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MIRROR SYMMETRY AND CLUSTER ALGEBRAS

We explain our proof, joint with Mark Gross and Maxim Kontsevich, of conjectures of Fomin–Zelevinsky and Fock–Goncharov on canonical bases of cluster algebras. We interpret a cluster algebra as the ring of global functions on a non-compact Calabi-Yau variety obtained from a toric variety by a blow up construction. We describe a canonical basis of a cluster algebra determined by tropical counts of holomorphic discs on the mirror variety, using the algebraic approach to the Strominger–Yau–Zaslow conjecture due to Gross and Siebert.