will then describe recent work of Lewis providing a common framework for both abelianizations, with applications including a partial answer to a question of Johnson.

Let C,D be classes of finitely presented groups. The epimorphism problem from C to D is the following decision problem:

Input: Finite descriptions (presentation, multiplication table, other) for groups  G in C and H in D

Question: Is there an epimorphism from G to H?

I will discuss some cases where it is decidable and where it is NP-complete. Spoiler alert: it is undecidable for C=D=the class of 2-step nilpotent groups (Remeslennikov).

This is joint work with Jerry Shen (UTS) and Armin Weiss (Stuttgart).