Positivity results for cluster algebras arising from surfaces

I will begin with an introduction to cluster algebras and the positivity conjecture of Fomin and Zelevinsky, assuming no prior background. Examples of cluster algebras include geometric objects such as the coordinate ring of the Grassmannian as well as polynomial analogues of numerical sequences such as the Somos sequences, Fibonacci numbers, and Markoff numbers.

In work of Fomin, Shapiro, and Thurston, they presented a construction for cluster algebras of certain types, those arising from triangulated surfaces. The class of such cluster algebras contains "almost all" cluster algebras of finite mutation type. I will conclude with recent joint work with Ralf Schiffler and Lauren Williams which proves the positivity conjecture for cluster algebras from surfaces.