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

## Bertrand Eynard

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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.

Les informations fournies dans la section « Synopsis » peuvent faire référence à une autre édition de ce titre.

Review :

“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)

Les informations fournies dans la section « A propos du livre » peuvent faire référence à une autre édition de ce titre.

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## 1.Counting Surfaces

Edité par Springer Basel 2016-03-31, [Switzerland] (2016)
ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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Description du livre Springer Basel 2016-03-31, [Switzerland], 2016. hardback. État : New. N° de réf. du libraire 9783764387969

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## 2.Counting Surfaces

Edité par Birkhauser Verlag AG (2016)
ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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Description du livre Birkhauser Verlag AG, 2016. HRD. État : New. New Book. Shipped from UK in 4 to 14 days. Established seller since 2000. N° de réf. du libraire GB-9783764387969

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## 3.Counting Surfaces: CRM Aisenstadt Chair lectures (Progress in Mathematical Physics) (Hardcover)

ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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## 4.Counting Surfaces: CRM Aisenstadt Chair lectures (Progress in Mathematical Physics)

Edité par Birkh?user (2016)
ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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Description du livre Birkh?user, 2016. État : New. This book explains the "matrix model" method developed by physicists to address the problem of enumerating maps and compares it with other methods. It includes proofs, examples and a general formula for the enumeration of maps on surfaces of any topology. Series: Progress in Mathematical Physics. Num Pages: 150 pages, biography. BIC Classification: PBW; PHU. Category: (P) Professional & Vocational. Dimension: 235 x 155. . . 2016. 1st ed. 2016. Hardcover. . . . . . N° de réf. du libraire V9783764387969

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## 5.Counting Surfaces: CRM Aisenstadt Chair lectures (Progress in Mathematical Physics)

Edité par BirkhÃ¤user
ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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## 6.Counting Surfaces: CRM Aisenstadt Chair lectures (Progress in Mathematical Physics)

Edité par Birkhäuser (2016)
ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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## 7.Counting Surfaces: CRM Aisenstadt Chair lectures (Progress in Mathematical Physics) (Hardcover)

ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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## 8.Counting Surfaces: CRM Aisenstadt Chair lectures (Progress in Mathematical Physics)

Edité par Birkhäuser
ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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Description du livre Birkhäuser. État : New. This book explains the "matrix model" method developed by physicists to address the problem of enumerating maps and compares it with other methods. It includes proofs, examples and a general formula for the enumeration of maps on surfaces of any topology. Series: Progress in Mathematical Physics. Num Pages: 150 pages, biography. BIC Classification: PBW; PHU. Category: (P) Professional & Vocational. Dimension: 235 x 155. . . 2016. 1st ed. 2016. Hardcover. . . . . Books ship from the US and Ireland. N° de réf. du libraire V9783764387969

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## 9.Counting Surfaces. CRM Aisenstadt Chair lectures

Edité par BirkhÃ¤user (2016)
ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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## 10.Counting Surfaces : CRM Aisenstadt Chair lectures

Edité par BirkhÃ¤user (2016)
ISBN 10 : 3764387963 ISBN 13 : 9783764387969
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