When you smooth a singularity in a symplectic manifold, you introduce some extra topology in its place: a Lagrangian vanishing cycle. Hence questions about degenerations of algebraic surfaces can be rephrased in terms of Lagrangian embeddings of 2D cell complexes. We will meet certain cell complexes called “pinwheels” (vanishing cycles of Wahl singularities), whose Lagrangian embeddings in CP^2 are classified by so-called Markov numbers (by work of Evans-Smith). I will talk a bit about my work on extending this to other surfaces, and give an idea of the proof in the case of P^1 x P^1, which uses holomorphic curve techniques. The talk should be accessible to all, regardless of symplectic background (or lack thereof!).