Topological recursion for Hurwitz type theories

Topological recursion is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani’s hyperbolic volumes of moduli spaces, knot polynomials. It is a recursion on the Euler characteristic, hence the name «topological» recursion. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve.
In the talk I will define the topological recursion, spectral curves and their invariants, and illustrate it with examples; I will introduce the Fock space formalism which proved to be very efficient for computing TR-invariants for the various classes of Hurwitz type problems and I will state recent results on the duality of «mixed» TR-invariants and the new recursion on the mixed invariants. The talk is based on the series of joint works with P. Dunin-Barkowski, M. Kazarian and S. Shadrin.