The topological recursion


The topological recursion

A "topological recursion" is a formalism which computes quantities F_{g,n} indexed by integers g and n, by induction on 2g - 2 + n in a way that resembles glueing rules for topological surfaces of genus g with n boundaries. Such a structure describes for instance 2d topological quantum field theory amplitudes, governs many problems in the enumerative geometry of surfaces (discretized surfaces, intersection theory on the moduli space of punctured Riemann surfaces and volumes of the moduli space of bordered Riemann surfaces, semi-simple Gromov-Witten theories), asymptotics of matrix models. It also captures some aspects of WKB expansions, knot theory, conformal field theory, etc. and provides a bridge between different areas of mathematics.

In the first lecture I will introduce one of the basic formalism of topological recursion, based on the study of quadratic differential constraints in symplectic vector spaces, and its first applications.


  •  Gaetan Borot
    Gaetan Borot, Max Planck Institute for Mathematics, Bonn