Topics in cluster algebras

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:

Cluster Algebras Portal
Java applets for quiver mutations and clusters/exchange graphs by B. Keller

Some references:

S. Fomin, L. Williams, and A. Zelevinsky, Introduction to cluster algebras, Chapters 1-3.
S. Fomin and A. Zelevinsky, Cluster algebras I, II, IV.
S. Fomin and A. Zelevinsky, Cluster algebras: Notes for the CDM-03 conference.
S. Fomin, Total positivity and cluster algebras.
R. Marsh, Lecture Notes on Cluster Algebras.
L. Williams, Cluster algebras: an introduction.

Lectures

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!