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 airplane 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 on. This challenge has been studied through many groups of researchers, generally combinatorists, probabilists, and physicists. seeing that 1978, physicists have invented a mode referred to as "matrix versions" to handle that challenge, and plenty of effects were obtained.

Besides, one other vital challenge in arithmetic and physics (in specific string theory), is to count number Riemann surfaces. Riemann surfaces of a given topology are parametrized by means of 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 quantity of that moduli area. extra usually, an incredible challenge in algebraic geometry is to signify the moduli areas, through computing not just their volumes, but in addition different attribute numbers referred to as intersection numbers.

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

In this publication, we exhibit how that restrict occurs. The aim of this booklet is to give an explanation for the "matrix version" strategy, to teach the most effects received with it, and to match it with tools utilized in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions).

The ebook intends to be self-contained and available to graduate scholars, and offers finished proofs, numerous examples, and provides the final formulation for the enumeration of maps on surfaces of any topology. in spite of everything, the hyperlink with extra normal subject matters resembling algebraic geometry, string conception, is mentioned, and particularly an explanation of the Witten-Kontsevich conjecture is provided.

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**Example text**

And there are four quadrangulations of genus g D 1 with n4 D 2 quadrangles, one has symmetry factor 8, one has symmetry factor 4, one has symmetry factor 2, one has symmetry factor 1. t43 /: F D t t4 2 4 8 8 4 2 Exercise 2 Find all planar maps with one marked face of arbitrary length l, and whose unmarked faces are only pentagons, and with up to five vertices. t6 / C C C C x5 x2 x7 x4 x9 x6 x3 Chapter 2 Formal Matrix Integrals In this chapter we introduce the notion of a formal matrix integral, which is very useful for combinatorics, as it turns out to be identical to the generating function of maps of Chap.

Tr M 4 /k e N Tr 2 k kŠ 4 HN © Springer International Publishing Switzerland 2016 B. N/ invariant Lebesgue measure on HN dM D N Y Y 1 dMii dReMij dImMij 2 =2 N=2 N 2 . e. N/ D 1. N/ is a polynomial in N and 1=N, so it can be analytically continued to any N 2 C . N/ D kD0 1 X t4k kD0 Nk kŠ 4k Z dM . t4 / D dM e 2 N Tr . M2 4 t4 M4 / : formal (as if we could exchange the order of sum and integral). t4 /. t4 /, we actually mean properties of the coefficients Ak of the t4 expansion. N/ 8k. t4 / is never convergent (in the Borel sense for instance).

G/ • If we choose all tj D 0 except t4 ¤ 0, we count only quadrangulations, and T4 is the number of rooted quadrangulations of genus g, where all faces (including the one on the right of the marked edge) are quadrangles. The total number of faces is n D n4 C 1, and [thanks to Eq. 1)] the number of vertices is v D nC2 2g. In the 60’s, Tutte (this is the famous Tutte’s formula [84, 85]) computed that (and we shall prove it in Chap. n C 2/Š In this formula, the coefficient of t40 t3 is 2. g/ and T3 is the number of rooted triangulations of genus g, where all faces (including the one on the right of the marked edge) are triangles.