Week 20.10.2025 – 26.10.2025

Models of random matrices can be viewed as zero-dimensional analogs of usual field theory. Despite decades of exploration, matrix models remain at the forefront of intensive research, motivated by a rich web of connections to string theory, quantum gravity, integrability, Yang-Mills theory, combinatorics, geometry and representation theory. These lectures will present a pedagogical introduction to the subject.

​Lecture 1. Motivation and basic definitions. Hermitian matrix models: Feynman rules, ribbon graphs, large N genus expansion.

We study the response of an infinite system of point particles on the real line, initially at rest, to the instantaneous release of energy in a localized region. The blast generates shock fronts that travel sub-ballistically. The density, velocity, and temperature profiles in the growing region between the shock fronts can be obtained from exact self-similar solutions of the Euler equations. We compare these with the results of microscopic simulations. At long times, the results obtained from the simulations show a remarkable agreement with the predictions from Euler hydrodynamics except at the blast core where we show that corrections due to Navier-Stokes terms are important. We will also discuss the "splash" problem where energy is injected from one side of a semi-infinite cold gas.

Consider the first passage percolation distance on random planar maps, which is obtained by putting i.i.d. exponential random lengths on each (dual) edge. The study of this distance is often simpler than the study of the (dual) graph distance. I will describe a time-reversal of the uniform peeling exploration, which enables me to obtain the scaling limit of the number of faces along the geodesics to the root, to compare the metric balls for the first passage percolation and the dual graph distance and to upperbound the diameter of large random maps. Then, I will obtain the scaling limit of the tree of first passage percolation geodesics to the root via a stochastic coalescing flow of pure jump diffusions. This stochastic flow is also a tool to construct some random metric spaces which I conjecture to be the scaling limit of random planar maps with high degrees.

Solutions to general relativity with a negative cosmological constant have received significant attention due to the conjectured AdS/CFT correspondence, a particularly well-understood example of which is exhibited in 2+1 dimensions. I will review known vacuum solutions to general relativity with a negative cosmological constant in 2+1 dimensions and discuss the difficulties in defining mass, which are resolved via minimisation using a positive energy theorem. I will present a gluing theorem for vacuum time-symmetric general-relativistic initial data sets in two spatial dimensions. By gluing two given time-symmetric vacuum initial data sets at conformal infinity, we obtain new time-symmetric vacuum initial data sets. I will sketch the derivation of the mass formulae of the resulting manifolds. Our gluing theorem yields complete manifolds with any mass aspect function, which are smooth except for one conical singularity.