Abstract: 
The SYZ conjecture suggests a folklore that ``Lagrangian multisections are mirror to holomorphic vector bundles". In this talk, we prove this folklore for Lagrangian multisections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of FangLiuTreumannZaslow. We will also introduce the notion of tropical Lagrangian multisections and the Lagrangian realization problem. The latter asks whether one can construct an unobstructed Lagrangian multisection with asymptotic conditions prescribed by a given tropical Lagrangian multisection. In dimension 2, we solve the realization problem for those tropical Lagrangian multisections that satisfy the socalled Ngeneric condition. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a simply connected Lagrangian multisection. This is a joint work with YongGeun Oh.
