11-14【Hang Yuan】On a mathematical formulation of SYZ conjecture

Given a Lagrangian fibration, we propose that the dual torus fibration, for a precise formulation of the SYZ conjecture, should be interpreted as a non-Archimedean torus fibration defined over the Novikov field. This proposal comes from a toy model of the SYZ duality between the complex logarithm map and the non-Archimedean tropicalization map. Both are viewed as torus fibrations over Euclidean space, equipped with the trivial integral affine structure. A canonical globalization of this toy model from A-side to B-side, which preserves the integral affine structure, is constructed via (family) Floer theory applied to the family of Lagrangian fibers. Time permitting, we will briefly explain some basic aspects of non-archimedean geometry, explore explicit SYZ examples with extensions to singular fibers, and discuss relevant applications to symplectic geometry.