page_title: Topological recursion
section_title: Introduction
hierarchical_section_title: Topological recursion / Introduction
caption_attribution_description: English: Schematic illustration of the topological recursion: recursively adding pairs of pants to build a surface of genus g with n boundaries
context_page_description: In mathematics, Topological Recursion is a recursive definition of invariants of spectral curves.
  It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory.
context_section_description: The topological recursion is a construction in algebraic geometry. It takes as initial data a spectral curve: the data of , where:  is a covering of Riemann surfaces with ramification points;  is a meromorphic differential 1-form on , regular at the ramification points;  is a symmetric meromorphic bilinear differential form on  having a double pole on the diagonal and no residue.
  The topological recursion is then a recursive definition of infinite sequences of symmetric meromorphic n-forms  on , with poles at ramification points only, for integers g≥0 such that 2g-2+n>0. The definition is a recursion on the integer 2g-2+n.
  In many applications, the n-form  is interpreted as a generating function that measures a set of surfaces of genus g and with n boundaries. The recursion is on 2-2g-n the Euler characteristics, whence the name "topological recursion".
caption: Schematic illustration of the topological recursion: recursively adding pairs of pants to build a surface of genus g with n boundaries
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