Explore the fascinating world of meandric systems in this 55-minute lecture presented by Jacopo Borga from Stanford University at IPAM's Vertex Models workshop. Delve into Henri Poincaré's century-old question about the number of ways a simple loop can cross a line in a plane. Discover the latest research on the geometry of uniform meandric systems, including a conjecture describing their large-scale structure and rigorous results supporting this hypothesis. Gain insights into this complex mathematical problem and its implications for algebraic and probabilistic aspects of universality.