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The theory of cluster algebras emerged in the year 2000 as a combinatorial approach to study bases for quantum groups. Since then, commutative cluster algebras and their quantum deformations have been present in very diverse areas of mathematics and theoretical physics. Constructing bases for both commutative and quantum cluster algebras and understanding their multiplicative structure is a meeting point of techniques and ideas from combinatorics, mirror symmetry, representation theory and quantum groups.
