Introduction

Matrix models are ubiquitous both in Mathematics and in Theoretical Physics. The aim of this course is to introduce the main approaches to U(N)-invariant random matrix models: Coulomb gas, orthogonal polynomials, loop equations, topological recursion, tau-functions and integrable hierarchies. Most of these techniques are based on the fact that the eigenvalues of the random matrix behave as free fermions in a confining potential. The collective theory for these fermions in the large N limit is formulated in terms of a special Riemann surface - the spectral curve of the matrix model. Once the spectral curve is known, the topological recursion generates the whole 1/N expansion.

Keywords: U(N) matrix integrals, planar graphs, tau-functions, Hirota equations, Virasoro constraints, Coulomb gas, free fermions, bosonisation, 1/N, topological recursion.

Lecturer Intro

Ivan Kostov obtained his PhD in 1982 from the Moscow State University, with scientific advisers Vladimir Feinberg and Alexander Migdal. Then he worked in the group of Ivan Todorov at the INRNE Sofia, and since 1990 as a CNRS researcher at the IPhT, CEA-Saclay, France. Currently he is emeritus DR CNRS at IPhT and a visiting professor at UFES, Vitoria, Brazil.