Symplectic Geometry and Mirror Symmetry

Mirror symmetry is a duality between pairs of Calabi-Yau manifolds that exchanges symplectic and complex geometry. On the symplectic side, the Fukaya category $\mathcal{F}(X)$ — whose objects are Lagrangian submanifolds and whose morphisms are Floer cochain complexes — captures the enumerative geometry of pseudo-holomorphic discs. Kontsevich's homological mirror symmetry conjecture asserts an equivalence $D^b\mathcal{F}(X) \simeq D^b\text{Coh}(\check{X})$ between the derived Fukaya category and the derived category of coherent sheaves on the mirror. We survey recent progress and open problems.