Positivity results for cluster algebras from surfaces

We give combinatorial formulas for cluster algebras with principal coefficients coming from triangulated surfaces (with or without punctures), as well as some cluster algebras obtained by ``folding''. In particular, this proves the positivity conjecture of Fomin and Zelevinsky for such cluster algebras, including those of classical type.

This is joint work with Ralf Schiffler and Lauren Williams.