Alon Levy: The Dynamical Mordell-Lang Problem in Dimension Two, part 2.

Let \phi be a morphism from a variety X to itself, let x be some point
of X, and let Y be some subvariety. If Y intersects the orbit of x in
infinitely many points, does there have to be a special reason? The
dynamical Mordell-Lang conjecture asserts that the answer is yes: Y,
or some subvariety of Y containing the intersection of Y with the
orbit of x, must be a periodic subvariety. The problem has remained
unsolved except in special cases, and has been attacked with tools
including algebraic geometry, nonarchimedean analysis, and model
theory. In a groundbreaking paper, Junyi Xie proves that the
conjecture is indeed correct if X = A^2. We give an overview of Xie's
proof, which splits into two cases, one of which is solved using
algebro-geometric techniques (namely, the concept of algebraic
stability), and the other of which is solved using a combination of
algebraic geometry and analysis (the so-called valuative tree at
infinity).