Abstract: The Arnold-Liouville theorem, roughly speaking, takes a dynamical system with lots of conserved quantities, and decomposes an open dense subset of its phase space into a family of tori on which time-evolution is linear. Mathematically, it gives a Lagrangian torus fibration, whose base  B  is an integral affine manifold.  Geometric prequantization then upgrades  B  into an "integral-integral affine manifold", locally modelled on   R^n  with its lattice  Z^n. When  B  is compact, its volume is equal to its number of lattice points. This fact is related to "independence of polarization" phenomena of geometric quantization.  It is true as expected, but its proof is surprisingly tricky.  We now have two proofs, one with Mark Hamilton and Takahiko Yoshida, and one with Yiannis Loizides and Oded Elisha.