Topological recursion is an ubiquitous and universal recursive relationship that has appeared in various domains of mathematics and physics: volumes of moduli spaces, coefficients of asymptotic expansions in random matrix theory, Hurwitz numbers, Jones polynomials, Gromov-Witten invariants, and many other combinatorial objects, all mysteriously satisfy the same relation. Moreover, this recursion relation is effective: it allows an actual computation. This recursion has been axiomatized into a definition of some "new invariants" of curves. In this lecture we shall introduce the topological recursion, illustrate it on examples and mention its beautiful properties.
Group for Research in Decision Analysis