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. 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 numbers and I will describe our results on explicit closed algebraic
formulas for generating functions of generalized double Hurwitz numbers, and how this
allows to prove topological recursion for a wide class of problems.
If time permits I'll talk about the implications for the so-called ELSV-type formulas
(relating Hurwitz-type numbers to intersection numbers on the moduli spaces of
algebraic curves); in particular, I'll explain how this almost immediately gives
proofs (of a purely combinatorial-algebraic nature) of the original ELSV formula and
of its r-spin generalization (originally conjectured by D.Zvonkine).
The talk is based on the series of joint works with P. Dunin-Barkowski, M. Kazarian
and S. Shadrin.
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