By Bertrand Eynard
The challenge of enumerating maps (a map is a collection of polygonal "countries" on an international of a undeniable topology, now not inevitably the aircraft or the sector) 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, ordinarily combinatorists, probabilists, and physicists. due to the fact that 1978, physicists have invented a mode known 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 via a finite variety of genuine parameters (called moduli), and the moduli area is a finite dimensional compact manifold or orbifold of complex topology. The variety of Riemann surfaces is the quantity of that moduli area. extra mostly, a huge challenge in algebraic geometry is to symbolize the moduli areas, through computing not just their volumes, but additionally different attribute numbers referred to as intersection numbers.
Witten's conjecture (which was once first proved via Kontsevich), was once the statement that Riemann surfaces should be acquired as limits of polygonal surfaces (maps), made up of a truly huge variety of very small polygons. In different phrases, the variety of maps in a definite restrict, may still supply the intersection numbers of moduli spaces.
In this ebook, we convey how that restrict occurs. The aim of this publication is to give an explanation for the "matrix version" approach, to teach the most effects bought with it, and to check it with equipment utilized in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions).
The ebook intends to be self-contained and obtainable to graduate scholars, and gives accomplished 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 basic subject matters equivalent to algebraic geometry, string conception, is mentioned, and specifically an evidence of the Witten-Kontsevich conjecture is provided.
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Extra info for Counting Surfaces: CRM Aisenstadt Chair lectures (Progress in Mathematical Physics)
Counting Surfaces: CRM Aisenstadt Chair lectures (Progress in Mathematical Physics) by Bertrand Eynard