By Bertrand Eynard

The challenge of enumerating maps (a map is a collection of polygonal "countries" on an international of a definite topology, now not inevitably the aircraft or the field) is a vital challenge in arithmetic and physics, and it has many functions starting from statistical physics, geometry, particle physics, telecommunications, biology, ... and so forth. This challenge has been studied by way of many groups of researchers, usually combinatorists, probabilists, and physicists. considering that 1978, physicists have invented a mode known as "matrix types" to handle that challenge, and lots of effects were obtained.

Besides, one other very important challenge in arithmetic and physics (in specific string theory), is to count number Riemann surfaces. Riemann surfaces of a given topology are parametrized via a finite variety of actual parameters (called moduli), and the moduli house is a finite dimensional compact manifold or orbifold of complex topology. The variety of Riemann surfaces is the amount of that moduli house. extra regularly, a major challenge in algebraic geometry is to represent the moduli areas, by means of computing not just their volumes, but additionally different attribute numbers referred to as intersection numbers.

Witten's conjecture (which was once first proved by means of Kontsevich), used to be the statement that Riemann surfaces should be received as limits of polygonal surfaces (maps), made from a truly huge variety of very small polygons. In different phrases, the variety of maps in a undeniable restrict, may still provide the intersection numbers of moduli spaces.

In this publication, we convey how that restrict happens. The objective of this publication is to provide an explanation for the "matrix version" strategy, to teach the most effects got with it, and to match it with equipment utilized in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions).

The publication intends to be self-contained and obtainable to graduate scholars, and offers finished proofs, numerous examples, and offers the overall formulation for the enumeration of maps on surfaces of any topology. in spite of everything, the hyperlink with extra common subject matters corresponding to algebraic geometry, string thought, is mentioned, and particularly an evidence of the Witten-Kontsevich conjecture is provided.

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### Counting Surfaces: CRM Aisenstadt Chair lectures (Progress in Mathematical Physics) by Bertrand Eynard

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