Bridging Random Geometries: The Directed Landscape and Liouville Quantum Gravity

In recent years, random geometry has developed into a subject in its own right, with the directed landscape and Liouville quantum gravity serving as central examples. Despite their very different origins, these models share striking similarities, particularly in their coalescent geodesic geometry. In this presentation, we discuss a collection of works motivated by viewing these models through a common lens, highlighting the transfer of ideas and techniques between the two settings. We conclude by considering dynamically evolving random geometries and describing recent results on the exceptional behaviour arising in the directed landscape under such dynamics.