Abstract: K3 surfaces are compact complex surfaces with a nowhere vanishing holomorphic 2-form, and thus a two-dimensional generalization of an elliptic curve. The moduli/parameter space of elliptic curves is the quotient of a symmetric space (the hyperbolic plane) by the action of a discrete group (SL_2(Z)). The same is true for K3 surfaces: Their moduli space is a quotient of a symmetric space of complex dimension 19 by the action of a discrete group.

In both cases, the moduli space is not compact. Just as the moduli space of elliptic curves has its Deligne-Mumford compactification parameterizing stable curves, moduli spaces of K3 surfaces have compactifications parameterizing singular surfaces, called log stable pairs. I will describe joint work with V. Alexeev on some of the beautiful combinatorics involved in understanding these log stable pairs, and the boundary of the compactification.