Week 10.11.2025 – 16.11.2025

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.
Lecture 2. Reduction to eigenvalues. Large N limit, Coulomb gas approach, saddle point equations.
Lecture 3. Continuum limit of saddle point equations. Eigenvalue density and spectral curve. Examples.
Lecture 4. Orthogonal polynomials. Relation to 2d gravity and phase transitions (sketch). Outlook: loop equations, topological recursion, integrability.

Despite recent progress, a complete formulation of a holographic correspondence for flat spacetimes remains elusive.
Any viable formulation of flat space holography should be based on a correspondence between bulk and boundary states built upon the equivalence of unitary irreducible representations (UIRs) of the asymptotic symmetry group of flat spacetimes, the BMS group. In this talk, I will present explicit wavefunctions for the UIRs of the BMS group (the so-called BMS particles). These are functions on supermomentum space that generalize the familiar notion of Poincaré particles by incorporating additional soft degrees of freedom. I will discuss their connections to infrared physics and outline prospects for defining an S-matrix for BMS states that is free from infrared divergences. This talk is based on joint work with X. Bekaert and Y. Herfray.