I will explain how the combinatorics of (1) crystals, and (2) paths in alcove geometries, arises in representation theory of the rational Cherednik algebra of the symmetric group. Roughly speaking, (1) describes branching rules, while (2) provides a way to tell when there is a homomorphism between Verma modules (on a certain poset of partitions). Combining these two types of combinatorics, we prove that the character formula of a simple unitary module L of the Cherednik algebra is categorified, so to speak, in the following way: L has a resolution whose terms are direct sums of Verma modules (a "BGG resolution"). These resolutions were conjectured by Berkesch-Griffeth-Sam. This is joint work with Chris Bowman and José Simental.