Orateur: Nicolas
Orentin (EPFL)
Titre: Topological
recursion, dessins d'enfants and other applications.
Résumé:
The topological recursion method is a formalism developed in the
context of random matrix theories in order to solve an associated problem of
combinatorics consisting in the enumeration of discrete surfaces. This
inductive procedure allows to enumerate such surfaces of arbitrary topology
out of the genus 0 data. This theory has further been formalized out of the
context of random matrices and mysteriously solved many problems of
enumerative geometry using a universal inductive procedure.
In the first part of this talk, which only uses elementary combinatorics, I
will present this topological recursion procedure in a simple example
consisting in the enumeration of dessins d'enfants, i.e. the enumeration of
clean Belyi maps.
In the second part of the talk, I will present some of the applications of
the general formalism such as the enumeration of simple Hurwitz covers of
the sphere, the computation of Gromov-Witten invariants of Toric Calabi-Yau
threefolds or of the Weil-Petersson volume of the moduli space of Riemann
surfaces as well as some related conjectures. If times allows, I will
explain one of the reasons for the universality of this inductive procedure
from the point of view of cohomological field theories with an application
to the computation of the total Chern class of the Verlinde bundle
generalizing the Verlinde formula giving the dimension of the space of
conformal blocks.