Explore the conditions under which geodesic flow becomes ergodic in this advanced mathematics seminar from the Institute for Advanced Study. Discover the fundamental relationship between ergodic geodesic flow on infinite Riemann surfaces and Brownian motion recurrence, which is equivalent to the divergence of Poincaré series. Learn how surfaces with ergodic geodesic flows share key similarities with compact surfaces and examine various sufficient conditions on Fenchel-Nielsen parameters that guarantee a surface belongs to this class. Delve into a proven version of the Kahn-Markovic conjecture, which demonstrates that surfaces with arbitrarily large cuff lengths and one topological end will exhibit ergodic geodesic flow when twists are appropriately selected. Gain insights from collaborative research findings involving Ara Basmajian, Hrant Hakobyan, and Michael Pandazis in this comprehensive exploration of geometric dynamics and hyperbolic geometry.