The mirror to a log Calabi-Yau surface -- Seán Keel, October 16, 2009

I'll explain joint work with Hacking and Gross, where we construct from a pair (Y, B=B_1+..+ B_n) of an anti-canonical cycle of rational curves on a smooth projective surface, a canonical smoothing of the cone over B, parametrized by the affine toric variety for Nef(Y). If you know how to get toric varieties from fans and from lattice polytopes, you know more than enough to follow the details.