  We explain the proof, obtained in collaboration with Chenyang Xu, of a 1999
conjecture of Veys about poles of maximal order of Igusa zeta functions. The
proof technique is based on the Minimal Model Program in birational geometry,
but the proof was heavily inspired by ideas coming from non-archimedean
geometry and mirror symmetry; we will outline these relations at the end of the
paper. This text is intended to be a low-tech introduction to these topics; we
only assume that the reader has a basic knowledge of algebraic geometry.
