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. Definition of cluster algebras and cluster varieties.
  3. The Laurent phenomenon.
  4. Total positivity.
  5. Double Bruhat cells and upper cluster algebras. Starfish lemma. Locally acyclic cluster algebras.
  6. Finite type classification (statement). Example: type An, root system and cluster variables, associahedron.
  7. Tropical semifields and principal coefficients. c-vectors, g-vectors, F-polynomials. Separation formulas.
  8. Sign-coherence and Laurent positivity.
  9. Cluster X-varieties. Fock–Goncharov dual bases conjecture.
  10. Scattering diagrams, broken lines, and theta functions.

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. 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.
  4. B. Keller, Java applets for quiver mutations and exchange graphs: webusers.imj-prg.fr/~bernhard.keller/quivermutation/.
  5. Cluster Algebras Portal, by Sergey Fomin: https://dept.math.lsa.umich.edu/~fomin/cluster.html.