How do simple local interactions combine to produce complex large-scale structure and patterns? The abelian sandpile model provides a beautiful test case. I'll discuss a pair of conjectures about the scale invariance and dimensional reduction of the patterns formed. A new perspective on sandpiles views them as free boundary problems for the discrete Laplacian with an extra integrality condition. The talk will contain theorems, conjectures, proofs and pictures in about equal proportion.
Joint work with Anne Fey and Yuval Peres.