Description
The study of random matrices has a fascinating connection with different fields of mathematics and physics ranging from number theory to topological string theories. Among such applications, the computation of moments of random matrices can be interpreted as defining generating series of discrete surfaces built by gluing together polygons by their edges. Trying to solve such a combinatorial problem, together with Chekhov and Eynard, we found a beautiful formula allowing to enumerate such surfaces by induction on their Euler characteristic. This inductive procedure, later called topological recursion, turns out to provide a universal solution to a large class of enumerative problems such as the study of statistical systems on a random lattice or the computation of integrals over the moduli space of Riemann surfaces with respect to different measures (including Gromov-Witten invariants or volumes of such moduli spaces).
In a first part of these lectures, I will introduce the formalism of topological recursion by explaining how one can transform the combinatorial problem of enumerating discrete surfaces into the
computation of residue integrals on an associated Riemann surface thanks to the definition of good generating series. The general idea is that, if one can find a generating series for the number of discs such that it defines a function on a Riemann surface, then applying topological recursion with this Riemann surface as initial data produces generating functions for the number of surfaces with any topology by induction on their Euler characteristic.
I will then give some more examples of application of the recursive formula discovered in this context to other enumeration problems including Hurwitz numbers, some volumes of the moduli space of Riemann surfaces and the expression of correlators of any semi-simple cohomological field theories.
In a second part of these lecture, I will present a completely different flavour of the topological recursion. Taking as input an algebraic curve defined by a polynomial equation E(x,y) = 0, topological recursion allows to build an eigenfunction of the operator
