Math 665: Cluster algebras
Course meets: Tuesday and Thursday 11:40-1:00 in 3088 East Hall.
Sergey Fomin, 4868 East Hall, 764-6297,
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.
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,
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
No special background in commutative algebra, representation theory,
or combinatorics is required.
References (none will be followed closely):
- Total positivity.
Grassmannians of planes.
- Base affine spaces. Wiring diagrams.
- Total positivity criteria for square matrices.
- Triangulations of unpunctured surfaces.
- Matrix mutations.
- Cluster algebras of geometric type.
- Examples of small rank.
- The Laurent phenomenon.
- Connections to number theory.
- Guest lecture by D. Thurston: Lambda lengths.
- Configurations in P1. The pentagram map.
- Tropical semifields. Changing the coefficients.
- Seed patterns of finite type. Rank 2.
- Seed patterns of type An.
- Seed patterns of type Dn.
- Seed patterns of type Dn (continued).
- Seed patterns of type Bn and Cn.
- Cartan matrices, Dynkin diagrams, and root systems.
- Classification of seed patterns of finite type.
- 2-finite exchange matrices. Quasi-Cartan companions.
- Cluster structures in commutative rings. Examples of
- The Starfish Lemma. Cluster structures for base affine spaces.
- Survey lecture #1.
Cluster structures in Grassmannians.
- Survey lecture #2.
Cluster structures in classical rings of invariants.
Tensor diagrams and webs.
- Survey lecture #3.
Cluster complexes and exchange graphs.
Cluster algebras from surfaces.
Finite mutation type classification.
Growth rate classification.