This mini workshop is devoted to the search for a mathematical foundation of a newly discovered topological recursion by Eynard and Orantin, which was further generalized by Bouchard, Klemm, Marino and Pasquetti, in the context of topological string theory. The recursion is originated in Random Matrix Theory as an effective tool to calculate free energies and n-point correlation functions of the resolvent of Hermitian matrix models. Here the technique involved is a simple complex analysis on a hyperelliptic curve that appears as the resolvent set of Hermitian random matrices.

When applied as an axiom to an arbitrary plane analytic curve, the topological recursion calculates an infinite sequence of differentials and symplectic invariants of the curve. A natural question arises: What are these quantities calculating?

Miraculously, a particular choice of the curve, the spectral curve of the theory, re-constructs the Witten-Kontsevich theory. Moreover, by deforming this spectral curve, the recursion reproduces the generalizations of the Witten-Kontsevich theory due to Mulase-Safnuk and Liu-Xu. Amazingly, a particular specialization of the deformation also gives the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli spaces of bordered hyperbolic surfaces.

BKMP further show that when the curve is defined by the Lambert W-function, the recursion provides an effective formula to compute linear Hodge integrals and Hurwitz numbers. This formula has been unknown to the mathematics community.

Even more miraculous is the claim of BKMP that the Eynard-Orantin recursion computes the Gromov-Witten invariants of toric Calabi-Yau 3-folds through mirror symmetry. An attempt of understanding the BCOV anomaly equation is also proposed by Dijkgraaf-Vafa.

In this mini workshop we provide an ample time to learn the origin of the topological recursion from its discoverer, Bertrand Eynard. He will also talk about the new derivation of Witten-Kontsevich theory and its various generalizations, as well as his most recent results.

Mulase will talk about an integrable system approach to the topological recursion using a deformation theory of tau-functions (joint work with Safnuk, in preparation).