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Progress in Mathematical Physics Ser.: Counting Surfaces : Combinatorics, Matrix Models and Algebraic Geometry by Bertrand Eynard (2016, Hardcover)
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About this product
Product Identifiers
PublisherSpringer Basel A&G
ISBN-103764387963
ISBN-139783764387969
eBay Product ID (ePID)71128186
Product Key Features
Number of PagesXvii, 414 Pages
LanguageEnglish
Publication NameCounting Surfaces : Combinatorics, Matrix Models and Algebraic Geometry
Publication Year2016
SubjectGeometry / Algebraic, Mathematical Analysis, Discrete Mathematics
TypeTextbook
Subject AreaMathematics
AuthorBertrand Eynard
SeriesProgress in Mathematical Physics Ser.
FormatHardcover
Dimensions
Item Weight271.1 Oz
Item Length9.3 in
Item Width6.1 in
Additional Product Features
Intended AudienceScholarly & Professional
LCCN2016-934704
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)
Dewey Edition23
Series Volume Number70
Number of Volumes1 vol.
IllustratedYes
Dewey Decimal514.2
Table Of ContentI Maps and discrete surfaces.- II Formal matrix integrals.- III Solution of Tutte-loop equations.- IV Multicut case.- V Counting large maps.- VI Counting Riemann surfaces.- VII Topological recursion and symplectic invariants.- VIII Ising model.- Index.- Bibliography.
SynopsisThis 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., 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., 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, informatics, biology, etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. In 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 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 their intersection numbers. The so-called Witten's conjecture (which was first proved by Kontsevich) asserts 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 pedagogical, and will provide comprehensive proofs, several examples, and will give the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics as algebraic geometry, string theory, will be discussed, and in particular we give a proof of the Witten-Kontsevich conjecture.
LC Classification NumberQA564-609
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