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Monday, August 23, 2021

**Abstract:** One of the most celebrated properties of cluster algebras is the Laurent Phenomenon, which states that every cluster variable can be written as a Laurent polynomial with integral coefficients in terms of any choice of cluster. Although the Laurent Phenomenon itself was proved in the original paper of Fomin and Zelevinsky, in 2002, the accompanying positivity property, which states that the expansion coefficients are in fact non-negative, was not proven in full generality until 2018. In 2011, Musiker, Schiffler, and Williams offered the first proof of positivity for the subclass of cluster algebras from surfaces via the construction of snake graphs, which can be used to give explicit combinatorial cluster expansion formulas. In this talk, we will use concrete examples to define cluster algebras from surfaces (both unpunctured and punctured), explain how to construct snake graphs, and then state the cluster expansion formula(s) of Musiker, Schiffler, and Williams. We will largely treat material found in "Positivity for cluster algebras from surfaces", https://arxiv.org/abs/0906.0748.