This course will survey some of the most important developments in Cluster algebras Theory and related fields such as Total Positivity in general linear group, Lusztig's theory of canonical and crystal bases. 

Cluster algebras were introduced in 2001 by Fomin and Zelevinsky in order to study  total positivity in algebraic groups and canonical bases in their representations. A cluster algebra is a commutative algebra given by combinatorial data as a subalgebra of the field of fractions of finitely many indeteminants. A spectacular ``Laurent phenomenon'' asserts that each cluster algebra, is, in fact, a subalgebra of the algebra of Laurent polynomials in those indeterminates. Cluster algebras have links with a variety of other fields, including Stasheff polytopes (associahedra), the Bethe ansatz, toric varieties, and representation theory. In particular, all cluster algebras of finite type have been classified by the Dynkin diagrams and are strongly related to finite root systems.

I am planning to cover the following topics:

• total positivity in reductive groups;
  
• canonical and crystal bases;
• clusters and cluster algebras;
• Laurent phenomenon;
  
• Finite type classfication;
• Cluster varieties: Grassmannians, double Bruhat cells;
  
• quantum cluster algebras.
  

I then plan to emphasize the following additional topics as time permits:

       cluster algebras emerging from triangulated surfaces, Y,Q,T-systems, and cluster categories;

Pre-requisites: 600 algebra. I will assume some familiarity with root systems, semisimple Lie algebras and highest weight representations.
However, I will spend a few lectures reviewing this material at the beginning of the course. I will not assume prior knowledge of cluster algebras.