# Topological recursion and enumeration of surfaces

Algebra/Geometry/Topology Seminar

#### by Prof Bertrand Eynard

Institution: IPhT, CEA Saclay
Date: Wed 7th July 2010
Time: 2:15 PM
Location: Room 213, Richard Berry Building, The University of Melbourne

Abstract: The topological recursion is a universal relationship between generating functions of various sorts of surfaces, which increases the Euler characteristics by 1 unit, and which thus allows to compute every given topology surfaces generating function, in terms of discs and cylinders.

For instance generating functions of maps (discrete surfaces possibly carrying a Ising model or O(n) model), number of Riemann surfaces counted with Weil-Petersson volume form, Witten-Kontsevich intersection numbers, Hurwitz numbers (enumeration of branched coverings of Riemann surfaces) or Gromov-Witten numbers (topological strings) do obey that recursion. Only initial conditions (disc amplitude) differ from one enumeration problem to another.

Beside, the recursion itself implies many interesting properties, like symplectic invariance, deformation theory, modularity, or integrability.