Random matrix universality as a tool in two-dimensional quantum gravity

The quest for a theory of quantum gravity remains one of the central challenges in theoretical physics. A promising approach to this problem in our four-dimensional universe lies in first understanding lower-dimensional theories. This thesis therefore focuses on studying quantum gravity in two dimensions, primarily considering Jackiw-Teitelboim (JT) gravity, for which a connection with random matrix ensembles was recently uncovered. The central idea is to investigate the implications of "random matrix universality"-a property of such ensembles-in JT gravity and related theories. These implications manifest as specific mathematical properties of the geometric objects underlying the studied quantum gravity theories and can be understood as imprints of quantum chaoticity of those theories. To pursue this program across all varieties of the considered gravitational theories, this thesis extends and implements computational methods for these geometric objects, particularly to settings involving unorientable geometries.