A short overview of the "Topological recursion"

TL;DR

This review provides an overview of topological recursion, a method connecting asymptotic analysis of integrable systems with geometric invariants like Gromov-Witten invariants and knot polynomials.

Contribution

It summarizes the definitions, properties, and applications of topological recursion, highlighting its role in unifying various geometric and combinatorial invariants.

Findings

Topological recursion links asymptotic expansions to geometric invariants.

Specializations recover classical invariants like Gromov-Witten and knot polynomials.

The method has broad applications in enumerative geometry and mathematical physics.

Abstract

This review is an extended version of the Seoul ICM 2014 proceedings.It is a short overview of the "topological recursion", a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall how computing large size asymptotics in random matrices, has allowed to discover some fascinating and ubiquitous geometric invariants. Specializations of this method recover many classical invariants, like Gromov--Witten invariants, or knot polynomials (Jones, HOMFLY,...). In this short review, we give some examples, give definitions, and review some properties and applications of the formalism.