18.217   M.I.T.   Fall 2018

## 18.217    Combinatorial Theory: Cluster Algebras

Class meets: MWF 1-2 pm   Room 2-147

Instructor: Alexander Postnikov

Course webpage: http://math.mit.edu/~apost/courses/18.217/

Synopsis: We will discuss combinatorial topics related to cluster algebras and total positivity. The course will be loosely based on several chapters of the forthcoming textbook "Introduction to Cluster Algebras" by Sergey Fomin, Lauren Williams, and Andrei Zelevinsky; as well as other sources.

Keywords: Cluster algebras, totally positive matrices, Laurent phenomenon, frieze patterns, triangulations of polygons, Ptolemy's relation, associahedra, mutations, quivers, Somos sequences, octahedron recurrence, double Bruhat cells, Grassmannian, positroids, plabic graphs, and more...

The course should be accessible to first year graduate students.

Problem Set: Problem Set (due December 3)

Lectures:

1. W 09/05/2018. Forerunners of cluster algebras: Coxeter's frieze patterns. Glide reflections and Laurent phenomenon. [Coxeter], [Conway-Coxeter].

2. F 09/07/2018. Ptolemy's relation and triangulations of an n-gon.

3. M 09/10/2018. Grassmannian Gr(2,n). Proofs of n-periodicity, glide reflection symmetry, and the positive Laurent phenomenon for Ptolemy's algebra and frieze patterns.

4. W 09/12/2018. Matchings in graphs. A combinatorial formula for Laurent polynomials. Somos sequences. [Carroll-Price], [Propp], [Schiffler].

Bibliography:

• [Carroll-Price]   G. D. Carroll, G. Price: Two new combinatorial models for the Ptolemy recurrence, 2003: paper

• [Conway-Coxeter]   J. H. Conway and H. S. M. Coxeter. Triangulated polygons and frieze patterns. Math. Gaz., 57(400):87-94, 175-183, 1973.

• [Coxeter]   H. S. M. Coxeter, Frieze patterns, Acta Arithmetica XVII (1971), 297-310: paper.

• [Fomin-Williams-Zelevinsky]   S. Fomin, L. Williams, A. Zelevinsky: Introduction to Cluster Algebras: arXiv:1608.05735 (chapters 1-3), arXiv:1707.07190 (chapters 4-5).

• [Propp]   J. Propp: The combinatorics of frieze patterns and Markoff numbers: arXiv:0511633, 2005.

• [Schiffler]   R. Schiffler: A cluster expansion formula (A_n case): Electronic Journal of Combinatorics, arXiv:0611956.

Lecture Notes:

Related course:

This webpage will be updated periodically. All information related to the course (list of lectures, references, problem sets, etc.) will be posted here.

last updated: September 12, 2018