Abstract: We study the Strominger-Yau-Zaslow (SYZ) conjecture, which provides a geometric framework for mirror symmetry in Calabi-Yau manifolds. We propose a mathematically precise statement for SYZ fibration duality based on a toy local model for integrable systems in symplectic and non-archimedean contexts. We explain the globalization of this toy model using quantum correction data. A focus will be given to a down-to-earth example of our SYZ duality, addressing the transition between the smoothing and crepant resolution of A_n singularity, accompanied by certain geometric phenomena related to the braid group action.
