From Catalan numbers to knot invariants: a short introduction to topological recursion

25.04.2024 14:15 – 16:00

Catalan numbers are integers which appear in various counting problems. In particular, they can be defined as the cardinal of a set of graphs drawn on the sphere. From this perspective, they admit a natural generalisation to higher genera as the cardinal of a specific class of graphs embedded on a surface of fixed topology. In this talk, I will use the computation of such numbers as an opportunity to present how one can transform such a combinatorial problem into a very simple problem of complex analysis on a Riemann surface through the introduction of generating series. We will see that the solution of this problem is given by an inductive procedure known as topological recursion.

In the second part of this talk, I will explain how this topological recursion gives rise to a universal solution to many problems of enumerative geometry and low dimensional topology. In particular, I will explain its application to the computation of Gromov-Witten invariants and volumes of moduli spaces of Riemann surfaces. Finally, I will briefly explain how it is conjectured to give rise to knot invariants and its relation to the AJ conjecture.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-15, Séminaire "Topologie et Géométrie"

entrée libre