Math 8680, Topics in Combinatorics: Cluster Algebras and their Variations (Fall 2018)

Lectures: MWF 11:15-12:05 in Vincent Hall 311.

Instructor: Gregg Musiker (musiker "at"

Office Hours (in Vincent Hall 251)
          Monday 10:00-11:00
          Wednesday 12:20 - 1:00
          Friday 2:30 - 3:20
Also, by appointment, or feel free to knock. I usually keep my door open if I'm in.

Course Description:

This is a graduate level topics course in algebraic combinatorics providing an introduction to cluster algebras as well as how these combinatorial structures connect to many different areas of mathematics and physics. Additionally, we will discuss extensions of the original definitions, including quantum and noncommutative cluster algebras, generalized cluster algebras, and higher cluster categories. Along the way, we will introduce and use techniques from tropical geometry, Teichmuller theory, and the theory of Grassmannians as we focus on certain classes of cluster algebras, such as those from surfaces or positroids. This course will also highlight recent developments such as maximal green sequences, greedy bases, and scattering diagrams.

While there is no required textbook, I recommend "Cluster Algebras and Poisson Geometry'', which we will draw from throughout the course. These will be supplemented with survey articles, including excerpts of the forthcoming book by Fomin, Williams, and Zelevinsky, and a variety of research articles.

Prerequisites: No prior knowledge of cluster algebras will be assumed; although familiarity with groups, rings, and modules, as in Math 8202, will be helpful.

Recommended (but not required) Texts:
Cluster Algebras and Poisson Geometry by Michael Gekhtman, Michael Shapiro, and Alek Vainshtein (2010, AMS Monograph).
Lecture Notes on Cluster Algebras by Robert Marsh (2013, EMS Zurich Lectures in Advanced Mathematics).
On reserve in the math library

Recommended Survey Articles:
  • [FWZ16] Introduction to Cluster Algebras (Chaps 1-3 Prelim Version)
  • [FWZ17] Introduction to Cluster Algebras (Chaps 4-5 Prelim Version)
  • [GR18] Introduction to Cluster Algebras
  • [Sch16] Cluster algebras from surfaces - Lecture notes for the CIMPA School
  • [M-G15] Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics
  • [H14] Periodic cluster mutations and related integrable maps
  • [N11] Tropicalization method in cluster algebras
  • [K12] Cluster algebras and derived categories
  • [FR04] Root systems and generalized associahedra (by Sergey Fomin and Nathan Reading, IAS/Park City 2004)
  • [McD09] What is Symplectic Geometry (by Dusa McDuff)

  • Recommended Research Articles:
  • [FZ02] Y-Systems and Generalized Associahedra
  • [FG05] Dual Teichmuller and Lamination Spaces
  • [FZ06] Cluster algebras IV: Coefficients
  • [FM09] Cluster Mutation-Periodic Quivers and Associated Laurent Sequences
  • [FST08] Cluster algebras and triangulated surfaces. Part I: Cluster complexes
  • [Sch08] A cluster expansion formula (An case)
  • [Sp08] Perfect matchings and the Octahedron Recurrence
  • [MS10] Cluster expansion formulas and perfect matchings
  • [Gl10] The pentagram map and Y-patterns
  • [GK11] Dimers and cluster integrable systems
  • [MSW11] Positivity for cluster algebras from surfaces
  • [Mul16] Skein algebras and cluster algebras of marked surfaces
  • [FT18] Cluster algebras and triangulated surfaces. Part II: Lambda lengths

  • Suggestions for Twenty-five Minute Presentations:
  • [FZ02] The Laurent Phenomenon
  • [SZ03] Positivity and Canonical Bases in Rank 2 Cluster Algebras of Finite and Affine Types
  • [CZ05] Laurent expansions in cluster algebras via quiver representations
  • [MW11] Matrix formulae and skein relations for cluster algebras from surfaces
  • [FST11] Cluster algebras and triangulated orbifolds
  • [CS11] Teichmuller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables
  • [GSTV11] Higher pentagram maps, weighted directed networks, and cluster dynamics
  • [FSTT12] Growth rate of cluster algebras
  • [LLZ12] Greedy elements in Rank 2 cluster algebras
  • [CGMMRSW15] The Greedy Basis Equals the Theta Basis: A Rank Two Haiku
  • [BM15] Maximal green sequences for cluster algebras associated to the n-torus with arbitrary punctures
  • [Ra16] F-polynomial formula from continued fractions
  • [K17] Quiver mutation and combinatorial DT-invariants
  • [GW17] Gale-Robinson Quivers: From Representations to Combinatorial Formulas
  • [Re18] A combinatorial approach to scattering diagrams

  • Helpful Resources:
    Sage Math via CoCalc

    Keller's Quiver Applet in Java

    My previous graduate topics courses on cluster algebras

    MSRI Graduate Summer School on Cluster Algebras (from Summer 2011)

    Grading: There will be no exams, but registered students are expected to attend, and should hand in the homework assignments. There will be two to three assignments over the course of the semester. I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page their collaborators. Depending on interest, there may be an in-class presentation on a research paper instead of one of the homework assignments.

