Andrei Okounov on Dimers and Amoebas
At low temperatures, many substances are made of crystals with specific shapes--for instance, a salt crystal is a cube. But when the temperature increases, the cube's corners get eaten away, leaving a rounded object with six flat patches. The goal of Okounov's talk was to come up with a mathematical description of how the corners get eaten away, and to relate the geometry of the resulting rounded "corner" to other mathematical objects.
If you look locally at the corner--that is, imagine it as a corner of a cube made of infinitely many small blocks--the eating away happens randomly. It's the same as randomly removing N blocks from the corner, with the ones closer to the corner more likely to be removed. In other words, it's the same as choosing a weighted random 3-D partition of N elements. As N goes to infinity, a limit shape emerges, and when viewed head-on it looks like the Mercedez-Benz logo with each of the three prongs stretching out to infinity.
This figure occurs elsewhere in math--it's the amoeba of the straight line. For a projective plane curve that's the zero set of some polynomial P(z,w), its amoeba is the image of the curve under the map (z,w) ---> (ln |z|, ln |w|). Its moniker comes from the fact that the graph in general has holes (the number of holes is related to the genus of the curve given by P) and strange stretchings, and so resembles an amoeba. The connection of amoebas to the limit shape of weighted random 3-D partitions goes through dimers. Given a graph, a dimer is a length two chain, i.e. a way of associating every vertex with a neighbor. If the vertices are assigned weights, then the study of random dimers is essentially the same as the study of weighted random 3-D partitions. Using this relationship, one can show that the limit shape of any such partition is the amoeba of some explicit polynomial P(z,w). Moreover, the curve P(z,w) = 0 must be a real algebraic curve of a very special and well-behaved kind, known as Harnack.
A final interesting note is that as certain crystals become rounded under high heat, seemingly mysterious flat spots remain. These spots correspond to holes in the associated amoeba. These holes in turn correspond exactly to the genus of the function that produces the amoeba. So the genus of the associated curve has direct physical consequences!
--- Rafe Jones, Brown University (AAAS-AMS Media Fellow, 2001)
[Andrei Okounov (University of California Berkeley) gave this AMS Invited Address on January 16.]