In this course we will explore cluster algebras and their connections to combinatorics, geometry, topology, representation theory, and string theory. Cluster algebras were discovered by Fomin and Zelevinsky in 2000. Since then, they have been shown to be related to a wide variety of branches of mathematics and mathematical physics. These fundamental objects can be constructed in an elementary manner. No background is required, as we will start from scratch.

Roughly one-third of the lectures will be devoted to combinatorics including graph theory and algebraic combinatorics, and another one-third to geometry/topology including algebraic geometry, hyperbolic geometry, Teichm\"uller theory and knot theory, and the rest to representation theory as well as homological algebra.

- Introduction to Cluster Algebras (Chaps 1-3 Prelim Version) (by Sergey Fomin, Andrei Zelevinsky, and Lauren Williams)
- Cluster algebras: an introduction (by Lauren Williams)
- Root systems and generalized associahedra (by Sergey Fomin and Nathan Reading, IAS/Park City 2004)
- Tilting theory and cluster algebras (by Idun Reiten)
- Cluster algebras, quiver representations and triangulated categories (by Bernhard Keller 2008)
- More articles available at the Cluster Algebras Portal.

- (Jan 10) Lecture 1: Introduction to the Course and Quiver Mutations
- (Jan 12) Lecture 2: Seed Mutations
- (Jan 19) Lecture 3: Mutation Invariants, Quivers of Finite Mutation Type, and the Path Algebras
- (Jan 24) Lecture 4: Mutation Invariants
- (Jan 26) Lecture 5: Introduction to Quiver Representations
- (Jan 31) Lecture 6: T-path formulas for type A cluster algebras, and Grassmannians
- (Feb 2) Lecture 7: Cluster algebras of type D, and Coxeter groups (as mutation invariants) [reference : Reflection group presentations arising from cluster algebras (by Michael Barot and Robert J. Marsh)]
- (Feb 7) Lecture 8: Snake gaphs [reference : Positivity for cluster algebras from surfaces (by Gregg Musiker, Ralf Schiffler, and Lauren Williams)]

Cluster categories [reference : Cluster algebras and cluster categories (by Ralf Schiffler)] - (Feb 9) Lecture 9: Cluster categories
- (Feb 14) Lecture 10: Knot theory [reference : Braids, Complex Volume, and Cluster Algebra (by Kazuhiro Hikami, Rei Inoue)]

Maximal green sequences [reference : Quiver mutation and quantum dilogarithm identities (by Bernhard Keller)] - (Feb 16) Lecture 11: Maximal green sequences (continued)

Minimal mutation-infinite quivers [reference : Maximal green sequences of minimal mutation-infinite quivers (by John Lawson and Matt Mills)] - (Feb 21) Lecture 12: Quiver Grassmannians
- (Feb 23) Lecture 13: Root system for Kac-Moody algebras
- (Feb 28) Lecture 14: Root system for Kac-Moody algebras (continued)

Frieze Patterns - (Mar 2) Lecture 15: Projective resolutions for quiver representations, and quadratic forms

Canonical bases for cluster algebras - (Mar 7) Lecture 16: Canonical bases
- (Mar 9) Lecture 17: Introduction to A-infinity algebras

Bases for cluster algebras - (Mar 14) Lecture 18: Introduction to A-infinity algebras (continued) [reference : Introduction to A-infinity algebras and modules (by Bernhard Keller)]

Bases for cluster algebras (continued) - (Mar 28) Lecture 19: Introduction to symplectic forms
- (Mar 30) Lecture 20: reflection groups, revisited
- (April 4) Lecture 21: Vector bundles
- (April 6) Lecture 22: guest lecture by Gregg Musiker (University of Minnesota)
- (April 11) Lecture 23: Vector bundles, Del Pezzo surfaces, and reflexive polygons
- (April 13) Lecture 24: Hilbert modules, and derived category
- (April 18) Lecture 25: continued fractions, snake graphs, and F-polynomials [reference : F-polynomial formula from continued fractions (by Michelle Rabideau)]
- (April 20) Lecture 26: continued fractions, snake graphs, and F-polynomials (continued)
- (April 25) Lecture 27: Introduction to quantum cluster algebras
- (April 27) Lecture 28: greedy basis