9:35am - 10:50am on Mondays and Wednesdays in 245 Kaprielian Hall (KAP).
We will assume familiarity with basic algebraic topology (e.g. Math 540a), smooth manifold theory (e.g. Math 535a), and commutative algebra (e.g. Math 510a). Some familiarity with algebraic geometry (especially toric geometry) will be very helpful but will not be assumed.
This course is an introduction to cluster varieties from the geometric viewpoint. In the first part of course, we will introduce the formalism of cluster algebras and discuss their basic features, including mutation graphs, examples, classification results, and so on. In the second part of the course, we will study cluster varieties via geometric lens of Gross-Hacking-Keel as log Calabi-Yau varieties. In the third part of the course, we will sample various advanced topics, including Fock-Goncharov duality and mirror symmetry, scattering diagrams, and the construction of canonical bases by Gross-Hacking-Keel-Kontsevich. In the final weeks of the course, each registered student will be expected to give a short presentation on a chosen topic, along with an accompanying writeup.
| $\#$ | Date | Material | References | Problem set |
|---|---|---|---|---|
| 1 | Monday 1/12/26 | Heuristic introduction to cluster algebras and the results of Gross-Hacking-Keel-Kontsevich. Total positivity for $n \times n$ matrices and Grassmannians. Plücker coordinates on Grassmannians. The Plücker ring and Grassmann-Plücker relations. Cluster variables, frozen variables, and extended cluster charts $\widetilde{x}(T)$ on $\operatorname{Gr}_{2,m}$ for each triangulation $T$ of the $m$-gon. | [FWZ] §1.1,§1.2. | |
| 2 | Wednesday 1/14/26 | More on positivity and "efficient testing" for $Gr_{2,m}$. Triangulations of the regular $m$-gon and flips between triangulations. The associated mutation graph as the $1$-skeleton of the associahedron, and the examples $m=5,6$. Cluster monomials as a linear basis for the Plücker ring. Flag minor and flag positivity for $G = \operatorname{SL}_n$. Wiring diagrams and efficient testing for flag positivity. Braid moves between wiring diagrams. | [FWZ] §1.1,§1.2. | |
| Monday 1/19/26: MLK Day - no class | ||||
| 3 | Friday 1/23/26 | The Borel subgroup $B \subset G := \operatorname{SL}_n(\mathbb{C})$ of lower triangular matrices and the subgroup $U \subset G$ of unipotent lower triangular matrices. The full flag variety $B\backslash G$ and basic affine space $U\backslash G$. The ring of invariant polynomials $\mathbb{C}[G]^U$ is generated by flag minors. Total positivity testing for $n \times n$ matrices and double wiring diagrams. Quivers, ice quivers, and mutations. Basic properties of quiver mutations. | [FWZ] §1.3,§1.4,§2.1. | |
| 4 | Monday 1/26/26 | Quivers from triangulations, wiring diagrams, and double wiring diagrams. Quiver mutations correspond to flips, braid moves, and the "local moves". Plabic graphs and their associated quivers. The (de)contraction move and spider move. The mutation equivalence class of a quiver. Examples of acyclic quivers and quivers of finite mutation type. Mutation equivalent acyclic quivers have the same underlying undirected graphs. | ||
| 5 | Wednesday 1/28/26 | |||
| 6 | Wednesday 2/4/26 | |||
| 7 | Friday 2/6/26 | |||
| 8 | Monday 2/9/26 | |||
| 9 | Wednesday 2/11/26 | |||
| Monday 2/16/26: Presidents' Day - no class | ||||
| 10 | Wednesday 2/18/26 | |||
| 11 | Monday 2/23/26 | |||
| 12 | Wednesday 2/25/26 | |||
| 13 | Monday 3/2/26 | |||
| 14 | Wednesday 3/4/26 | |||
| 15 | Monday 3/9/26 | |||
| 16 | Wednesday 3/11/26 | |||
| Monday 3/16/26: Spring Break - no class | ||||
| Wednesday 3/18/26: Spring Break - no class | ||||
| 17 | Monday 3/23/26 | |||
| 18 | Wednesday 3/25/26 | |||
| 19 | Monday 3/30/26 | |||
| 20 | Wednesday 4/1/26 | |||
| 21 | Monday 4/6/26 | |||
| 22 | Wednesday 4/8/26 | |||
| 23 | Monday 4/13/26 | |||
| 24 | Wednesday 4/15/26 | |||
| 25 | Monday 4/20/26 | |||
| 26 | Wednesday 4/22/26 | |||
| 27 | Monday 4/27/26 | |||
| 28 | Wednesday 4/29/26 |