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:
Lectures
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.)