SYZ Fibration
Mirror symmetry: pairs CY manifolds X,X̌ with h^{1,1}(X)=h^{2,1}(X̌). For elliptic curve E_τ: mirror is E_{−1/τ} (S-duality).
SYZ conjecture (Strominger-Yau-Zaslow 1996): X fibers over base B with special Lagrangian T^n fibers; X̌ = fiberwise dual torus. T-duality exchanges fiber with dual fiber.
Homological MS (Kontsevich 1994): D^b(Coh(X)) ≅ D^b(Fuk(X̌)) — derived categories of coherent sheaves ↔ Fukaya category (Lagrangians + HF*).