Explore geometric inequalities in spaces of nonpositive curvature through this 54-minute lecture delivered by Mohammad Ghomi at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Part of the Thematic Programme on "Geometry beyond Riemann: Curvature and Rigidity," this talk delves into advanced mathematical concepts, offering insights into the intricate relationships between geometry and curvature in non-Euclidean spaces. Gain a deeper understanding of how geometric principles apply in environments characterized by nonpositive curvature, and discover the implications of these concepts for broader mathematical and physical theories.