Symplectic geometry arose from the study of classical mechanics, and later many interesting symplectic invariants has been found since Gromov introduced techniques of J-holomorphic curves.

Miraculously, such invariants are closely related with complex geometry information of a certain complex manifold, called the mirror manifold, as mirror symmetry conjecture predicts.

In this talk, we discuss a few approaches to construct a mirror manifold of a given symplectic manifold, and investigate how mirror symmetry can be used to discover new interesting geometries.