Elba Garcia-Failde: Introduction to topological recursion - Lecture 1

Mini course of the conference "Noncommutative geometry meets topological recursion", August 2021, University of Münster. Abstract: In this mini-course I will introduce the universal procedure of topological recursion, both by treating examples and by presenting the general formalism. We will study the classical case of the Hermitian matrix model in detail, which combinatorially corresponds to ribbon graphs, beginning from the loop equations, which correspond to Tutte’s recursion in the combinatorial setting. This will be the starting point to make the connection to free probability, which moreover provides a combinatorial way of exploring the variation of the topological recursion output when applying a symplectic transformation to the input. Apart from the (conjectural) property of symplectic invariance, topological recursion has many other interesting features and, together with its generalizations, has established connections to various domains of mathematics and physics, like intersection theory of the moduli space of curves and integrability. We will explain some of these properties and connections, giving several ideas why this is worth considering, and is the starting or gluing point of an active field of research, and finally hoping to instigate the search of new beautiful connections.