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:

S. Fomin, L. Williams, and A. Zelevinsky,
Introduction to cluster algebras, Chapters 13.
 S. Fomin and A. Zelevinsky, Cluster
algebras I,
II, IV.

S. Fomin and A. Zelevinsky,
Cluster algebras: Notes for the CDM03 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. Ypatterns.
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 nonnegative Grassmannian
and its cell decomposition.
Lecture 23 (Wednesday November 30): The totally nonnegative 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!