Chekhov–Eynard–Orantin spectral curve topological recursion and some of its applications

In this talk I will give an overview of the theory of Chekhov–Eynard–Orantin spectral curve topological recursion. Spectral curve topological recursion is an abstract procedure producing the so-called $n$-point differentials from certain initial data (the spectral curve), generalizing the procedure of computing matrix model resolvents via loop equations. It turns out that this one and same procedure, with a correct choice of initial data, produces generating functions for answers to various seemingly unrelated problems in mathematics and mathematical physics, including, but not limited to, Gromov–Witten invariants (and, more generally, cohomological field theory correlators), Weil–Petersson volumes of moduli spaces, various types of Hurwitz numbers and counts of maps and hypermaps, certain knot invariants and so on. In the talk I will describe the procedure itself and some of its generalizations, and I will describe the ideas of how to prove that generating functions for various numbers (some of the mentioned above ones) satisfy this recursion, and I will talk about what applications this has: for instance, how one can prove ELSV-type formulas, relating Hurwitz-type numbers to CohFT correlators. Spectral curve topological recursion also has interesting connections to integrability (specifically, to KP and BKP hierarchies), which I will briefly mention in the talk.