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Positivity for cluster algebras from surfaces

Algebra & Discrete Mathematics

Speaker: Lauren Williams, UC Berkeley
Location: 2112 MSB
Start time: Fri, Sep 25 2009, 4:10PM

I'll start by introducing the cluster algebras of Fomin and Zelevinsky, then present a construction of the cluster algebra associated to a Riemann surface with marked points (based on Fomin-Shapiro-Thurston). By work of Felikson-Shapiro-Tumarkin, this construction realizes all but finitely many (= eleven) of the cluster algebras of finite mutation type. Then I will explain a combinatorial formula for the Laurent expansion of each cluster variable in any such cluster, with respect to an arbitrary seed. An immediate corollary of our formula is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces.

This is joint work with Gregg Musiker and Ralf Schiffler.