Explore the geometric structures associated with Anosov representations in this 59-minute lecture by Sara Maloni from the University of Virginia. Delve into the generalization of convex-cocompact representations to higher-rank groups, focusing on connected components of Anosov representations from the fundamental group of closed hyperbolic manifolds. Examine the domains of discontinuity and their quotients, which form smooth fiber bundles over the manifold. Investigate the challenges in determining the topology of the fiber, with specific insights into representations in Sp(4,C) and the domain of discontinuity in the space of complex Lagrangians in C^4. Learn about the application of smooth 4-manifold classification in this context. This talk, presented at the Institut des Hautes Etudes Scientifiques (IHES), is based on joint work with Daniele Alessandrini, Nicolas Tholozan, and Anna Wienhard.