Topics in Cluster Algebras - Fall 2016

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 11:40am until 12:55pm on Mondays and Wednesdays, in the Mathematics building room 520, Columbia University. Here is the syllabus for the course.

Helpful links:

Some references:


  • Lecture 1 (Wednesday September 7): Some motivating examples from total positivity.
  • Lecture 2 (Monday September 12): Total positivity tests and quiver mutation.
  • Note: no lecture on Wednesday September 14 (make up class TBA)
  • Lecture 3 (Monday September 19): Mutation.
  • Lecture 4 (Wednesday September 21): Cluster algebras of geometric type
  • Lecture 5 (Monday September 26): The Laurent phenomenon
  • Lecture 6 (Wednesday September 28): Applications of Laurent phenomenon. Y-patterns.
  • Lecture 7 (Monday October 3): New cluster algebras from old (subcluster algebras; start folding).
  • Lecture 8 (Wednesday October 5): Folding.
  • Lecture 9 (Monday October 10): Start finite type classification.
  • Lecture 10 (Wednesday October 12): Type A cluster algebras (finite type; frieze patterns etc).
  • Lecture 11 (Monday October 17): Type D cluster algebra (preview of cluster algebras from surfaces).
  • Lecture 12 (Wednesday October 19): Proof that Type B, C, E cluster algebras are of finite type.
  • Lecture 13 (Monday October 24): If a cluster algebra is of finite type it must come from a Dynkin diagram.
  • Lecture 14 (Wednesday October 26): Alternative criteria for finite type; cluster complexes.
  • Lecture 15 (Monday October 31): Cluster structures in commutative rings.
  • Lecture 16 (Wednesday November 2): The Starfish Lemma.
  • Monday November 7: Columbia holiday, no class!
  • Lecture 17 (Wednesday November 9): Cluster structures in the coordinate rings of matrices and Grassmannians.
  • Lecture 18 (Monday November 14): Cluster algebras from surfaces.
  • Lecture 19 (Wednesday November 16): Formulas for cluster variables in surface cluster algebras (in terms of matchings and matrix products).
  • Lecture 20 (Monday November 21): Decorated Teichmuller theory and cluster algebras from surfaces (or, where do tagged arcs come from?)
  • Lecture 21 (Wednesday November 23): Cluster varieties (of Fock and Goncharov).
  • Lecture 22 (Monday November 28): The totally non-negative Grassmannian and its cell decomposition.
  • Lecture 23 (Wednesday November 30): The totally non-negative Grassmannian and soliton solutions to the KP equation.
  • Lecture 24 (Monday December 5): The amplituhedron (total positivity and scattering amplitudes in N=4 SYM).
  • Lecture 25 (Wednesday December 7): Student presentations.
  • Lecture 26 (Monday December 12): Student presentations. FINAL PAPERS DUE!