Counting Surfaces
CRM Aisenstadt Chair lectures
Authors: Eynard, Bertrand
 First book on explaining the random matrix method to enumerate maps and Riemann surfaces
 The method has been discovered recently (between 2004 and 2007) and is presently explained only in very few specialized articles
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 About this book

The problem of enumerating maps (a map is a set of polygonal "countries" on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, telecommunications, biology, ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. Since 1978, physicists have invented a method called "matrix models" to address that problem, and many results have been obtained.
Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold or orbifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. More generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also other characteristic numbers called intersection numbers.
Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces.
In this book, we show how that limit takes place. The goal of this book is to explain the "matrix model" method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions).
The book intends to be selfcontained and accessible to graduate students, and provides comprehensive proofs, several examples, and gives the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, is discussed, and in particular a proof of the WittenKontsevich conjecture is provided.
 Reviews

“The author explains how matrix models and counting surfaces are related and aims at presenting to mathematicians and physicists the random matrix approach to quantum gravity. … The book is an outstanding monograph of a recent research trend in surface theory.” (Gert Roepstorff, zbMATH 1338.81005, 2016)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Counting Surfaces
 Book Subtitle
 CRM Aisenstadt Chair lectures
 Authors

 Bertrand Eynard
 Series Title
 Progress in Mathematical Physics
 Series Volume
 70
 Copyright
 2016
 Publisher
 Birkhäuser Basel
 Copyright Holder
 Springer International Publishing Switzerland
 eBook ISBN
 9783764387976
 DOI
 10.1007/9783764387976
 Hardcover ISBN
 9783764387969
 Series ISSN
 15449998
 Edition Number
 1
 Number of Pages
 XVII, 414
 Number of Illustrations and Tables
 62 b/w illustrations, 47 illustrations in colour
 Topics