Topological recursion, enumerative geometry and mirror symmetry
Event details
| Date | 23.02.2016 |
| Hour | 15:15 › 17:00 |
| Speaker | Nicolas Orantin (EPFL) |
| Location | |
| Category | Conferences - Seminars |
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 addition, 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.
In the second part of the talk, using the formalism of Frobenius manifolds, I will explain how the topological recursion formalism is a generalization of Givental\Teleman's theory. This point of view should make clear the relation between the topological recursion, mirror symmetry and the theory of vanishing cycles. As an application, I will show how this provides us with an effective method for computing the Gromov-Witten invariants of projective spaces through the study of their mirror Landau-Ginzburg model.
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 addition, 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.
In the second part of the talk, using the formalism of Frobenius manifolds, I will explain how the topological recursion formalism is a generalization of Givental\Teleman's theory. This point of view should make clear the relation between the topological recursion, mirror symmetry and the theory of vanishing cycles. As an application, I will show how this provides us with an effective method for computing the Gromov-Witten invariants of projective spaces through the study of their mirror Landau-Ginzburg model.
Practical information
- General public
- Free