We survey recent developments in the Birational Anabelian Geometry program
aimed at the reconstruction of function fields of algebraic varieties over
algebraically closed fields from pieces of their absolute Galois groups.

We propose a general framework governing the intersection properties of
extremal rays of irreducible holomorphic symplectic manifolds under the
Beauville-Bogomolov form. Our main thesis is that extremal rays associated to
Lagrangian projective subspaces control the behavior of the cone of curves. We
explore implications of this philosophy for examples like Hilbert schemes of
points on K3 surfaces and generalized Kummer varieties. We also collect
evidence supporting our conjectures in specific cases.