曲面上のグラフとその応用<br>Graphs on Surfaces and Their Applications (Encyclopedia of Mathematical Sciences Vol.141) (2004. 463 p. w. 150 figs.)

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曲面上のグラフとその応用
Graphs on Surfaces and Their Applications (Encyclopedia of Mathematical Sciences Vol.141) (2004. 463 p. w. 150 figs.)

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  • 製本 Hardcover:ハードカバー版/ページ数 463 p., 150 figs.
  • 商品コード 9783540002031

基本説明

Provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs, and more.

Full Description


Graphs drawn on two-dimensional surfaces have always attracted researchers by their beauty and by the variety of difficult questions to which they give rise. The theory of such embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models, and has become a kind of a focus of a vast field of research. The book provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation theory. The presentation is concrete throughout, with numerous figures, examples (including computer calculations) and exercises, and should appeal to both graduate students and researchers.

Table of Contents

0 Introduction: What is This Book About            1  (6)
0.1 New Life of an Old Theory 1 (1)
0.2 Plan of the Book 2 (2)
0.3 What You Will Not Find in this Book 4 (3)
1 Constellations, Coverings, and Maps 7 (72)
1.1 Constellations 7 (6)
1.2 Ramified Coverings of the Sphere 13 (13)
1.2.1 First Definitions 13 (2)
1.2.2 Coverings and Fundamental Groups 15 (3)
1.2.3 Ramified Coverings of the Sphere and 18 (4)
Constellations
1.2.4 Surfaces 22 (4)
1.3 Maps 26 (13)
1.3.1 Graphs Versus Maps 26 (2)
1.3.2 Maps: Topological Definition 28 (5)
1.3.3 Maps: Permutational Model 33 (6)
1.4 Cartographic Groups 39 (4)
1.5 Hypermaps 43 (12)
1.5.1 Hypermaps and Bipartite Maps 43 (2)
1.5.2 Trees 45 (4)
1.5.3 Appendix: Finite Linear Groups 49 (1)
1.5.4 Canonical Triangulation 50 (5)
1.6 More Than Three Permutations 55 (8)
1.6.1 Preimages of a Star or of a Polygon 56 (1)
1.6.2 Cacti 57 (4)
1.6.3 Preimages of a Jordan Curve 61 (2)
1.7 Further Discussion 63 (7)
1.7.1 Coverings of Surfaces of Higher Genera 63 (2)
1.7.2 Rift's Theorem 65 (3)
1.7.3 Symmetric and Regular Constellations 68 (2)
1.8 Review of Riemann Surfaces 70 (9)
2 Dessins d'Enfants 79 (76)
2.1 Introduction: The Belyi Theorem 79 (1)
2.2 Plane Trees and Shabat Polynomials 80 (29)
2.2.1 General Theory Applied to Trees 80 (8)
2.2.2 Simple Examples 88 (6)
2.2.3 Further Discussion 94 (7)
2.2.4 More Advanced Examples 101(8)
2.3 Belyi Functions and Belyi Pairs 109(6)
2.4 Galois Action and Its Combinatorial 115(11)
Invariants
2.4.1 Preliminaries 115(3)
2.4.2 Galois Invariants 118(5)
2.4.3 Two Theorems on Trees 123(3)
2.5 Several Facets of Belyi Functions 126(20)
2.5.1 A Bound of Davenport-Stothers-Zannier 126(5)
2.5.2 Jacobi Polynomials 131(4)
2.5.3 Fermat Curve 135(2)
2.5.4 The abc Conjecture 137(2)
2.5.5 Julia Sets 139(3)
2.5.6 Pell Equation for Polynomials 142(4)
2.6 Proof of the Belyi Theorem 146(9)
2.6.1 The "Only If" Part of the Belyi 146(1)
Theorem
2.6.2 Comments to the Proof of the "Only 147(3)
If" Part
2.6.3 The "If", or the "Obvious" Part of 150(5)
