Mirror symmetry for log del Pezzo surfaces

Abstract: 

If X is a del Pezzo surface and D is a smooth anti-canonical divisor, we can regard the complement X\D as a non-compact Calabi-Yau surface.  I will discuss a proof of a strong form of the Strominger-Yau-Zaslow mirror symmetry conjecture for these non-compact surfaces.  It turns out the mirror Calabi-Yau is a rational elliptic surface (in particular, it has an elliptic fibration onto P^1) with a singular fiber which is a chain of nodal spheres.  I will discuss how we can construct special Lagrangian fibrations on these manifolds, as well as moduli of complex and symplectic structures and how hyper-Kahler rotation allows us to construct an identification of these moduli spaces.  This is joint work with A. Jacob and Y.-S. Lin.