Examples of n-dimensional Ricci flat manifolds are Riemannian manifolds whose holonomy groups Hol(g) are subgroups of SU(n), for n=2m, and subgroups of the exceptional Lie group G_2, for n=7. These are called Calabi-Yau and G_2 manifolds, respectively, and are also examples of manifolds with special holonomy.

Calibrated submanifolds of Calabi-Yau and G_2 manifolds are volume minimizing in their homology classes, and their moduli spaces have important applications in geometry, topology, and physics. In particular, they are believed to play a crucial role in explaining the mysterious “mirror symmetry” between pairs of Calabi-Yau and G_2 manifolds.

In this talk, we provide an introduction to manifolds with special holonomy and discuss some applications in symplectic and contact geometry. We expect these applications to shed light on the understanding of mirror symmetry.