Mirror symmetry and cluster algebras        
-- Paul Hacking, May 2, 2014

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We interpret a Fomin-Zelevinsky cluster algebra as
the ring of global functions on a non-compact Calabi-Yau
variety obtained from a toric variety by a blowup
construction. Motivated by mirror symmetry, we describe a
canonical basis of a cluster algebra defined via tropical
counts of holomorphic discs on the mirror, verifying
conjectures of Fomin-Zelevinsky and Fock-Goncharov. This is
joint work with Gross, Keel, and Kontsevich.