Math 665: Cluster algebras

Fall 2021

Course meets: Tuesday 11:30-12:50 in 1060 East Hall, Thursday 11:30-12:50 in 1096 East Hall.

Instructor: Sergey Fomin, 4868 East Hall, 764-6297, fomin@umich.edu

Course homepage: http://www.math.lsa.umich.edu/~fomin/665f21.html

Level: introductory graduate.

Prerequisites: none (for graduate students).

Synopsis: Cluster algebras are a class of commutative rings endowed with a particular kind of combinatorial structure. They arise in a variety of contexts including representation theory of Lie groups, Teichmüller theory, discrete integrable systems, classical invariant theory, and quiver representations. This course will provide an elementary introduction to the basic notions and results of the theory of cluster algebras, and present some of its most accessible applications. Combinatorial aspects will be emphasized. No special background in commutative algebra, representation theory, or combinatorics is required.

Main reference: S. Fomin, L. Williams, and A. Zelevinsky, Introduction to cluster algebras.

• Chapter 1. Total positivity
• Chapter 2. Mutations of quivers and matrices
• Chapter 3. Clusters and seeds
• Chapter 4. New patterns from old
• Chapter 5. Finite type classification
• Chapter 6. Cluster structures in commutative rings
• Chapter 7. Plabic graphs  

Additional references:

• S. Fomin and N. Reading, Root systems and generalized associahedra.
• S. Fomin and A. Zelevinsky, Cluster algebras I, II, IV.
• R. Marsh, Lecture notes on cluster algebras.
• L. Williams, Cluster algebras: an introduction.

Helpful links:

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

Lecture topics:

1. Grassmannians of planes. [slides]
2. Basic affine spaces. [slides]
3. In search of cluster structures. Total positivity. [slides]
4. Quiver mutations. Example: triangulated surfaces. [Sections 2.1-2.2]
5. Wiring diagrams. Mutation equivalence [Sections 2.3, 2.6].
6. Acyclic quivers. Quivers of finite mutation type.
7. Full subquivers [Section 4.1]. Exchange matrix of a quiver [Section 2.7].
8. Matrix mutations and their invariants [Sections 2.7-2.8].
9. Seeds and their mutations [Section 3.1].
10. Cluster algebras of rank 1 and 2 [Section 3.2].
11. The Laurent Phenomenon [Section 3.3].
12. Markov triples. Integrability and linearizability. Somos sequences [Section 3.4].
13. Y-patterns [Section 3.5].
14. Point configurations on a projective line [Section 3.5].
15. The pentagram map [Section 3.5]. Tropical semifields [Section 3.6].
16. Cluster algebras over semifields [Section 3.6].
17. Rescaling of cluster variables [Section 4.3].
18. Finite type classification in rank 2 [Section 5.1].
19. Cartan matrices, Dynkin diagrams, and root systems [Section 5.2].
20. Finite type classification (statement) [Section 5.2]. Seed patterns of type A_n [Section 5.3].
21. Examples of seed patterns of type A_n [Section 5.3].
22. Seed patterns of type D_n [Section 5.4].
23. Seed patterns of types E_n [Section 5.6]. Folding and unfolding [Section 4.4].
24. Seed patterns of types B_n and C_n [Section 5.5]. Seed patterns of types F₄ and G₂ [Section 5.7].
25. Cluster structures in commutative rings [Section 6.2].
26. The Starfish Lemma [Section 6.4].
27. Cluster structures in basic affine spaces [Section 6.5].
28. Cluster structures in C[SL_k] [Section 6.6]. Cluster structures in Grassmannians [Section 6.7].
    [slides for lectures 27-28]

• Problem Set #1   due 9/30
• Problem Set #2   due 11/16