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Explore the fascinating intersection of cluster algebras and hyperbolic geometry in this Aisenstadt Chair Lecture Series talk. Delve into the application of cluster algebra in studying two- and three-dimensional hyperbolic geometry. Learn how mutations in two dimensions correspond to flips in ideal triangulations of punctured surfaces, with cluster x-variables providing coordinates for the decorated Teichmuller space. Discover the three-dimensional perspective, where mutations produce ideal tetrahedra and cluster y-variables are interpreted as the modulus of these tetrahedra. Examine the octahedral braiding operator, composed of four mutations, and investigate its role in studying knot volumes. This hour-long lecture, part of the Workshop on Integrable systems, exactly solvable models and algebras, offers a deep dive into the intricate connections between algebraic structures and geometric concepts.
