Introduction to Cluster Algebras - Fall 2014 - Institut Henri Poincare

Course description

This course will survey one of the most exciting recent developments in algebraic combinatorics, namely, Fomin and Zelevinsky's theory of cluster algebras. Cluster algebras are a class of combinatorially defined commutative rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. Introduced in 2000, cluster algebras have already been shown to be related to a host of other fields of math, such as quiver representations, Teichmuller theory, Poisson geometry, total positivity, and statistical physics.

I will not assume prior knowledge of cluster algebras, though familiarity with root systems will be helpful for a few lectures.

Lectures will take place from 10am until 12pm on most Tuesdays during the fall, at the Institut Henri Poincare, Paris. (See below for the list of lectures, with dates.) Un grand merci a la Fondation Sciences Mathematiques de Paris pour avoir facilite la tenue de ce cours. Videos of the lectures will be posted here.

Helpful links:

Some references:


  • Lecture 1 (mardi 9 septembre, salle 314): Three motivating examples from total positivity.
  • Lecture 2 (mardi 16 septembre, salle 314): Mutation.
  • Lecture 3 (mardi 30 septembre, salle 314): Cluster algebras of geometric type.
  • Lecture 4 (mardi 14 octobre, salle 201): The Laurent phenomenon.
  • Lecture 5 (mardi 28 octobre, salle 314): Rank 2 cluster algebras of finite type. Type A cluster algebras in nature: frieze patterns, etc.
  • Lecture 6 (mardi 4 novembre, salle 201): Type D cluster algebras (first non-trivial cluster algebra from a surface).
  • Lecture 7 (mardi 18 novembre, salle 314): Finite type classification of cluster algebras; folding of cluster algebras.
  • Lecture 8 (mardi 25 novembre, salle 314): Cluster structures in coordinate rings.
  • Lecture 9 (mardi 2 decembre, salle 314): Cluster structures on Grassmannians.
  • Lecture 10 (mardi 16 decembre, salle 201): Cluster algebras from surfaces and Teichmuller theory (first part of lecture given by Sergey Fomin.)