Mirror symmetry and cluster algebras - Paul Hacking & Sean Keel - ICM2018

Algebraic and Complex Geometry
Invited Lecture 4.15
Mirror symmetry and cluster algebras
Paul Hacking & Sean Keel
Abstract: 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.
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