    Tentative Lecture Schedule

  • (Sep 5) Lecture 1: Introduction to the Course; First Examples (Sections 1-2 of [GR18])
  • (Sep 7) Lecture 2: Seeds and Mutation (Section 3 of [GR18] or Section 3.1 of [GSV10])
  • (Sep 10) Lecture 3: Cluster Structure of Plucker coordinates (Section 2.1 of [GSV10] or Section 1.2 of [FWZ16])
  • (Sep 12) Lecture 4: Cluster Structures associated to triangulations (Chapter 2 of [Sch16] or Lecture 1 of [MSRI-Mus11]) (See also Lecture 21 from 2011)
  • (Sep 14) Lecture 5: Cluster Structures associated to triangulations II (See also [FST08])
  • (Sep 17) Lecture 6: Cluster algebras from surfaces III: Combinatorial formulas (Chapter 3 of [Sch16] or Section 7 of [MS10] or Lecture 1 and Lecture 3 of [MSRI-Mus11])
  • (Sep 19) Lecture 7: Cluster algebras from surfaces IV: Punctured Surfaces and Tagged Arcs (Lecture 2 and Lecture 3 of [MSRI-Mus11]) (See also Lecture 22 and Lecture 23 from from 2011)
  • (Sep 21) Lecture 8: Cluster algebras from surfaces V: Sketch of Positivity Proof ([Sch08], [MS10], [MSW11])
  • (Sep 24) Lecture 9: Crash Course on Poisson Geometry (Section 1.3 of [GSV10] or Section 1 of [McD09])
  • (Sep 26) Lecture 10: Cluster algebras of geometric type via Poisson geometry (Section 4.1 of [GSV10] or Section 5 of [GR18]; also see [Mul16, Sec 6.2])
  • (Sep 28) Lecture 11: Cluster algebra via Poisson geometry II (Sec 4.1 of [GSV10])
  • (Oct 1) Lecture 12: Cluster algebra via Poisson geometry III (Sec 4.1 of [GSV10])
  • (Oct 3) Lecture 13: Pre-symplectic structures (Sec 6.1 of [GSV10])
  • (Oct 5) Lecture 14: Teichmuller space and Hyperbolic Geometry (Sec 6.2 of [GSV10], Lecture 26 from from 2011, and Lecture 4 of [MSRI-Mus11]) )
  • (Oct 8) Lecture 15: Decorated Teichmuller space and lambda lengths (Sec 6.2 of [GSV10], Lecture 27 from from 2011, and Lecture 5 of [MSRI-Mus11]))
    (See Mobius Functions Revealed by our very own Doug Arnold and Jon Rogness)
  • (Oct 10) Lecture 16: Cross Ratio Coordinates, Lambda Lengths and Mutation (Sec 6.2 of [GSV10] and [FT18])
  • (Oct 12) Lecture 17: Cluster Algebras from Surfaces with Punctures via Lambda Lengths (Sec 6.2 of [GSV10] and [FT18])
  • (Oct 15) Lecture 18: Cross Ratio Coordinates under Mutation (Sec 6.2 of [GSV10] and [FT18])
  • (Oct 17) Lecture 19: Shear Coordinates, Laminations, and Reconstructing Arbitrary Extended Exchange Matrices ([FT18] and [FG05])
  • (Oct 19) Lecture 20: Tropical Versions of Lambda Lengths and towards Y-Systems ([FG05] and Sections 3.3 and 6 of [GR18], [FZ02])
  • (Oct 22) Lecture 21: Zamolodchikov and Keller Periodicity (Sections 3.3-3.6 of [FWZ16], [FZ02])
  • (Oct 24) Lecture 22: Cluster Algebras with Auxiliary Additions; Seed Data Triples (Sections 3 and 4 of [FZ06])
  • (Oct 26) Lecture 23: F-polynomials and g-vectors (Sections 5 and 6 of [FZ06] and [N11])
  • (Oct 29) Lecture 24: Periodic Quivers and Somos Sequences ([FM09])
  • (Oct 31) Lecture 25: The Classification of 1-Periodic Quivers and The Pentagram Map (Section 6 of [FM09] and Section 6.2 of [GR18])
  • (Nov 2) Lecture 26: Aztec Diamonds, F-polynomials and the Pentagram Map ([Gl10])
  • (Nov 5) Lecture 27: The Octahedron Recurrence ([Sp08])
  • (Nov 7) Lecture 28: A Combinatorial Interpretation of the Octahedron Recurrence (See Lecture 27 from 2015)
  • (Nov 9) No class due to Fomin Fest
  • (Nov 12) Lecture 29: Gale-Robinson Sequences; other Discrete Integrable Systems
  • (Nov 14) Lecture 30: Combinatorics of Gale-Robinson Sequences I
  • (Nov 16) Lecture 31: Combinatorics of the Octahedron Recurrence and Gale-Robinson Sequences II
  • (Nov 19) Lecture 31: Cluster Integrable Systems I ([GK11])
  • (Nov 21) Lecture 32: Cluster Integrable Systems II
  • (Nov 23) Happy Thanksgiving!
  • (Nov 26) Lecture 33: Cluster Integrable Systems III ([GK11])
  • (Nov 28) Lecture 34: Cluster Integrable Systems IV ([GK11])
  • (Nov 30) Lecture 35: Cluster Integrable Systems V ([GK11])
  • (Dec 3) Lecture 36: In Class Presentations: The Laurent Phenomenon / F-polynomials and Continued Fractions
  • (Dec 5) Lecture 37: In Class Presentations: Growth Rates of Cluster Algebras from Surfaces / Cluster Algebras from Orbifolds
  • (Dec 7) Lecture 38: In Class Presentations: Maximal Green Sequences / Scattering Diagrams
  • (Dec 10) Lecture 39: In Class Presentations: Cluster Algebras and Root Systems / Gale-Robinson Quiver Representations
  • (Dec 12) Lecture 40: In Class Presentations: Resistor Networks / Birational Rowmotion and R-systems

    Homework assignments

    Assignment Due date
    HW 1 Monday October 1
    HW 2 Monday November 19
    HW 3 25 Minute Presentations Dec 3rd, 5th, 7th, 10th, and 12th