The Eynard-Orantin recursion is a universal mechanism of constructing a (re-modeld) B-model topological string theory on any given Riemann surface. The physics predictions tell us that its mirror symmetric partner, the A-model topological string theory corresponding to the starting B-model, should give us most everything about Gromov-Witten invariants of certain target spaces. Most recent speculatons from physics include formulas for quantum knot invariants in terms of the starting Riemann surface (the A-polynomial). Surprisingly, from a purely algebro-goemmetric point of view, the Eynard-Orantin recursion appears often as the Laplace transform of the natural degeneration formulas of algebraic curves. This mysterious B-model formula is therefore a completely familiar equation to algebraic geometers. The only difference here is the Laplace transform, which plays the role of mirror symmetry. These talks are aimed at explaining this relation between algebraic geometry of curve degenerations and the re-modeled B-model topological string theory. We will present mathematically rigorous examples to illustrate the new developments and the exciting picture that is emerging now.

Since its introduction in the context of random matrix theory for the enumeration of maps, the topological recursion formalism mysteriously appeared as a common solution to a large class of problems of enumerative geometry. In particular, as many examples suggest, Gromov-Witten invariants enumerating holomorphic maps from a surface of given topology to a given ambient space seem to be computable by this formalism by induction on the Euler characteristic of the embedded surfaces. In these lectures I will show how localization computations allow, using some combinatorics, to reduce the computation of these Gromov-Witten invariants to a sum over graphs which can then be obtained by a local version of the topological recursion. I will first present a few explicit examples before explaining the general setup based on a theory developed by Givental.

Based on joint works with Dunin-Barkowski, Eynard, Shadrin and Spitz.

Since its introduction in the context of random matrix theory for the enumeration of maps, the topological recursion formalism mysteriously appeared as a common solution to a large class of problems of enumerative geometry. In particular, as many examples suggest, Gromov-Witten invariants enumerating holomorphic maps from a surface of given topology to a given ambient space seem to be computable by this formalism by induction on the Euler characteristic of the embedded surfaces. In these lectures I will show how localization computations allow, using some combinatorics, to reduce the computation of these Gromov-Witten invariants to a sum over graphs which can then be obtained by a local version of the topological recursion. I will first present a few explicit examples before explaining the general setup based on a theory developed by Givental.

Based on joint works with Dunin-Barkowski, Eynard, Shadrin and Spitz.

The Eynard-Orantin recursion is a universal mechanism of constructing a (re-modeld) B-model topological string theory on any given Riemann surface. The physics predictions tell us that its mirror symmetric partner, the A-model topological string theory corresponding to the starting B-model, should give us most everything about Gromov-Witten invariants of certain target spaces. Most recent speculations from physics include formulas for quantum knot invariants in terms of the starting Riemann surface (the A-polynomial). Surprisingly, from a purely algebro-goemmetric point of view, the Eynard-Orantin recursion appears often as the Laplace transform of the natural degeneration formulas of algebraic curves. This mysterious B-model formula is therefore a completely familiar equation to algebraic geometers. The only difference here is the Laplace transform, which plays the role of mirror symmetry. These talks are aimed at explaining this relation between algebraic geometry of curve degenerations and the re-modeled B-model topological string theory. We will present mathematically rigorous examples to illustrate the new developments and the exciting picture that is emerging now.