SYZ Mirror Symmetry for Del Pezzo and Rational Elliptic Surfaces

In this talk I will demonstrate the construction of mirror special Lagrangian torus fibrations on two non-compact spaces: A Del Pezzo surface with a smooth anticanonical divisor removed, and a rational

elliptic surface minus a singular fiber of Kodaira type I_k. Special emphasis will be given to local geometric models, and how the mean curvature flow provides a key step in the construction. This is joint work with T.C. Collins and Y.-S. Lin.