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:

Additional references:

Helpful links:

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. Finite mutation type. Restrictions and embeddings. [Sections 2.6, 4.1]