Explore the fascinating world of random planar geometry and the directed landscape in this 56-minute colloquium talk by Duncan Dauvergne from the University of Toronto. Delve into the concept of assigning random weights to edges in a lattice Z^2 to create a random planar metric. Discover how this model, along with other natural models of random planar metrics and random interface growth, is expected to converge to a universal scaling limit known as the directed landscape. Learn about the properties of this object and examine at least one model where convergence can be proven. Gain insights into the KPZ universality class and its connection to random planar geometry. This talk, part of the University of Chicago Department of Mathematics colloquium series, offers a deep dive into cutting-edge research in mathematical physics and probability theory.