Cluster Algebras

Cluster algebras were introduced in 2001 by Fomin and Zelevinsky in order to study the canonical basis of a quantum group and total positivity in algebraic groups. A cluster algebra is a commutative algebra given by combinatorial data as a subalgebra of the field of rational polynomials in finitely many indeterminates. Cluster algebras have links with a variety of other fields, including Stasheff polytopes (associahedra), Y-systems, the Bethe ansatz, recurrence problems, Teichmueller theory, toric varieties and representation theory. In particular, the cluster algebras of finite type have been classified by the Dynkin diagrams are are strongly connected to root systems; in representation theory cluster algebras have motivated the definition of a new kind of tilting theory (cluster-tilting theory) and a family of algebras, known as the cluster-tilted algebras. The talks will be an introduction to the theory of cluster algebras and will also consider some of the links with the representation theory of algebras.