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Cluster algebras are a class of commutative rings with a remarkable combinatorial structure, which were introduced by Fomin and Zelevinsky around 2000. I will give a gentle introduction to cluster algebras, and then explain how Grassmannians and more generally their Schubert varieties have a cluster algebra structure (joint work with Khrystyna Serhiyenko and Melissa Sherman-Bennett). If time permits, I will also discuss applications to toric degenerations and mirror symmetry (joint work with Konstanze Rietsch).
