Higher-dimensional analogues of the Deligne-Mumford moduli space of stable curves were introduced by Koll?r, Shepherd-Barron, and Alexeev. I will give a survey of their
applications to topology of smooth algebraic surfaces. One example will be a recent joint result with Julie Rana and Giancarlo Urzua that the Craighero?Gattazzo surface, the minimal
resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured in 1997 by Dolgachev and Werner, who proved that its
fundamental group has a trivial profinite completion. This makes the Craighero?Gattazzo surface the only explicitly known example of a smooth simply-connected complex surface of geometric
genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface, we use an algebraic reduction mod p
technique and deformation theory.