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. Joint work with Anne Fey and Yuval Peres.
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Lionel Levine (Massachusetts Institute of Technology)