the Belyi Theorem
3 Introduction to the Matrix Integrals Method 155(68)
3.1 Model Problem: One-Face Maps 155(5)
3.2 Gaussian Integrals 160(19)
3.2.1 The Gaussian Measure on the Line 160(2)
3.2.2 Gaussian Measures in Rk 162(1)
3.2.3 Integrals of Polynomials and the Wick 163(1)
Formula
3.2.4 A Gaussian Measure on the Space of 164(3)
Hermitian Matrices
3.2.5 Matrix Integrals and Polygon Gluings 167(4)
3.2.6 Computing Gaussian Integrals. Unitary 171(5)
Invariance
3.2.7 Computation of the Integral for One 176(3)
Face Gluings
3.3 Matrix Integrals for Multi-Faced Maps 179(6)
3.3.1 Feynman Diagrams 179(1)
3.3.2 The Matrix Integral for an Arbitrary 180(3)
Gluing
3.3.3 Getting Rid of Disconnected Graphs 183(2)
3.4 Enumeration of Colored Graphs 185(7)
3.4.1 Two-Matrix Integrals and the Ising 185(3)
Model
3.4.2 The Gauss Problem 188(2)
3.4.3 Meanders 190(1)
3.4.4 On Enumeration of Meanders 191(1)
3.5 Computation of Matrix Integrals 192(7)
3.5.1 Example: Computing the Volume of the 192(3)
Unitary Group
3.5.2 Generalized Hermite Polynomials 195(2)
3.5.3 Planar Approximations 197(2)
3.6 Korteweg-de Vries (KdV) Hierarchy for the 199(11)
Universal One-Matrix Model
3.6.1 Singular Behavior of Generating 200(2)
Functions
3.6.2 The Operator of Multiplication by a 202(2)
in the Double Scaling Limit
3.6.3 The One-Matrix Model and the KdV 204(2)
Hierarchy
3.6.4 Constructing Solutions to the KdV 206(4)
Hierarchy from the Sato Grassmanian
3.7 Physical Interpretation 210(5)
3.7.1 Mathematical Relations Between 211(1)
Physical Models
3.7.2 Feynman Path Integrals and String 211(2)
Theory
3.7.3 Quantum Field Theory Models 213(1)
3.7.4 Other Models 214(1)
3.8 Appendix 215(8)
3.8.1 Generating Functions 215(2)
3.8.2 Connected and Disconnected Objects 217(2)
3.8.3 Logarithm of a Power Series and 219(4)
Wick's Formula
4 Geometry of Moduli Spaces of Complex Curves 223(46)
4.1 Generalities on Nodal Curves and Orbifolds 223(9)
4.1.1 Differentials and Nodal Curves 223(3)
4.1.2 Quadratic Differentials 226(1)
4.1.3 Orbifolds 227(5)
4.2 Moduli Spaces of Complex Structures 232(2)
4.3 The Deligne-Mumford Compactification 234(3)
4.4 Combinatorial Models of the Moduli Spaces 237(6)
of Curves
4.5 Orbifold Euler Characteristic of the 243(6)
Moduli Spaces
4.6 Intersection Indices on Moduli Spaces and 249(7)
the String and Dilaton Equations
4.7 KdV Hierarchy and Witten's Conjecture 256(1)
4.8 The Kontsevich Model 257(6)
4.9 A Sketch of Kontsevich's Proof of 263(6)
Witten's Conjecture
4.9.1 The Generating Function for the 263(1)
Kontsevich Model
4.9.2 The Kontsevich Model and Intersection 264(2)
Theory
4.9.3 The Kontsevich Model and the KdV 266(3)
Equation
5 Meromorphic Functions and Embedded Graphs 269(68)
5.1 The Lyashko-Looijenga Mapping and Rigid 270(7)
Classification of Generic Polynomials
5.1.1 The Lyashko-Looijenga Mapping 270(1)
5.1.2 Construction of the LL Mapping on the 271(2)
Space of Generic Polynomials
5.1.3 Proof of the Lyashko-Looijenga Theorem 273(4)
5.2 Rigid Classification of Nongeneric 277(11)
Polynomials and the Geometry of the
Discriminant
5.2.1 The Discriminant in the Space of 277(2)
Polynomials and Its Stratification
