Cluster Algebras — 2019


Lectures: Monday, Tuesday 11.30am-1pm.

Lecturer: Jon Wilson (jon.wilson@im.unam.mx)

Location: Second floor lounge of the new IMUNAM building.


Course Description:

Fomin and Zelevinsky's cluster algebras are a 21st centrury construct which propogate through vast areas of mathematics.
The general set-up of their cluster algebra is incredibly simple; one starts with a finite number of variables, called an initial cluster,
and then iteratively creates more using only high school algebra.
In this course we will explore some of the fundamental properties of cluster algebras, and tour combinatorial, geometric and algebraic concepts
which exhibit this newfound structure.

Lecture Schedule

Lecture 1: Motivation for cluster algebras Part I — positivity tests of Gr(2,n).

Lecture 2: Motivation for cluster algebras Part II — positivity tests of flag minors.

Lecture 3: Motivation for cluster algebras Part III — total positivity tests of nxn matrices.

Lecture 4: Definition of a cluster algebra and examples.

Lecture 5: Diagram mutation and the link between diagrams and skew-symmetrizable matrices.

Lecture 6: Proof of the Laurent phenomenon Part I.

Lecture 7: Proof of the Laurent phenomenon Part II.

Lecture 8: Y-seed patterns, cross ratios, and the Pentegram Map.

Lecture 9: Moving towards finite type classification Part I — tropical semi-fields and Y-dynamics in replace of extended part of exchange matrices.

Lecture 10: Moving towards finite type classification Part II — coverings of exchange graphs via change of coefficients, and the classification of rank 2 cluster algebras.

Lecture 11: Classification of type An seed patterns.

Lecture 12: Classification of type Dn seed patterns.

Lecture 13: Folding quivers to get new seed patterns.

Lecture 14: Classification of type Bn, Cn, F4, G2 seed patterns via folding, and a remark on the finiteness of E6, E7, E8.

Lecture 15: Classification of finite type cluster algebras via 2-finiteness.

Lecture 16: Basics of bordered surfaces and their ideal triangulations.

Lecture 17: Connectivity of ideal triangulations via flips.

Lecture 18: Cluster algebras from surfaces Part I — tagged triangulations and their associated quivers.

Lecture 19: Cluster algebras from surfaces Part II — the decorated Teichmüller space.

Lecture 19: Cluster algebras from surfaces Part III — keeping track of coefficients via laminations.

Lecture 20: Mutation finite cluster algebras (of skew-symmetric type).

Lecture 21: Denominator vectors, g-vectors and c-vectors.

Lecture 22: F-polynomials and 'separation of additions'.

Student Presentations

Edgar Vázquez : Cluster variable expansions via perfect matchings of snake graphs.

Andrés Flores : Properties of Laurent phenomenon algebras.