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Wednesday, October 22, 2014

**Abstract:** Cluster Algebras were invented by Fomin and Zelevinsky in order to better understand Lusztig's canonical basis. However, the Cluster Algebra structure turned out to be a rather ubiquitous thing in various areas of Mathematics. They appear in Algebraic Geometry, Integrable Combinatorics, Representation Theory and the theory of Teichmüller spaces to name a few. We will introduce Cluster algebras and state their important properties and results through examples. Finally, we will expose an old friend from Algebraic Geometry to have been a Cluster Algebra all along.