5.2.2 Statement of the Enumeration Theorem 279(1)
5.2.3 Primitive Strata 280(2)
5.2.4 Proof of the Enumeration Theorem 282(6)
5.3 Rigid Classification of Generic 288(16)
Meromorphic Functions and Geometry of Moduli
Spaces of Curves
5.3.1 Statement of the Enumeration Theorem 288(1)
5.3.2 Calculations: Genus 0 and Genus 1 289(3)
5.3.3 Cones and Their Segre Classes 292(2)
5.3.4 Cones of Principal Parts 294(3)
5.3.5 Hurwitz Spaces 297(2)
5.3.6 Completed Hurwitz Spaces and Stable 299(1)
Mappings
5.3.7 Extending the LL Mapping to Completed 300(2)
Hurwitz Spaces
5.3.8 Computing the Top Segre Class; End of 302(2)
the Proof
5.4 The Braid Group Action 304(23)
5.4.1 Braid Groups 304(5)
5.4.2 Braid Group Action on Cacti: 309(3)
Generalities
5.4.3 Experimental Study 312(6)
5.4.4 Primitive and Imprimitive Monodromy 318(7)
Groups
5.4.5 Perspectives 325(2)
5.5 Megamaps 327(10)
5.5.1 Hurwitz Spaces of Coverings with Four 328(1)
Ramification Points
5.5.2 Representation of H as a Dessin 329(2)
d'Enfant
5.5.3 Examples 331(6)
6 Algebraic Structures Associated with Embedded 337(62)
Graphs
6.1 The Bialgebra of Chord Diagrams 337(13)
6.1.1 Chord Diagrams and Arc Diagrams 337(2)
6.1.2 The 4-Term Relation 339(3)
6.1.3 Multiplying Chord Diagrams 342(1)
6.1.4 A Bialgebra Structure 343(3)
6.1.5 Structure Theorem for the Bialgebra M 346(1)
6.1.6 Primitive Elements of the Bialgebra 347(3)
of Chord Diagrams
6.2 Knot Invariants and Origins of Chord 350(9)
Diagrams
6.2.1 Knot Invariants and their Extension 350(3)
to Singular Knots
6.2.2 Invariants of Finite Order 353(2)
6.2.3 Deducing 1-Term and 4-Term Relations 355(2)
for Invariants
6.2.4 Chord Diagrams of Singular Links 357(2)
6.3 Weight Systems 359(8)
6.3.1 A Bialgebra Structure on the Module V 359(1)
of Vassiliev Knot Invariants
6.3.2 Renormalization 360(2)
6.3.3 Weight Systems 362(2)
6.3.4 Vassiliev Knot Invariants and Other 364(3)
Knot Invariants
6.4 Constructing Weight Systems via 367(17)
Intersection Graphs
6.4.1 The Intersection Graph of a Chord 367(1)
Diagram
6.4.2 Tutte Functions for Graphs 368(1)
6.4.3 The 4-Bialgebra of Graphs 369(10)
6.4.4 The Bialgebra of Weighted Graphs 379(4)
6.4.5 Constructing Vassiliev Invariants 383(1)
from 4-Invariants
6.5 Constructing Weight Systems via Lie 384(9)
Algebras
6.5.1 Free Associative Algebras 385(2)
6.5.2 Universal Enveloping Algebras of Lie 387(3)
Algebras
6.5.3 Examples 390(3)
6.6 Some Other Algebras of Embedded Graphs 393(6)
6.6.1 Circle Diagrams and Open Diagrams 393(2)
6.6.2 The Algebra of 3-Graphs 395(1)
6.6.3 The Temperley-Lieb Algebra 395(4)
A Applications of the Representation Theory of 399(30)
Finite Groups (by Don Zagier)
A.1 Representation Theory of Finite Groups 399(9)
A.1.1 Irreducible Representations and 399(4)
Characters
A.1.2 Examples 403(3)
A.1.3 Frobenius's Formula 406(2)
A.2 Applications 408(21)
A.2.1 Representations of Sn and Canonical 409(6)
Polynomials Associated to Partitions
A.2.2 Examples 415(1)
A.2.3 First Application: Enumeration of 416(2)
Polygon Gluings
A.2.4 Second Application: the 418(5)
Goulden-Jackson Formula
A.2.5 Third Application: "Mirror Symmetry" 423(6)
in Dimension One
References 429(16)
Index 445