Mikhail Bershtein

Cluster Algebras 2025-2026 - Course Plan

Cluster algebras and cluster varieties were introduced in the early 2000s by Fomin and Zelevinsky. Almost simultaneously, related geometric objects, the cluster varieties, appeared in the work of Fock and Goncharov. These notions found numerous applications across mathematics and mathematical physics, and over the last years their popularity has not decreased in any way.

This course begins with a gradual introduction to cluster algebras. In the second half we discuss (partially in survey style) more recent fundamental results in the subject.

Some familiarity with the following notions is desirable: unique factorization domains, Grassmannians, convex cones, and quivers.

Plan of the Course

  1. Motivating examples: Somos sequences, configuration spaces, Grassmannians.
  2. Seeds, mutations, geometric coefficients.
  3. The Laurent phenomenon.
  4. Tropical semifields and principal coefficients. c-vectors, g-vectors, F-polynomials. Separation formulas.
  5. Sign-coherence and Laurent positivity. G-fan
  6. Finite type classification (statement). Example: type An, root system and cluster variables, associahedron.
  7. Cluster algerbas, cluster varietries, upper cluster algebras. Cluster presymplectic form and Poisson bracket.
  8. Quantum cluster algebras. Quantum dilogarith. Fock-Goncharov factorizaion. One step mutation property (starfish lemma).
  9. Scattering diagrams, broken lines, and theta functions.
  10. Reddering sequinces. dilogarithm idenitities. Fock-Goncharov dual basis conjecture

References

  1. S. Fomin, L. Williams, and A. Zelevinsky, Introduction to Cluster Algebras.
    Chapters 1-3 (arXiv:1608.05735) - Chapters 4-5 (arXiv:1707.07190) - Chapter 6 (arXiv:2008.09189).
  2. T. Nakanishi, Cluster Algebras and Scattering Diagrams, Mathematical Society of Japan, (2023).
    Part I (arXiv:2201.11371) - Part II (arXiv:2103.16309) - Part III (arXiv:2111.00800).
  3. V. Fock and A. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Ecole Norm. Super. 42 (2009), 865-930; arXiv:math/0311245.
  4. A. Berenstein and A. Zelevinsky, Quantum Cluster Algebras, Adv. Math. 195 (2005), 405-455; arXiv:math/0404446.
  5. M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical Bases for Cluster Algebras, J. Amer. Math. Soc. 31 (2018), 497-608; arXiv:1411.1394.
  6. C. Hohlweg, V. Pilaud, and S. Stella, Polytopal Realizations of Finite Type g-Vector Fans, Adv. Math. 328 (2018), 713-749; arXiv:1703.09551.
  7. Bethany Marsh and Jeanne Scott, Twists of Plücker Coordinates as Dimer Partition Functions, Commun. Math. Phys. 341 (2016), 821-884; arXiv:1309.6630.
  8. B. Keller, Java applets for quiver mutations and exchange graphs: webusers.imj-prg.fr/~bernhard.keller/quivermutation/.
  9. Cluster Algebras Portal, by Sergey Fomin: https://dept.math.lsa.umich.edu/~fomin/cluster.html.