- large N asymptotic expansions in 1d Coulomb gases of N particles;
- counting surfaces of an arbitrary (but fixed) topology;
- studying the geometry of (families of) algebraic varieties;
- (toy models, discretizations, etc.) of quantum field theories.

abstract loop equations. Each problem has its own set of applications (and in particular, of universality questions) driving the interest:

(1) repulsive particle systems, random tilings,

(2) combinatorics of maps, random geometry of surfaces, enumerative geometry.

The relation between (2) and (3) is one of the avatar of mirror symmetry.

In (4) one is interested in studying some aspects (or perhaps constructing) a QFT by first studying a well-defined toy model (or very fine discretization) for it.

The name

topological expansionwhich one commonly uses for all 4 themes comes from the fact, via (2), that coefficients in these expansions have 'something to do' with bordered surfaces sorted out by their topology.

The lectures will present some of the mathematical progress of the last 15 years, which tends (in the big picture) towards a better understanding of all the relations and common structures between the above problems. Remaining down-to-earth, our practical goal will be

- explaining a general strategy to establish the existence of all-order asymptotic expansions for the macroscopic properties of a class of models in (1), based on large deviations and the analysis of Schwinger–Dyson equations. As a spinoff, we can catch central limit theorems and Gaussian Free Field behaviors when applicable.
- explaining how to enumerate maps (possibly carrying an O(n)-loop model) using Tutte's recursion, which produces a tower of equations identical to the above Schwinger–Dyson tower (but now in the context of formal series).
- explaining how in both cases these Schwinger–Dyson equations imply abstract loop equations, and their solution via the Eynard–Orantin recursion. This exploits the geometry of the spectral curve defined from the large N spectral density (in a) or the generating series of disks (in b).

equations with catalytic variables(a term coined by D. Zeilberger). The catalytic variables are parameters in which one is not interested a priori, but are needed to write down a closed system of equations. They are to be discarded (i.e. taken to be equal to some trivial value like 0 or 1) at the end of the

reaction.

For maps, one typically encounters equations which involve one catalytic variable and are quadratic in the unknown series. W.T. Tutte and W.G. Brown were able to solve the first instances of such equations by

guessing and checking, before developing some sort of systematic method called the

quadratic method.

In parallel (and somewhat independently), D.E. Knuth initiated the development of the

kernel methodthat allows to solve linear equations with one catalytic variable, as one may encounter in the classical ballot problem and other problems related to 1D lattice walks.

After several decades, in 2006, M. Bousquet-Mélou and A. Jehanne were able to merge both approaches into a general effective method to handle polynomial equations with

This is however not the end of the story: when studying

I will attempt to review this fascinating history by explaining the effective methods developed so far and illustrating them on concrete examples from enumerative combinatorics.

In free probability, those laws give rise to von Neumann algebras, a non-commutative analogue of algebras of bounded measurable functions. In this setting, the gaussian model, independent GUE matrices, give rise to free semicircular variables as limiting objects, and Voiculescu first identified the von Neumann algebras they generate with von Neumann algebras of non-commutative free groups. Schwinger–Dyson equations are then the starting point to see how much the properties of free group factors extend to algebras generated by a different limiting non-commutative law.

In this minicourse, we will explain several trends in getting inspiration of classical probability to use limiting Schwinger–Dyson equations (in short SD equation) in order to understand algebraic or analytic properties of the algebras they generate. We will roughly follow the following plan:

- First, we will explain the random matrix origin and perturbative expansions of the SD equation, and the most common convexity assumptions used to get limiting laws satisfying an SD equation beyond the perturbative regime. We will then explain the most basic property that can be shown on the algebra using SD equation, namely that the corresponding algebra is a factor: it has a center reduced to complex numbers.
- We will give several insigths in understanding SD equations as a starting point for Fisher information theory and entropy theory in free probability.
- We will explain Guionnet-Shlyakhtenko free transport. This is a limit of classical optimal transport applied to random matrix models which enables to prove isomorphisms between algebras generated by laws satisfying SD equations. SD equation is crucially used there as a substitute to equations on densities in order to start an analytic machinery.

- Lecture 1: Random matrices and lattice gauge theories

In order to make contact with other lectures at this school, we first introduce loop equations for random matrices in the form of Virasoro constraints and relate them to the Tutte recursion. We also give a similar formulation for lattice gauge theories. -
Lecture 2: Random tensors

Random tensors are natural generalisations of random matrices with more than two indices and can be used to generate higher dimensional random geometries. We derive the associated loop equations, involving differential operators that represent Lie algebras of graphs. -
Lecture 3: The Kontsevich–Soibelman approach

Recently, a general formalism based on certain differential operators has been introduced by Kontsevich and Soibelman. We describe this formalism and show how a solution of the loop equations can be constructed using suitable graphs.