Math 665: Cluster algebras

Fall 2014

Course meets: Tuesday and Thursday 11:40-1:00 in 3088 East Hall.

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

Course homepage: http://www.math.lsa.umich.edu/~fomin/665f14.html

Level: introductory graduate.

Prerequisites: none (for graduate students).

Student work expected: several problem sets.

Synopsis: Cluster algebras are a class of commutative rings constructed from a certain kind of combinatorial data via a recursive “mutation” procedure. 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 fundamental notions and results of the theory of cluster algebras, and its most basic applications. Combinatorial aspects will be emphasized throughout. No special background in commutative algebra, representation theory, or combinatorics is required.

Helpful links:

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

References (none will be followed closely):

• S. Fomin and N. Reading, Root systems and generalized associahedra.
• S. Fomin and A. Zelevinsky, Cluster algebras I, II, IV.
• S. Fomin and A. Zelevinsky, Cluster algebras: Notes for the CDM-03 conference.
• S. Fomin, Total positivity and cluster algebras.
• R. Marsh, Lecture Notes on Cluster Algebras.
• L. Williams, Cluster algebras: an introduction.

Lecture topics:

1. Total positivity.
   Grassmannians of planes.
2. Base affine spaces. Wiring diagrams.
3. Total positivity criteria for square matrices.
   Quiver mutation.
4. Triangulations of unpunctured surfaces.
   Mutation equivalence.
5. Matrix mutations.
6. Cluster algebras of geometric type.
7. Examples of small rank.
8. The Laurent phenomenon.
9. Connections to number theory.
10. Guest lecture by D. Thurston: Lambda lengths.
11. Y-patterns.
12. Configurations in P¹. The pentagram map.
13. Tropical semifields. Changing the coefficients.
14. Seed patterns of finite type. Rank 2.
15. Seed patterns of type A_n.
16. Seed patterns of type D_n.
17. Seed patterns of type D_n (continued).
18. Folding.
19. Seed patterns of type B_n and C_n.
20. Cartan matrices, Dynkin diagrams, and root systems.
21. Classification of seed patterns of finite type.
22. 2-finite exchange matrices. Quasi-Cartan companions.
23. Cluster structures in commutative rings. Examples of types ABCD.
24. The Starfish Lemma. Cluster structures for base affine spaces.
25. Survey lecture #1. Cluster structures in Grassmannians.
26. Survey lecture #2. Weak separation. Cluster structures in classical rings of invariants. Tensor diagrams and webs.
27. Survey lecture #3. Zamolodchikov periodicity. Cluster complexes and exchange graphs. Cluster algebras from surfaces. Finite mutation type classification. Growth rate classification.