Math 8680, Topics in Combinatorics: Cluster Algebras, Tilings, and Physics (Spring 2015)

Lectures: MW 4:00-5:30 in Vincent Hall 311.

Instructor: Gregg Musiker (musiker "at"

Office Hours: TBA. 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. The topic for this semester is cluster algebras, tilings, and physics. Cluster algebras are a class of combinatorially defined rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. A partial list of related areas includes quiver representations, statistical physics, Teichmuller theory, and string theory. This course will focus on combinatorial aspects of cluster algebras, as well as their connections to dimer models and gauge theories in physics. Besides providing background on the fundamentals of cluster theory, we will discuss Kastelyn matrices, Pfaffian orientations, Kuo condensation, Dodgson condensation, and other techniques for enumerating domino tilings (i.e. dimers). While there is no required textbook, we will begin with "Lecture Notes on Cluster Algebras" by Robert Marsh, followed by selections from "Lectures on Dimers" by Richard Kenyon and "Dimer Models and Calabi-Yau Algebras" by Nathan Broomhead. These will be supplemented with articles from both mathematics and the physics literature.

Part 1 of this course will closely follow my Spring 2011 Course on Cluster Algebras and Quiver Representations.

Parts 2 and 3 will include cutting edge connections to dimer models and string theory.

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

Recommended (but not required) Texts:
Lecture Notes on Cluster Algebras by Robert Marsh (2013, EMS Zurich Lectures in Advanced Mathematics).
Dimer Models and Calabi-Yau Algebras by Nathan Broomhead (2012, Memoir of the AMS).
Cluster Algebras and Poisson Geometry by Michael Gekhtman, Michael Shapiro, and Alek Vainshtein (2010, AMS Monograph).

Recommended Survey Articles:
  • [F10] Total Positivity and cluster algebras (by Sergey Fomin, ICM 2010)
  • [W10] Cluster algebras: an introduction (by Lauren Williams)
  • [FR04] Root systems and generalized associahedra (by Sergey Fomin and Nathan Reading, IAS/Park City 2004)
  • [K09] Lectures on Dimers (by Rick Kenyon)
  • [B09] Dimer Models and Calabi-Yau Algebras (by Nathan Broomhead)
  • [HK05] Dimer Models and Toric Diagrams (by Amihay Hanany and Kristian Kennaway)
  • [FHMSVW05] Gauge Theories from Toric Geometry and Brane Tilings (by Sebastian Franco, Amihay Hanany, Dario Martelli, James Sparks, David Vegh, and Brian Wecht)
  • [MR08] On the noncommutative Donaldson-Thomas invariants arising from brane tilings (by Sergey Mozgovoy and Markus Reineke)
  • [K07] Brane Tilings (Survey) (by Kristian Kennaway)
  • [GK11] Dimers and cluster integrable systems (by Alexander Goncharov and Rick Kenyon)
  • [DiF13] T-systems, networks and dimers (by Philippe Di Francesco)
  • [UY11] A note on dimer models and McKay quivers (by Kazushi Ueda and Masahito Yamazaki)

  • More articles available at the Cluster Algebras Portal.

