opological recursion (TR) is a universal procedure that helps connect diverse areas of mathematics and physics. Starting from a spectral curve—a Riemann surface equipped with additional data—it produces a family of differentials that often encode enumerative invariants, such as volumes of moduli spaces, matrix model correlators, and intersection numbers. After a brief introduction, we explore two stories, in principle unrelated to, yet guided by, TR—almost in a whisper—towards their resolution:
Witten conjectured that the generating series of psi-class intersection numbers is a tau function of the KdV hierarchy, a result first proved by Kontsevich. Norbury later conjectured an analogous statement for intersection numbers of psi-classes times a negative square root of the canonical bundle. We prove Norbury’s conjecture and derive polynomial relations among kappa-classes. We further introduce a new cohomological field theory (CohFT): the negative analogue of Witten’s r-spin CohFT. We show that its intersection numbers are recursively computable via W-constraints and obtain new tautological relations on the moduli space of curves.
The generating series of maps—graphs embedded on surfaces—obeys TR. Exchanging the roles of x and y in the initial data of TR transforms these maps into fully simple maps. Ordinary and fully simple maps are related via monotone Hurwitz numbers, unveiling a universal duality. This duality translates into functional relations between moments and free cumulants, resolving an open problem in free probability that generalises Voiculescu’s R-transform. In this way, we uncover a general theory of freeness which takes into account higher genus corrections and captures the full, all-order asymptotic behaviour of random matrices.