| Abstract: The nth iterate of the map (x,y) ->
(y,(y^2+1)/x) is of the form (x,y) -> (f(x,y), g(x,y)), where the rational
functions f and g turn out to belong to the ring Z[x,1/x,y,1/y] of Laurent
polynomials. This is just one example of a very widespread phenomenon that
governs iteration of many multivariate rational maps. I'll talk about how
combinatorial models can help one understand specific instances of this "Laurent
phenomenon"; e.g., for the specific mapping described above, the relevant
combinatorial model is the dimer model on a 2-by-n grid. Such models can
yield positivity theorems that seem hard to prove by purely algebraic means. |