Quelques probl\`emes de g\'eom\'etrie \'enum\'erative, de matrices al\'eatoires, d'int\'egrabilit\'e, \'etudi\'es via la g\'eometrie des surfaces de Riemann

TL;DR

This paper explores the use of complex analysis and algebraic geometry to solve loop equations in integrable systems, random matrices, and topological string theory, emphasizing the role of spectral curves and Riemann surfaces.

Contribution

It develops techniques connecting differential geometry of Riemann surfaces with solutions to loop equations across various mathematical physics topics.

Findings

Solution of loop equations via spectral curves

Application to random matrices and topological recursion

Insights into large N asymptotics and Gromov-Witten invariants

Abstract

Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or "Virasoro constraints". In the simplest case, the complete solution of those equations was found recently: it can be expressed in the framework of differential geometry over a certain Riemann surface which depends on the problem : the "spectral curve". This thesis is a contribution to the development of these techniques, and to their applications. Keywords: random matrices, random maps, integrable systems, algebraic geometry, loop equations, topological recursion, Hurwitz numbers, Gromov-Witten invariants, Tracy-Widom laws, beta ensemble, large N asymptotics in random matrix theory.