Local and global behavior of arboreal Galois representations — Joachim Koenig (Korea National University of Education)

Arithmetic dynamics deals with arithmetic properties of iterates of maps such as polynomials (or rational functions)  $f: \mathbb{Q} \to \mathbb{Q}$. Many of these properties are encoded in the ``arboreal Galois group", i.e., the projective limit of Galois groups $Gal(f^{\circ n} / \mathbb{Q})$ ($n\to \infty$).
One key problem is the question whether, or under what conditions, a polynomial map can be expected to be dynamically irreducible, meaning that all of its iterates are irreducible. This question is relevant both from the global and the local viewpoint. Several previous works have given rise to the expectation that for ``most polynomials" $f$, dynamical irreducibility over $\mathbb{Q}$ holds, and more strongly that the arboreal Galois group is close to ``largest possible"; and that on the other hand the set of ``stable primes", i.e., primes  modulo which $f$ remains dynamically irreducible is usually a ``small" set. 
In this talk, I will present some new results on both problems, in particular results on the ``largeness" of dynamical monodromy groups and applications thereof, as well as a group-theoretical approach which shows for the first time that for "most" polynomials of a given degree, the set of stable primes is of density zero.