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Explore the mathematical connections between SL2-tilings and Coxeter frieze patterns through the lens of Farey complexes in this comprehensive lecture. Begin with an examination of tame frieze patterns over the integers, then delve into the Farey tessellation of the hyperbolic plane, drawing inspiration from the theory of dessins d'enfants. Discover how the geometric and numeric properties of the Farey tessellation illuminate known results on classifying frieze patterns while providing a framework for new mathematical insights. Learn about the foundational work of Morier-Genoud, Ovsienko, and Tabachnikov, along with recent generalizations of their ideas. Engage with numerous diagrams, practical exercises, and open questions that demonstrate the visual and computational aspects of this mathematical theory. Gain understanding of how hyperbolic geometry intersects with algebraic structures and combinatorial patterns, making this material accessible through clear geometric visualization and concrete examples.
