Manifolds with special holonomy, such as Calabi–Yau and G₂ manifolds, are Riemannian manifolds whose holonomy groups are contained in SU(n) (for n = 2m) and the exceptional Lie group G₂ (for n = 7), respectively. These spaces arise naturally as Ricci-flat examples in differential geometry and play a central role in theoretical physics, particularly in M-theory compactifications. M-theory, often described as a candidate for a "theory of everything'', aims to unify the fundamental forces of nature: electromagnetism, gravity, and the strong and weak nuclear forces.

Calibrated submanifolds within Calabi–Yau and G₂ manifolds are volume minimizing in their homology classes, and their moduli spaces have deep applications in geometry, topology, and physics. In this talk, we will begin with a brief introduction to Calabi–Yau and G₂ manifolds, and then report on recent research exploring the rich interplay between symplectic, contact, and calibrated structures in manifolds with special holonomy.