Mikhail Bershtein

Cluster Algebras. 2026 (SISSA)

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 algebras, cluster varieties, upper cluster algebras. Cluster presymplectic form and Poisson bracket.
  8. Quantum cluster algebras. Quantum dilogarithm. Fock-Goncharov factorization. One step mutation property (starfish lemma).
  9. Scattering diagrams, broken lines, and theta functions.
  10. Reddening sequences. Dilogarithm identities. Fock-Goncharov dual basis conjecture

Cluster Integrable systems. 2024 (Skoltech)

Lecture notes. arXiv:2503.18573.

Recordings of the lectures. playlist.

Introduction to cluster algebras and varieties. 2022 (Skoltech)

Plan of the Course

  1. Preliminary examples Notes.
  2. Seeds. Mutations. Laurent phenomenon Notes.
  3. Total positivity. Notes.
  4. Double Bruhat cells. X coordinates Notes.
  5. Cluster Poisson structure. Poisson Lie groups Notes.
  6. Cluster Poisson structure. Integrable systems Notes.
  7. Upper cluster algebra. Coordinate rings on double Bruhat cells Notes.
  8. Starfish Lemma. Grassmannians Notes.
  9. Plabic graphs. Grassmannians. Notes.
  10. Poisson structures on Grassmannians. Notes.
  11. Geometric approach to cluster varieties. Notes.
  12. Cluster structure on the space of framed local systems. PGL_2 case Notes.
  13. Pinnings Notes.
  14. Cluster structure on the space of framed local systems. PGL_m case Notes.
Problems. pdf.

Recordings of the lectures. playlist.