  • Research Articles (Some Suggestions for Presentations):
  • [DG14] Arctic curves of the octahedron equation (by Philippe Di Francesco and Rodrigo Soto-Garrido)
  • [S11] Polyhedral models for generalized associahedra via Coxeter elements (by Salvatore Stella)
  • [N14] Structure of seeds in generalized cluster algebras (by Tomoki Nakanishi)
  • [BGH14] Gauge Theories and Dessins d'Enfants:Beyond the Torus (by Sownak Bose, James Gundry, and Yang-Hui He)
  • [GSTV14] Integrable cluster dyanmics of directed networks and pentagram maps (by Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, and Alex Vainshtein)
  • [GMT14] Anatomy of the Amplituhedron (by Sebastian Franco, Daniele Galloni, Alberto Mariotti, and Jaroslav Trnka)
  • [PS14] Cluster algebras and the Positive Grassmannian (by Miguel F. Paulos, Burkhard U. W. Schwab)
  • [FM14] Loop groups, Clusters, Dimers and Integrable Systems (by Vladmir Fock and Alexander Marshakov)
  • [BSW08] Superpotentials and Higher Order Derivations (by Ralf Bocklandt, Travis Schedler and Michael Wemyss)
  • [JS08] A theory of generalized Donaldson-Thomas invariants (by Dominic Joyce and Yinan Song) - See Section 7
  • [Bo11] Generating toric noncommutative crepant resolutions (by Ralf Bocklandt)
  • [V07] Mutations versus Seiberg duality (by Jorge Vitoria)
  • [D08] Consistency conditions for brane tilings (by Ben Davison)
  • [G06] Calabi-Yau algebras (by Victor Ginzburg)
  • [HHV06] Brane Tilings and Exceptional Collections (by Amihay Hanany, Christopher P. Herzog, and David Vegh)

  • Older Research Articles relevant for lectures:
  • [IU09] Dimer models and the special McKay correspondence (by Akira Ishii and Kazushi Ueda)
  • [FV06] Moduli Spaces of Gauge Theories from Dimer Models: Proof of the Correspondence (by Sebastian Franco and David Vegh)
  • [PSW07] Matching polytopes, toric geometry, and the non-negative part of the Grassmannian (by Alex Postnikov, David Speyer, and Lauren Williams)
  • [HV05] Quivers, Tilings, Branes and Rhombi (by Amihay Hanany and David Vegh)
  • [FHH00] D-Brane Gauge Theories from Toric Singularities and Toric Duality (by Bo Feng, Amihay Hanany, and Yang-Hui He)
  • [BP01] Toric Duality is Seiberg Duality (by Chris E. Beasley and M. Ronen Plesser)
  • [B13] Morita equivalences from Higgsing Toric Superpotential Algebras (by Charlie Beil) - especially the appendix
  • [B11] Consistency conditions for dimer models (by Ralf Bocklandt)
  • [Sz07] Non-commutative Donaldson-Thomas theory and the conifold (by Balazs Szendroi)
  • [Y07] Computing a pyramid partion generating function with dimer shuffling (by Benjamin Young)
  • [K04] Applications of Graphical Condensation for Enumerating Matchings and Tilings (by Eric Kuo)
  • [Sp04] Perfect Matchings and the Octahedron Recurrence (by David Speyer)
  • [E10] Brane Tilings and Non-Commutative Geometry (by Richard Eager)
  • [EF11] Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations (by Richard Eager and Sebastian Franco)

  • See further articles in the Special Issue in Journal of Physics A on Cluster Algebras (2014)

  • More to be added later.

  • Other Helpful Resources:
    A Compendium on the Cluster Algebra and Quiver Package in SAGE (with Christian Stump)
    Sage-Combinat Server (for Cluster Algebra and other calculations)

    Also Available via the Sage Math Cloud
    Keller's Quiver Applet in Java

    My Spring 2011 Course on Cluster Algebras and Quiver Representations.

    MSRI Graduate Summer School on Cluster Algebras (from 2012)


    There will be no exams, but registered students are expected to attend, and should hand in the homework assignments. There will be homework every three weeks or so, tentatively three assignments over 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 student presentations instead of homework.

    In lieau of the third assignment, an in-class presentation on a research paper is encouraged.

