Counting Surfaces

Counting Surfaces
Author :
Publisher : Springer Science & Business Media
Total Pages : 427
Release :
ISBN-10 : 9783764387976
ISBN-13 : 3764387971
Rating : 4/5 (971 Downloads)

Book Synopsis Counting Surfaces by : Bertrand Eynard

Download or read book Counting Surfaces written by Bertrand Eynard and published by Springer Science & Business Media. This book was released on 2016-03-21 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.


Counting Surfaces Related Books

Counting Surfaces
Language: en
Pages: 427
Authors: Bertrand Eynard
Categories: Mathematics
Type: BOOK - Published: 2016-03-21 - Publisher: Springer Science & Business Media

GET EBOOK

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 import
Laboratory Apparatus and Reagents Selected for Laboratories of Chemistry and Biology
Language: en
Pages: 676
Authors: Thomas, Arthur H., Company, Philadelphia
Categories:
Type: BOOK - Published: 1914 - Publisher:

GET EBOOK

Mirzakhani’s Curve Counting and Geodesic Currents
Language: en
Pages: 233
Authors: Viveka Erlandsson
Categories: Mathematics
Type: BOOK - Published: 2022-09-20 - Publisher: Springer Nature

GET EBOOK

This monograph presents an approachable proof of Mirzakhani’s curve counting theorem, both for simple and non-simple curves. Designed to welcome readers to th
A Treatise on Diagnostic Methods of Examination
Language: en
Pages: 1332
Authors: Hermann Sahli
Categories: Diagnosis
Type: BOOK - Published: 1920 - Publisher:

GET EBOOK

Protocols for Neural Cell Culture
Language: en
Pages: 366
Authors: Sergey Fedoroff
Categories: Medical
Type: BOOK - Published: 2001 - Publisher: Springer Science & Business Media

GET EBOOK

Sergey Fedoroff and Arleen Richardson extensively revise, update, and expand their best-selling and highly praised collection of readily reproducible neural tis