Mirror symmetry is a duality transformation between the complex geometry and the symplectic geometry of two Calabi-Yau manifolds. It gives highly non-trivialpredictions in the enumerative geometry.
In this talk, we explain this conjectural duality in terms of the Fouriertransformation and the Legendre transformation. This verifies the conjecture for semi-flat Calabi-Yau manifolds.
We will discuss such duality transformations on G2 manifolds and Hyperkahler manifolds.In the latter case, we obtain a general Plucker formula for dual varieties.