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. | [FWZ] §2.2,§2.3,§2.4,§2.5,§2.6 | |
| 5 | Wednesday 1/28/26 | The matrix associated to a quiver. Skew-symmetrizable matrices and their mutations. Uniqueness of the skew-symmetrizing vector when the diagram of the matrix is connected. Rank and determinant are mutation invariants of a mutation. Definition of a labeled seed. | [FWZ] §2.7,§2.8 | |
| 6 | Wednesday 2/4/26 | Labeled seeds and their mutations. The $A_2$ example and its mutation graph. Revisting the exchange relations for triangulations of polygons (Plücker coordinates), wiring diagrams (flag minors), and double wiring diagrams (genera minors). Seed patterns. Definition of cluster algebras. | [FWZ] §3.1 | |
| 7 | Friday 2/6/26 | Examples of rank $1$, e.g. the algebra of invariant polynomials $\mathbb{C}[G]^U$ for $G = \operatorname{SL}_3(\mathbb{C})$ and $U \subset G$ the subgroup of unipotent lower triangular $3 \times 3$ matrices. Analysis of rank two cluster algebras, and the general case $\mathcal{A}(b,c)$ with no frozen variables. Revisiting the $A_2$ cluster algebra and its $5$-periodic exchange graph. The case of rank $2$ with one frozen variable. | [FWZ] §3.2 | |
| 8 | Monday 2/9/26 | More rank $2$ examples with no frozens: $\mathcal{A}(1,2),\mathcal{A}(1,3),\mathcal{A}(1,4)$. Precise statement of the Laurent phenomenon. Proof sketch under a simplifying assumption. Markov triples and the Markov equation. | [FWZ] §3.3, §3.4 | |
| 9 | Wednesday 2/11/26 | More on the Markov cluster algebra. The Somos-4 sequence and a proof of its integrality using cluster algebras. Introduction to Y-patterns. The tropical semifield. | [FWZ] §3.4, §3.5, §3.6 | |
| 10 | Friday 2/13/26 | More on Y-patterns and their mutations. Homomorphisms out of the universal semifield. Tropicalization of the $Y$-pattern mutation formula. Proof of $5$-periodicity of the $A_2$ cluster algebra with an arbitrary number of frozen variables. Introduction to the classification of finite type cluster algebras in the rank two case. | [FWZ] §3.5, §3.6, §5.1 | |
| Monday 2/16/26: Presidents' Day - no class | ||||
| 11 | Wednesday 2/18/26 | Finishing the proof of the finite type classification for rank two cluster algebras. Statement of the finite type classification theorem in general rank. Cartan matrices and Dynkin diagrams. The Cartan-Killing classification of Dynkin diagrams. Recalling the analogous statement for complex simple Lie algebras. | [FWZ] §5.2 | |
| 12 | Friday 2/20/26 | Introduction to surface cluster algebras and their connections with Teichmüller theory. | [W] §3 | |
| 13 | Monday 2/23/26 | More on surface cluster algebras. The once-punctured torus, the Farey tessellation, and the Markov quiver. The regular m-gon with zero, one, or two punctures. | [W] §3 | |
| 14 | Wednesday 2/25/26 | The $d$-vector of a cluster variable and its interpretation for surface cluster algebras. More examples, including annuli with one or two boundary punctures. The full list in ranks $2$ and $3$. An example of a $3$-vertex quiver of infinite mutation type. Conditions ensuring complexity growth under mutations for $3$-vertex cyclic quivers. | ||
| 15 | Friday 2/27/26 | Forks and the tree lemma. Conditions guaranteeing that an exchange graph is a tree. Diamonds and pentagons in the exchange graph. | [Wa] | |
| 16 | Monday 3/2/26 | Quiver representations and Gabriel's theorem. The path algebra of a quiver. Admissible orderings and canonical transjectives. Criterion for infinitely many canonical transjectives. Preliminary list of the $10$ possible exchange graphs for $3$-vertex quivers. | [Wa] | |
| 17 | Wednesday 3/4/26 | The root space, Coxeter group, and Coxeter element of a quiver. Characterization of quivers with finite Coxeter groups. The c-vectors, sign coherence, and mutation transformation rules. Proof that the canonical transjectives form an infinite chain for non-ADE quivers. The classification theorem for exchange graphs 3-vertex quivers and the description of the $10$ graphs. | [Wa] | |
| Week of March 9 - 13: student lectures on toric algebraic geometry | [F] Ch. 1-2 | |||
| March 16 - 20: Spring break - no class | ||||
| Week of March 23 - 27: student lectures on toric algebraic geometry | [F] Ch. 1-2 | |||
| 18 | Monday 3/30/26 | |||
| 19 | Wednesday 4/1/26 | |||
| 20 | Monday 4/6/26 | |||
| 21 | Wednesday 4/8/26 | |||
| 22 | Monday 4/13/26 | |||
| 23 | Wednesday 4/15/26 | |||
| 24 | Monday 4/20/26 | |||
| 25 | Wednesday 4/22/26 | |||
| 26 | Monday 4/27/26 | |||
| 27 | Wednesday 4/29/26 |