Explore the theory of divisible convex sets in this comprehensive lecture by Pierre-Louis Blayac from the Institut des Hautes Etudes Scientifiques (IHES). Delve into the rich mathematical concepts involving dynamical systems, geometric group theory, (G,X)-structures, and Riemannian geometry with nonpositive curvature. Discover how these objects serve as important examples of discrete subgroups of Lie groups that are not lattices and their connections to Anosov representations. Survey known examples of divisible convex sets and the discrete groups that divide them, then examine new examples of irreducible, non-symmetric, and non-strictly convex divisible convex sets in arbitrary dimensions (at least 3) obtained in collaboration with Gabriele Viaggi. Cover topics including hyperbolic surfaces, general definitions, dynamics, and construction methods throughout this 1-hour and 5-minute talk presented by Pierre-Louis Blayac from the University of Michigan.