Cluster algebras and integrable maps, Lecture 3

Abstract:

1) Background and examples of cluster algebras: Somos sequences in number theory; Laurent property; Abel pentagon identity, Lyness map and the dilogarithm; Zamolodchikov Y-systems; Plucker coordinates in Grassmanians; discrete Hirota equations.

2) Cluster algebras without coefficients: quivers and quiver mutation; exchange matrices and matrix mutation; cluster variables and cluster mutation.

3) Poisson and symplectic structures: Poisson brackets; symplectic forms; Gekhtman-Shapiro-Vainshtein Poisson structure for cluster algebras; examples of noninvariant symplectic leaves; compatible presymplectic forms and reduction to symplectic coordinates.

4) Cluster mutation-periodicity: Mutation-periodic quivers; Fordy & Marsh classification of period 1 and recurrence relations; primitives and affine Dynkin diagrams; Dodgson condensation; linear relations for cluster variables.

5) Tropical relations and algebraic entropy: Growth of denominators; max-plus tropical algebra; dynamics of tropical maps; algebraic entropy; entropy classification of cluster maps.

6) Discrete integrable systems: Affine A-type cluster algebras and dressing chain - monodromy matrix and Lenard-Magri chain; discrete Hirota and reduction to Somos/Gale-Robinson; connection with QRT maps.

About the speaker: Andy Hone is an EPSRC Advanced Research Fellow with broad research interests in integrable systems, cluster algebras and number theory. You can learn more about him here: https://www.kent.ac.uk/smsas/our-people/profiles/hone_andrew.html