Math 669: Cluster algebras

Winter 2018

Course meets: Tuesday and Thursday 1:10-2:30 in 3866 East Hall.

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

Course homepage: http://www.math.lsa.umich.edu/~fomin/669w18.html

Level: introductory graduate.

Prerequisites: none (for graduate students).

Synopsis: Cluster algebras are a class of commutative rings constructed via a recursive combinatorial process of "seed mutations." They arise in a variety of algebraic and geometric 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 throughout. 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. Chapters 1-3, Chapters 4-5.

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. Motivating examples: Grassmannians of planes, basic affine spaces. [Slides]
2. Quiver mutations. Examples: triangulations, wiring diagrams. [Sections 2.1-2.4]
3. Plabic graphs. [arXiv:1711.10598, Section 7]
4. Mutation equivalence [Section 2.6]. Triangulated surfaces [arXiv:math/0608367, Sections 2, 4].
5. Finite mutation type [arXiv:0811.1703]. Restriction and freezing for quivers [Sections 4.1-4.2].
6. Matrix mutations [Sections 2.7-2.8].
7. Folding and unfolding [Section 4.4].
8. Cluster algebras of geometric type [Section 3.1].
9. Cluster algebras of rank 2 [Section 3.2]. The Laurent Phenomenon (easy cases).
10. The Laurent Phenomenon [Section 3.3].
11. Somos sequences [Section 3.4]. Y-patterns [Section 3.5].
12. Point configurations on a projective line. The pentagram map [Section 3.5].
13. Tropical semifields [Section 3.6]. Rescaling [Section 4.3].
14. Seed patterns of finite type. Rank 2 [Section 5.1]. Cartan matrices and Dynkin diagrams [Section 5.2].
15. Finite type classification (statement) [Section 5.2]. Seed patterns of type A_n [Section 5.3].
16. Seed patterns of type D_n [Section 5.4].
17. Seed patterns of types B_n and C_n [Section 5.5]. Seed patterns of exceptional types [Sections 5.6-5.7].
18. Guest lecture #1 by D. Thurston: Cluster algebras from surfaces: hyperbolic geometry.
19. Guest lecture #2 by D. Thurston: Cluster algebras from surfaces: representation theory.
20. Cluster algebras from surfaces: recapitulation. Seed patterns of finite type: enumeration [Sections 5.8-5.9].
21. 2-finite exchange matrices. Quasi-Cartan companions [Sections 5.10-5.11]. Cluster structures in commutative rings.
22. The Starfish Lemma.
23. Cluster structures for base affine spaces.
24. Cluster structures in C[SL_k]. Cluster structures in Grassmannians.
25. Zamolodchikov periodicity (type A_k×A_l). Cluster structures in rings of invariants.
26. Guest lecture #3 by A. Leaf: Cluster algebras and dimer models.