My research interests are symplectic geometry, complex geometry and their close relations with modern Physics. More specifically, I work on mirror symmetry, which is a duality between symplectic and complex geometry discovered by string theorists. Its enumerative power astonished the mathematical society: it transforms quantum symplectic invariants, which are very difficult to compute, into certain classical integrals, which are much easier to handle. Strominger-Yau-Zaslow proposed that mirror symmetry can be understood geometrically by duality between tori. Their approach has to receive “quantum correction,” which is the main subject of my study. As an application I compute open Gromov- Witten invariants of Calabi-Yau and semi-Fano toric manifolds.