    Tentative Lecture Schedule

  • (Jan 21) Lecture 1: Introduction to the Course: Cluster Algebras, Dimers, and Quiver Gauge Theories
  • Part 1: Introduction to Cluster Algebras

  • (Jan 26) Lecture 2: Labelled seeds and general definition of a cluster algebra (Chapter 2 of Marsh)
  • (Jan 28) Lecture 3: Cluster complexes and exchange matrices as quivers (Lecture 3 of [FR04], Section 3 of [FR04])
  • (Feb 2) Lecture 4: Crash Course on Finite Reflection Groups and Coxeter Diagrams (Chapter 4 of Marsh, Section 1 of [FR04])
  • (Feb 4) Lecture 5: Root systems and the Finite type classification (Chapter 4-5 of Marsh, Section 2 of [FR04])
  • (Feb 9) Lecture 6: Cluster Complexes, Generalized Associahedra, and Polytopal Realizations (Chpater 5-6 of Marsh, Section 4 of [FR04])
  • (Feb 11) Lecture 7: The Laurent Phenomenon and the Caterpillar Lemma (Chapter 3 of Marsh)
  • (Feb 16) Lecture 8: Guest Lecturer (Pasha Pylyavskyy): LP Algebras
  • (Feb 18) Lecture 9: Applications of the Laurent Phenomenon to Somos Sequences
  • Part 2: Towards Dimer Models

  • (Feb 23) Lecture 10: Periodic Quivers (Chapter 7 of Marsh)
  • (Feb 25) Lecture 11: From Bipartite Tilings to Quivers and Superpotential Algebras (Section 2.1 of Broomhead [B09])
  • (Mar 2) Lecture 12: Toric Actions and Perfect Matchings on Dimer Models (Sections 2.2-2.3 of Broomhead)
  • (Mar 4) Lecture 13: Kastelyn Theory (Section 3 of [K09])
  • (Mar 9) Lecture 14: Kastelyn Theory II: Higher Genus Case
  • (Mar 11) Lecture 15: Back to Bipartite Tilings and the Superpotential Algebras (for triangles) (Section 2 of [GK], [HV05])
  • (Mar 16) Spring Break
  • (Mar 18) Spring Break
  • (Mar 23) Class Postponed
  • (Mar 25) Class Postponed
  • From March 30 - April 8, class will be 4:00-6:00

  • (Mar 30) Lecture 16-17: Triangular Toric Diagrams, Abelian Orbifolds, and the 3-dim McKay Correspondence ([UY11])
  • (Apr 1) Lecture 17-18: Triangular Toric Diagrams and Abelian Orbifolds II
  • (Apr 6) Lecture 19-20: F-flatness, D-Flatness, and the Forward Algorithm (Kennaway, [FV06], [HV05] also see [PSW07])
  • (Apr 8) Lecture 20-21: Zig Zag Fans and Extremal Perfect Matchings (Chapters 3-4 of Broomhead, [HV05])
  • We resume our regular 4:00-5:15 Schedule

  • (Apr 13) Lecture 22: Geometric consistency versus R-symmetry consistency (Chapters 3-4 of Broomhead, [HV05])
  • (Apr 15) Lecture 23: Toric duality and Seiberg duality versus Mutation of Quivers with Potential ([FHH00], [BP01])
  • (Apr 20) Lecture 24: Algebraic Consistency and the Centers of Toric Algebras ([IU09], [B11] and Chapter 5 of Broomhead)
  • Part 3: From Dimer Models back to Cluster Algebras and Combinatorial Formulas

  • (Apr 22) Lecture 25: Pyramid Partition Functions as a dimer model ([Sz07], [MR08], [Y07])
  • (Apr 27) Lecture 26: Octahedron Recurrence and Speyer's Crosses and Wrenches Graphs ([Sp04])
  • (Apr 29) Lecture 27: Speyer's proof by Kuo Condensation ([Sp04], [K04])
  • (May 4) Lecture 28: Urban Renewal and T-systems ([Sp04], [DiF13])
  • (May 6) Lecture 29: Aztec Castles and Beyond ([LMNT], [LM], [E10], [EF11])

  • Homework assignments

    Assignment Due date
    Homework 1 Wednesday 2/25
    Homework 2 Wednesday 4/8