Progress in Mathematical Physics

# Counting Surfaces

Authors: Eynard, Bertrand

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• 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

eBook $89.00 price for USA in USD • ISBN 978-3-7643-8797-6 • Digitally watermarked, DRM-free • Included format: EPUB, PDF • Immediate eBook download after purchase and usable on all devices • Bulk discounts from 10 eBooks Hardcover$119.99
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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. Mor

e 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 self-contained and accessible to graduate students, and provides comprehensive proofs, several examples, and give

s 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 Witten-Kontsevich conjecture is provided.

Reviews

“This book brings together details of topological recursion from many different papers and organizes them in an accessible way. … this book will be an invaluable resource for mathematicians learning about topological recursion.” (Daniel D. Moskovich, Mathematical Reviews, February, 2017)

“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)

• Maps and Discrete Surfaces

Pages 1-24

Eynard, Bertrand

• Formal Matrix Integrals

Pages 25-51

Eynard, Bertrand

• Solution of Tutte-Loop Equations

Pages 53-143

Eynard, Bertrand

• Multicut Case

Pages 145-168

Eynard, Bertrand

• Counting Large Maps

Pages 169-236

Eynard, Bertrand

eBook $89.00 price for USA in USD • ISBN 978-3-7643-8797-6 • Digitally watermarked, DRM-free • Included format: EPUB, PDF • Immediate eBook download after purchase and usable on all devices • Bulk discounts from 10 eBooks Hardcover$119.99
price for USA in USD

## Bibliographic Information

Bibliographic Information
Book Title
Counting Surfaces
Book Subtitle
Authors
Series Title
Progress in Mathematical Physics
Series Volume
70
2016
Publisher
Birkhäuser Basel
Springer International Publishing Switzerland
eBook ISBN
978-3-7643-8797-6
DOI
10.1007/978-3-7643-8797-6
Hardcover ISBN
978-3-7643-8796-9
Series ISSN
1544-9998
Edition Number
1
Number of Pages
XVII, 414
Number of Illustrations
62 b/w illustrations, 47 illustrations in colour
Topics