Cluster Algebras

Cluster algebras are commutative algebras defined by a recursive procedure called "mutation". They were introduced about 20 years ago by Fomin and Zelevisnky with the original aim to get a better understanding of Lusztig's dual canonical basis for quantum groups. However, the applications of cluster algebras rapidly grew to reach fields of mathematics that are surprisingly far from their origins. Today the theory of cluster algebras forms an active area of research and has lead to results in representation theory, homological algebra, total positivity, mirror symmetry, theoretical physics, combinatorics and polyhedral geometry, knot theory, Donadlson--Thomas theory, Teichmueller spaces and more. In this talk I aim to introduce cluster algebras and go through a few examples of their applications.