Watch a 58-minute conference talk exploring the cone of Jordan variations and their applications in hyperbolic geometry, presented by Andrés Sambarino from Sorbonne University. Part of a broader conference on hyperbolic geometry and hyperbolic group theory at IMPA in Rio de Janeiro, this presentation contributes to recent developments in understanding hyperbolic manifolds, their submanifolds, and fundamental groups. The talk is one of twenty featured presentations from leading mathematicians worldwide, complemented by a subsequent workshop offering in-depth exploration of key topics for graduate students. Organized in partnership with the Clay Mathematics Institute and the NSF, this academic event brings together experts to discuss cutting-edge research in hyperbolic geometry, group theory, and related mathematical areas.