Cluster Algebras and Hyperbolic Geometry

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.