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Explore the intricate connections between frieze patterns and representation theory of associative algebras in this mathematical conference talk. Begin with classical Conway-Coxeter friezes over positive integers and discover their correspondence with Jacobian algebras of type A, where frieze entries count submodules of indecomposable representations. Learn how this relationship can be reinterpreted through the Caldero-Chapoton map, establishing close connections to Fomin-Zelevinsky's cluster algebras. Extend beyond classical cases to examine higher dimensional friezes, specifically (tame) SLk friezes, and understand their relation to cluster algebras on coordinate rings of Grassmannians Gr(k,n) and their categorification. Investigate SLk friezes as special types of SLk tilings—integer tilings of the plane where every k×k square has determinant 1. Discover a characterization of SLk tilings in terms of pairs of bi-infinite sequences in Zk and explore applications to duality and positivity. This presentation was recorded during the thematic meeting "Frieze patterns in algebra, combinatorics and geometry" at the Centre International de Rencontres Mathématiques in Marseille, France.
