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N. Orantin, CERN

Enumerative geometry from topological recursion

The random matrix theory can be used to solve a problem of enumerative geometry: counting maps of arbitrary genus, i.e. counting surfaces composed of polygons glued by their edges. This issue has been extensively investigated by physicists for its possible applications such as quantum gravity. Following the work of Eynard, it was possible to compute such generating functions for different matrix models using a unique recursive procedure: a topological recursion allowing to build the generating function of surfaces with fixed Euler characteristic out of generating functions of surfaces with bigger Euler characteristic. The blind application of the same procedure to more general problems of enumeration such as topological string theories, Weyl-Peterson volume of moduli space of surfaces or Kontsevich-Witten theorem, proved to be very successful: one just has to change the basis ingredient of the procedure to go from one problem to the other, a plane curve referred to as the spectral curve to reflect it originates from integrable properties of the model considered. In this non-technical talk, I will give an overview of this procedure and its applications in mathematics and physics. I will also give a foretaste of the numerous open questions arising from these works.