Prof. Andrew N W Hone, from University of Kent at Canterbury, UK, gave a lecture titled “Cluster algebras and integrable maps (I-II)” at AMSS on 6, April, 2017.
In this talk, he talked about 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; Ploucker coordinates in Grassmanians; discrete Hirota equations. 2) Cluster algebras without coecients: 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 classifcation 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; experimental classification. 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.